pysr/sr.py

import os
import sys
import numpy as np
import pandas as pd
import sympy
from sympy import sympify
import re
import tempfile
import shutil
from pathlib import Path
from datetime import datetime
import warnings
from multiprocessing import cpu_count
from sklearn.base import BaseEstimator, RegressorMixin, MultiOutputMixin
from sklearn.utils import check_array, check_consistent_length, check_random_state
from sklearn.utils.validation import (
    _check_feature_names_in,
    check_is_fitted,
)

from .julia_helpers import (
    init_julia,
    _get_julia_project,
    is_julia_version_greater_eq,
    _escape_filename,
    _add_sr_to_julia_project,
    import_error_string,
)
from .export_numpy import CallableEquation
from .deprecated import make_deprecated_kwargs_for_pysr_regressor


Main = None  # TODO: Rename to more descriptive name like "julia_runtime"

already_ran = False

sympy_mappings = {
    "div": lambda x, y: x / y,
    "mult": lambda x, y: x * y,
    "sqrt_abs": lambda x: sympy.sqrt(abs(x)),
    "square": lambda x: x**2,
    "cube": lambda x: x**3,
    "plus": lambda x, y: x + y,
    "sub": lambda x, y: x - y,
    "neg": lambda x: -x,
    "pow": lambda x, y: abs(x) ** y,
    "cos": sympy.cos,
    "sin": sympy.sin,
    "tan": sympy.tan,
    "cosh": sympy.cosh,
    "sinh": sympy.sinh,
    "tanh": sympy.tanh,
    "exp": sympy.exp,
    "acos": sympy.acos,
    "asin": sympy.asin,
    "atan": sympy.atan,
    "acosh": lambda x: sympy.acosh(abs(x) + 1),
    "acosh_abs": lambda x: sympy.acosh(abs(x) + 1),
    "asinh": sympy.asinh,
    "atanh": lambda x: sympy.atanh(sympy.Mod(x + 1, 2) - 1),
    "atanh_clip": lambda x: sympy.atanh(sympy.Mod(x + 1, 2) - 1),
    "abs": abs,
    "mod": sympy.Mod,
    "erf": sympy.erf,
    "erfc": sympy.erfc,
    "log_abs": lambda x: sympy.log(abs(x)),
    "log10_abs": lambda x: sympy.log(abs(x), 10),
    "log2_abs": lambda x: sympy.log(abs(x), 2),
    "log1p_abs": lambda x: sympy.log(abs(x) + 1),
    "floor": sympy.floor,
    "ceil": sympy.ceiling,
    "sign": sympy.sign,
    "gamma": sympy.gamma,
}


def pysr(X, y, weights=None, **kwargs):  # pragma: no cover
    warnings.warn(
        "Calling `pysr` is deprecated. "
        "Please use `model = PySRRegressor(**params); model.fit(X, y)` going forward.",
        FutureWarning,
    )
    model = PySRRegressor(**kwargs)
    model.fit(X, y, weights=weights)
    return model.equations_


def _process_constraints(binary_operators, unary_operators, constraints):
    constraints = constraints.copy()
    for op in unary_operators:
        if op not in constraints:
            constraints[op] = -1
    for op in binary_operators:
        if op not in constraints:
            constraints[op] = (-1, -1)
        if op in ["plus", "sub", "+", "-"]:
            if constraints[op][0] != constraints[op][1]:
                raise NotImplementedError(
                    "You need equal constraints on both sides for - and +, "
                    "due to simplification strategies."
                )
        elif op in ["mult", "*"]:
            # Make sure the complex expression is in the left side.
            if constraints[op][0] == -1:
                continue
            if constraints[op][1] == -1 or constraints[op][0] < constraints[op][1]:
                constraints[op][0], constraints[op][1] = (
                    constraints[op][1],
                    constraints[op][0],
                )
    return constraints


def _maybe_create_inline_operators(binary_operators, unary_operators):
    global Main
    binary_operators = binary_operators.copy()
    unary_operators = unary_operators.copy()
    for op_list in [binary_operators, unary_operators]:
        for i, op in enumerate(op_list):
            is_user_defined_operator = "(" in op

            if is_user_defined_operator:
                Main.eval(op)
                # Cut off from the first non-alphanumeric char:
                first_non_char = [j for j, char in enumerate(op) if char == "("][0]
                function_name = op[:first_non_char]
                # Assert that function_name only contains
                # alphabetical characters, numbers,
                # and underscores:
                if not re.match(r"^[a-zA-Z0-9_]+$", function_name):
                    raise ValueError(
                        f"Invalid function name {function_name}. "
                        "Only alphanumeric characters, numbers, "
                        "and underscores are allowed."
                    )
                op_list[i] = function_name
    return binary_operators, unary_operators


def _check_assertions(
    X,
    use_custom_variable_names,
    variable_names,
    weights,
    y,
):
    # Check for potential errors before they happen
    assert len(X.shape) == 2
    assert len(y.shape) in [1, 2]
    assert X.shape[0] == y.shape[0]
    if weights is not None:
        assert weights.shape == y.shape
        assert X.shape[0] == weights.shape[0]
    if use_custom_variable_names:
        assert len(variable_names) == X.shape[1]


def best(*args, **kwargs):  # pragma: no cover
    raise NotImplementedError(
        "`best` has been deprecated. Please use the `PySRRegressor` interface. "
        "After fitting, you can return `.sympy()` to get the sympy representation "
        "of the best equation."
    )


def best_row(*args, **kwargs):  # pragma: no cover
    raise NotImplementedError(
        "`best_row` has been deprecated. Please use the `PySRRegressor` interface. "
        "After fitting, you can run `print(model)` to view the best equation, or "
        "`model.get_best()` to return the best equation's row in `model.equations_`."
    )


def best_tex(*args, **kwargs):  # pragma: no cover
    raise NotImplementedError(
        "`best_tex` has been deprecated. Please use the `PySRRegressor` interface. "
        "After fitting, you can return `.latex()` to get the sympy representation "
        "of the best equation."
    )


def best_callable(*args, **kwargs):  # pragma: no cover
    raise NotImplementedError(
        "`best_callable` has been deprecated. Please use the `PySRRegressor` "
        "interface. After fitting, you can use `.predict(X)` to use the best callable."
    )


# Class validation constants
VALID_OPTIMIZER_ALGORITHMS = ["NelderMead", "BFGS"]


class PySRRegressor(MultiOutputMixin, RegressorMixin, BaseEstimator):
    """
    High-performance symbolic regression.

    This is the scikit-learn interface for SymbolicRegression.jl.
    This model will automatically search for equations which fit
    a given dataset subject to a particular loss and set of
    constraints.

    Parameters
    ----------
    model_selection : str, default="best"
        Model selection criterion. Can be 'accuracy' or 'best'.
        `"accuracy"` selects the candidate model with the lowest loss
        (highest accuracy). `"best"` selects the candidate model with
        the lowest sum of normalized loss and complexity.

    binary_operators : list[str], default=["+", "-", "*", "/"]
        List of strings giving the binary operators in Julia's Base.

    unary_operators : list[str], default=None
        Same as :param`binary_operators` but for operators taking a
        single scalar.

    niterations : int, default=40
        Number of iterations of the algorithm to run. The best
        equations are printed and migrate between populations at the
        end of each iteration.

    populations : int, default=15
        Number of populations running.

    population_size : int, default=33
        Number of individuals in each population.

    max_evals : int, default=None
        Limits the total number of evaluations of expressions to
        this number.

    maxsize : int, default=20
        Max complexity of an equation.

    maxdepth : int, default=None
        Max depth of an equation. You can use both :param`maxsize` and
        :param`maxdepth`. :param`maxdepth` is by default not used.

    warmup_maxsize_by : float, default=0.0
        Whether to slowly increase max size from a small number up to
        the maxsize (if greater than 0).  If greater than 0, says the
        fraction of training time at which the current maxsize will
        reach the user-passed maxsize.

    timeout_in_seconds : float, default=None
        Make the search return early once this many seconds have passed.

    constraints : dict[str, int | tuple[int,int]], default=None
        Dictionary of int (unary) or 2-tuples (binary), this enforces
        maxsize constraints on the individual arguments of operators.
        E.g., `'pow': (-1, 1)` says that power laws can have any
        complexity left argument, but only 1 complexity in the right
        argument. Use this to force more interpretable solutions.

    nested_constraints : dict[str, dict], default=None
        Specifies how many times a combination of operators can be
        nested. For example, `{"sin": {"cos": 0}}, "cos": {"cos": 2}}`
        specifies that `cos` may never appear within a `sin`, but `sin`
        can be nested with itself an unlimited number of times. The
        second term specifies that `cos` can be nested up to 2 times
        within a `cos`, so that `cos(cos(cos(x)))` is allowed
        (as well as any combination of `+` or `-` within it), but
        `cos(cos(cos(cos(x))))` is not allowed. When an operator is not
        specified, it is assumed that it can be nested an unlimited
        number of times. This requires that there is no operator which
        is used both in the unary operators and the binary operators
        (e.g., `-` could be both subtract, and negation). For binary
        operators, you only need to provide a single number: both
        arguments are treated the same way, and the max of each
        argument is constrained.

    loss : str, default="L2DistLoss()"
        String of Julia code specifying the loss function. Can either
        be a loss from LossFunctions.jl, or your own loss written as a
        function. Examples of custom written losses include:
        `myloss(x, y) = abs(x-y)` for non-weighted, or
        `myloss(x, y, w) = w*abs(x-y)` for weighted.

        Among the included losses, these are as follows.
        Regression: `LPDistLoss{P}()`, `L1DistLoss()`,
        `L2DistLoss()` (mean square), `LogitDistLoss()`,
        `HuberLoss(d)`, `L1EpsilonInsLoss(ϵ)`, `L2EpsilonInsLoss(ϵ)`,
        `PeriodicLoss(c)`, `QuantileLoss(τ)`.
        Classification: `ZeroOneLoss()`, `PerceptronLoss()`,
        `L1HingeLoss()`, `SmoothedL1HingeLoss(γ)`,
        `ModifiedHuberLoss()`, `L2MarginLoss()`, `ExpLoss()`,
        `SigmoidLoss()`, `DWDMarginLoss(q)`.

    complexity_of_operators : dict[str, float], default=None
        If you would like to use a complexity other than 1 for an
        operator, specify the complexity here. For example,
        `{"sin": 2, "+": 1}` would give a complexity of 2 for each use
        of the `sin` operator, and a complexity of 1 for each use of
        the `+` operator (which is the default). You may specify real
        numbers for a complexity, and the total complexity of a tree
        will be rounded to the nearest integer after computing.

    complexity_of_constants : float, default=1
        Complexity of constants.

    complexity_of_variables : float, default=1
        Complexity of variables.

    parsimony : float, default=0.0032
        Multiplicative factor for how much to punish complexity.

    use_frequency : bool, default=True
        Whether to measure the frequency of complexities, and use that
        instead of parsimony to explore equation space. Will naturally
        find equations of all complexities.

    use_frequency_in_tournament : bool, default=True
        Whether to use the frequency mentioned above in the tournament,
        rather than just the simulated annealing.

    alpha : float, default=0.1
        Initial temperature for simulated annealing
        (requires :param`annealing` to be `True`).

    annealing : bool, default=True
        Whether to use annealing. You should (and it is default).

    early_stop_condition : { float | str }, default=None
        Stop the search early if this loss is reached. You may also
        pass a string containing a Julia function which
        takes a loss and complexity as input, for example:
        `"f(loss, complexity) = (loss < 0.1) && (complexity < 10)"`.

    ncyclesperiteration : int, default=550
        Number of total mutations to run, per 10 samples of the
        population, per iteration.

    fraction_replaced : float, default=0.000364
        How much of population to replace with migrating equations from
        other populations.

    fraction_replaced_hof : float, default=0.035
        How much of population to replace with migrating equations from
        hall of fame.

    weight_add_node : float, default=0.79
        Relative likelihood for mutation to add a node.

    weight_insert_node : float, default=5.1
        Relative likelihood for mutation to insert a node.

    weight_delete_node : float, default=1.7
        Relative likelihood for mutation to delete a node.

    weight_do_nothing : float, default=0.21
        Relative likelihood for mutation to leave the individual.

    weight_mutate_constant : float, default=0.048
        Relative likelihood for mutation to change the constant slightly
        in a random direction.

    weight_mutate_operator : float, default=0.47
        Relative likelihood for mutation to swap an operator.

    weight_randomize : float, default=0.00023
        Relative likelihood for mutation to completely delete and then
        randomly generate the equation

    weight_simplify : float, default=0.0020
        Relative likelihood for mutation to simplify constant parts by evaluation

    crossover_probability : float, default=0.066
        Absolute probability of crossover-type genetic operation, instead of a mutation.

    skip_mutation_failures : bool, default=True
        Whether to skip mutation and crossover failures, rather than
        simply re-sampling the current member.

    migration : bool, default=True
        Whether to migrate.

    hof_migration : bool, default=True
        Whether to have the hall of fame migrate.

    topn : int, default=12
        How many top individuals migrate from each population.

    should_optimize_constants : bool, default=True
        Whether to numerically optimize constants (Nelder-Mead/Newton)
        at the end of each iteration.

    optimizer_algorithm : str, default="BFGS"
        Optimization scheme to use for optimizing constants. Can currently
        be `NelderMead` or `BFGS`.

    optimizer_nrestarts : int, default=2
        Number of time to restart the constants optimization process with
        different initial conditions.

    optimize_probability : float, default=0.14
        Probability of optimizing the constants during a single iteration of
        the evolutionary algorithm.

    optimizer_iterations : int, default=8
        Number of iterations that the constants optimizer can take.

    perturbation_factor : float, default=0.076
        Constants are perturbed by a max factor of
        (perturbation_factor*T + 1). Either multiplied by this or
        divided by this.

    tournament_selection_n : int, default=10
        Number of expressions to consider in each tournament.

    tournament_selection_p : float, default=0.86
        Probability of selecting the best expression in each
        tournament. The probability will decay as p*(1-p)^n for other
        expressions, sorted by loss.

    procs : int, default=multiprocessing.cpu_count()
        Number of processes (=number of populations running).

    multithreading : bool, default=True
        Use multithreading instead of distributed backend.
        Using procs=0 will turn off both.

    cluster_manager : str, default=None
        For distributed computing, this sets the job queue system. Set
        to one of "slurm", "pbs", "lsf", "sge", "qrsh", "scyld", or
        "htc". If set to one of these, PySR will run in distributed
        mode, and use `procs` to figure out how many processes to launch.

    batching : bool, default=False
        Whether to compare population members on small batches during
        evolution. Still uses full dataset for comparing against hall
        of fame.

    batch_size : int, default=50
        The amount of data to use if doing batching.

    fast_cycle : bool, default=False (experimental)
        Batch over population subsamples. This is a slightly different
        algorithm than regularized evolution, but does cycles 15%
        faster. May be algorithmically less efficient.

    precision : int, default=32
        What precision to use for the data. By default this is 32
        (float32), but you can select 64 or 16 as well.

    random_state : int, Numpy RandomState instance or None, default=None
        Pass an int for reproducible results across multiple function calls.
        See :term:`Glossary <random_state>`.

    deterministic : bool, default=False
        Make a PySR search give the same result every run.
        To use this, you must turn off parallelism
        (with :param`procs`=0, :param`multithreading`=False),
        and set :param`random_state` to a fixed seed.

    warm_start : bool, default=False
        Tells fit to continue from where the last call to fit finished.
        If false, each call to fit will be fresh, overwriting previous results.

    verbosity : int, default=1e9
        What verbosity level to use. 0 means minimal print statements.

    update_verbosity : int, default=None
        What verbosity level to use for package updates.
        Will take value of :param`verbosity` if not given.

    progress : bool, default=True
        Whether to use a progress bar instead of printing to stdout.

    equation_file : str, default=None
        Where to save the files (.csv separated by |).

    temp_equation_file : bool, default=False
        Whether to put the hall of fame file in the temp directory.
        Deletion is then controlled with the :param`delete_tempfiles`
        parameter.

    tempdir : str, default=None
        directory for the temporary files.

    delete_tempfiles : bool, default=True
        Whether to delete the temporary files after finishing.

    julia_project : str, default=None
        A Julia environment location containing a Project.toml
        (and potentially the source code for SymbolicRegression.jl).
        Default gives the Python package directory, where a
        Project.toml file should be present from the install.

    update: bool, default=True
        Whether to automatically update Julia packages.

    output_jax_format : bool, default=False
        Whether to create a 'jax_format' column in the output,
        containing jax-callable functions and the default parameters in
        a jax array.

    output_torch_format : bool, default=False
        Whether to create a 'torch_format' column in the output,
        containing a torch module with trainable parameters.

    extra_sympy_mappings : dict[str, Callable], default=None
        Provides mappings between custom :param`binary_operators` or
        :param`unary_operators` defined in julia strings, to those same
        operators defined in sympy.
        E.G if `unary_operators=["inv(x)=1/x"]`, then for the fitted
        model to be export to sympy, :param`extra_sympy_mappings`
        would be `{"inv": lambda x: 1/x}`.

    extra_jax_mappings : dict[Callable, str], default=None
        Similar to :param`extra_sympy_mappings` but for model export
        to jax. The dictionary maps sympy functions to jax functions.
        For example: `extra_jax_mappings={sympy.sin: "jnp.sin"}` maps
        the `sympy.sin` function to the equivalent jax expression `jnp.sin`.

    extra_torch_mappings : dict[Callable, Callable], default=None
        The same as :param`extra_jax_mappings` but for model export
        to pytorch. Note that the dictionary keys should be callable
        pytorch expressions.
        For example: `extra_torch_mappings={sympy.sin: torch.sin}`

    denoise : bool, default=False
        Whether to use a Gaussian Process to denoise the data before
        inputting to PySR. Can help PySR fit noisy data.

    select_k_features : int, default=None
         whether to run feature selection in Python using random forests,
         before passing to the symbolic regression code. None means no
         feature selection; an int means select that many features.

    kwargs : dict, default=None
        Supports deprecated keyword arguments. Other arguments will
        result in an error.

    Attributes
    ----------
    equations_ : { pandas.DataFrame | list[pandas.DataFrame] }
        Processed DataFrame containing the results of model fitting.

    n_features_in_ : int
        Number of features seen during :term:`fit`.

    feature_names_in_ : ndarray of shape (`n_features_in_`,)
        Names of features seen during :term:`fit`. Defined only when `X`
        has feature names that are all strings.

    nout_ : int
        Number of output dimensions.

    selection_mask_ : list[int] of length `select_k_features`
        List of indices for input features that are selected when
        :param`select_k_features` is set.

    tempdir_ : Path
        Path to the temporary equations directory.

    equation_file_ : str
        Output equation file name produced by the julia backend.

    raw_julia_state_ : tuple[list[PyCall.jlwrap], PyCall.jlwrap]
        The state for the julia SymbolicRegression.jl backend post fitting.

    equation_file_contents_ : list[pandas.DataFrame]
        Contents of the equation file output by the Julia backend.

    Notes
    -----
    Most default parameters have been tuned over several example equations,
    but you should adjust `niterations`, `binary_operators`, `unary_operators`
    to your requirements. You can view more detailed explanations of the options
    on the [options page](https://astroautomata.com/PySR/#/options) of the
    documentation.

    Examples
    --------
    >>> import numpy as np
    >>> from pysr import PySRRegressor
    >>> randstate = np.random.RandomState(0)
    >>> X = 2 * randstate.randn(100, 5)
    >>> # y = 2.5382 * cos(x_3) + x_0 - 0.5
    >>> y = 2.5382 * np.cos(X[:, 3]) + X[:, 0] ** 2 - 0.5
    >>> model = PySRRegressor(
    ...     niterations=40,
    ...     binary_operators=["+", "*"],
    ...     unary_operators=[
    ...         "cos",
    ...         "exp",
    ...         "sin",
    ...         "inv(x) = 1/x",  # Custom operator (julia syntax)
    ...     ],
    ...     model_selection="best",
    ...     loss="loss(x, y) = (x - y)^2",  # Custom loss function (julia syntax)
    ... )
    >>> model.fit(X, y)
    >>> model
    PySRRegressor.equations_ = [
    0         0.000000                                          3.8552167  3.360272e+01           1
    1         1.189847                                          (x0 * x0)  3.110905e+00           3
    2         0.010626                          ((x0 * x0) + -0.25573406)  3.045491e+00           5
    3         0.896632                              (cos(x3) + (x0 * x0))  1.242382e+00           6
    4         0.811362                ((x0 * x0) + (cos(x3) * 2.4384754))  2.451971e-01           8
    5  >>>>  13.733371          (((cos(x3) * 2.5382) + (x0 * x0)) + -0.5)  2.889755e-13          10
    6         0.194695  ((x0 * x0) + (((cos(x3) + -0.063180044) * 2.53...  1.957723e-13          12
    7         0.006988  ((x0 * x0) + (((cos(x3) + -0.32505524) * 1.538...  1.944089e-13          13
    8         0.000955  (((((x0 * x0) + cos(x3)) + -0.8251649) + (cos(...  1.940381e-13          15
    ]
    >>> model.score(X, y)
    1.0
    >>> model.predict(np.array([1,2,3,4,5]))
    array([-1.15907818, -1.15907818, -1.15907818, -1.15907818, -1.15907818])
    """

    def __init__(
        self,
        model_selection="best",
        *,
        binary_operators=None,
        unary_operators=None,
        niterations=40,
        populations=15,
        population_size=33,
        max_evals=None,
        maxsize=20,
        maxdepth=None,
        warmup_maxsize_by=0.0,
        timeout_in_seconds=None,
        constraints=None,
        nested_constraints=None,
        loss="L2DistLoss()",
        complexity_of_operators=None,
        complexity_of_constants=1,
        complexity_of_variables=1,
        parsimony=0.0032,
        use_frequency=True,
        use_frequency_in_tournament=True,
        alpha=0.1,
        annealing=True,
        early_stop_condition=None,
        ncyclesperiteration=550,
        fraction_replaced=0.000364,
        fraction_replaced_hof=0.035,
        weight_add_node=0.79,
        weight_insert_node=5.1,
        weight_delete_node=1.7,
        weight_do_nothing=0.21,
        weight_mutate_constant=0.048,
        weight_mutate_operator=0.47,
        weight_randomize=0.00023,
        weight_simplify=0.0020,
        crossover_probability=0.066,
        skip_mutation_failures=True,
        migration=True,
        hof_migration=True,
        topn=12,
        should_optimize_constants=True,
        optimizer_algorithm="BFGS",
        optimizer_nrestarts=2,
        optimize_probability=0.14,
        optimizer_iterations=8,
        perturbation_factor=0.076,
        tournament_selection_n=10,
        tournament_selection_p=0.86,
        procs=cpu_count(),
        multithreading=None,
        cluster_manager=None,
        batching=False,
        batch_size=50,
        fast_cycle=False,
        precision=32,
        random_state=None,
        deterministic=False,
        warm_start=False,
        verbosity=1e9,
        update_verbosity=None,
        progress=True,
        equation_file=None,
        temp_equation_file=False,
        tempdir=None,
        delete_tempfiles=True,
        julia_project=None,
        update=True,
        output_jax_format=False,
        output_torch_format=False,
        extra_sympy_mappings=None,
        extra_torch_mappings=None,
        extra_jax_mappings=None,
        denoise=False,
        select_k_features=None,
        **kwargs,
    ):

        # Hyperparameters
        # - Model search parameters
        self.model_selection = model_selection
        self.binary_operators = binary_operators
        self.unary_operators = unary_operators
        self.niterations = niterations
        self.populations = populations
        self.population_size = population_size
        self.ncyclesperiteration = ncyclesperiteration
        # - Equation Constraints
        self.maxsize = maxsize
        self.maxdepth = maxdepth
        self.constraints = constraints
        self.nested_constraints = nested_constraints
        self.warmup_maxsize_by = warmup_maxsize_by
        # - Early exit conditions:
        self.max_evals = max_evals
        self.timeout_in_seconds = timeout_in_seconds
        self.early_stop_condition = early_stop_condition
        # - Loss parameters
        self.loss = loss
        self.complexity_of_operators = complexity_of_operators
        self.complexity_of_constants = complexity_of_constants
        self.complexity_of_variables = complexity_of_variables
        self.parsimony = parsimony
        self.use_frequency = use_frequency
        self.use_frequency_in_tournament = use_frequency_in_tournament
        self.alpha = alpha
        self.annealing = annealing
        # - Evolutionary search parameters
        # -- Mutation parameters
        self.weight_add_node = weight_add_node
        self.weight_insert_node = weight_insert_node
        self.weight_delete_node = weight_delete_node
        self.weight_do_nothing = weight_do_nothing
        self.weight_mutate_constant = weight_mutate_constant
        self.weight_mutate_operator = weight_mutate_operator
        self.weight_randomize = weight_randomize
        self.weight_simplify = weight_simplify
        self.crossover_probability = crossover_probability
        self.skip_mutation_failures = skip_mutation_failures
        # -- Migration parameters
        self.migration = migration
        self.hof_migration = hof_migration
        self.fraction_replaced = fraction_replaced
        self.fraction_replaced_hof = fraction_replaced_hof
        self.topn = topn
        # -- Constants parameters
        self.should_optimize_constants = should_optimize_constants
        self.optimizer_algorithm = optimizer_algorithm
        self.optimizer_nrestarts = optimizer_nrestarts
        self.optimize_probability = optimize_probability
        self.optimizer_iterations = optimizer_iterations
        self.perturbation_factor = perturbation_factor
        # -- Selection parameters
        self.tournament_selection_n = tournament_selection_n
        self.tournament_selection_p = tournament_selection_p
        # Solver parameters
        self.procs = procs
        self.multithreading = multithreading
        self.cluster_manager = cluster_manager
        self.batching = batching
        self.batch_size = batch_size
        self.fast_cycle = fast_cycle
        self.precision = precision
        self.random_state = random_state
        self.deterministic = deterministic
        self.warm_start = warm_start
        # Additional runtime parameters
        # - Runtime user interface
        self.verbosity = verbosity
        self.update_verbosity = update_verbosity
        self.progress = progress
        # - Project management
        self.equation_file = equation_file
        self.temp_equation_file = temp_equation_file
        self.tempdir = tempdir
        self.delete_tempfiles = delete_tempfiles
        self.julia_project = julia_project
        self.update = update
        self.output_jax_format = output_jax_format
        self.output_torch_format = output_torch_format
        self.extra_sympy_mappings = extra_sympy_mappings
        self.extra_jax_mappings = extra_jax_mappings
        self.extra_torch_mappings = extra_torch_mappings
        # Pre-modelling transformation
        self.denoise = denoise
        self.select_k_features = select_k_features

        # Once all valid parameters have been assigned handle the
        # deprecated kwargs
        if len(kwargs) > 0:  # pragma: no cover
            deprecated_kwargs = make_deprecated_kwargs_for_pysr_regressor()
            for k, v in kwargs.items():
                # Handle renamed kwargs
                if k in deprecated_kwargs:
                    updated_kwarg_name = deprecated_kwargs[k]
                    setattr(self, updated_kwarg_name, v)
                    warnings.warn(
                        f"{k} has been renamed to {updated_kwarg_name} in PySRRegressor. "
                        "Please use that instead.",
                        FutureWarning,
                    )
                # Handle kwargs that have been moved to the fit method
                elif k in ["weights", "variable_names", "Xresampled"]:
                    warnings.warn(
                        f"{k} is a data dependant parameter so should be passed when fit is called. "
                        f"Ignoring parameter; please pass {k} during the call to fit instead.",
                        FutureWarning,
                    )
                else:
                    raise TypeError(
                        f"{k} is not a valid keyword argument for PySRRegressor."
                    )

    def __repr__(self):
        """
        Prints all current equations fitted by the model.

        The string `>>>>` denotes which equation is selected by the
        `model_selection`.
        """
        if not hasattr(self, "equations_") or self.equations_ is None:
            return "PySRRegressor.equations_ = None"

        output = "PySRRegressor.equations_ = [\n"

        equations = self.equations_
        if not isinstance(equations, list):
            all_equations = [equations]
        else:
            all_equations = equations

        for i, equations in enumerate(all_equations):
            selected = ["" for _ in range(len(equations))]
            if self.model_selection == "accuracy":
                chosen_row = -1
            elif self.model_selection == "best":
                chosen_row = equations["score"].idxmax()
            else:
                raise NotImplementedError
            selected[chosen_row] = ">>>>"
            repr_equations = pd.DataFrame(
                dict(
                    pick=selected,
                    score=equations["score"],
                    equation=equations["equation"],
                    loss=equations["loss"],
                    complexity=equations["complexity"],
                )
            )

            if len(all_equations) > 1:
                output += "[\n"

            for line in repr_equations.__repr__().split("\n"):
                output += "\t" + line + "\n"

            if len(all_equations) > 1:
                output += "]"

            if i < len(all_equations) - 1:
                output += ", "

        output += "]"
        return output

    def __getstate__(self):
        """
        Handles pickle serialization for PySRRegressor.

        The Scikit-learn standard requires estimators to be serializable via
        `pickle.dumps()`. However, `PyCall.jlwrap` does not support pickle
        serialization.

        Thus, for `PySRRegressor` to support pickle serialization, the
        `raw_julia_state_` attribute must be hidden from pickle. This will
        prevent the `warm_start` of any model that is loaded via `pickle.loads()`,
        but does allow all other attributes of a fitted `PySRRegressor` estimator
        to be serialized. Note: Jax and Torch format equations are also removed
        from the pickled instance.
        """
        state = self.__dict__
        if "raw_julia_state_" in state:
            warnings.warn(
                "raw_julia_state_ cannot be pickled and will be removed from the "
                "serialized instance. This will prevent a `warm_start` fit of any "
                "model that is deserialized via `pickle.load()`."
            )
        pickled_state = {
            key: None if key == "raw_julia_state_" else value
            for key, value in state.items()
        }
        if "equations_" in pickled_state:
            pickled_state["output_torch_format"] = False
            pickled_state["output_jax_format"] = False
            if self.nout_ == 1:
                pickled_columns = ~pickled_state["equations_"].columns.isin(
                    ["jax_format", "torch_format"]
                )
                pickled_state["equations_"] = (
                    pickled_state["equations_"].loc[:, pickled_columns].copy()
                )
            else:
                pickled_columns = [
                    ~dataframe.columns.isin(["jax_format", "torch_format"])
                    for dataframe in pickled_state["equations_"]
                ]
                pickled_state["equations_"] = [
                    dataframe.loc[:, signle_pickled_columns]
                    for dataframe, signle_pickled_columns in zip(
                        pickled_state["equations_"], pickled_columns
                    )
                ]
        return pickled_state

    @property
    def equations(self):  # pragma: no cover
        warnings.warn(
            "PySRRegressor.equations is now deprecated. "
            "Please use PySRRegressor.equations_ instead.",
            FutureWarning,
        )
        return self.equations_

    def get_best(self, index=None):
        """
        Get best equation using `model_selection`.

        Parameters
        ----------
        index : int, default=None
            If you wish to select a particular equation from `self.equations_`,
            give the row number here. This overrides the :param`model_selection`
            parameter.

        Returns
        -------
        best_equation : pandas.Series
            Dictionary representing the best expression found.

        Raises
        ------
        NotImplementedError
            Raised when an invalid model selection strategy is provided.
        """
        check_is_fitted(self, attributes=["equations_"])
        if self.equations_ is None:
            raise ValueError("No equations have been generated yet.")

        if index is not None:
            if isinstance(self.equations_, list):
                assert isinstance(index, list)
                return [eq.iloc[i] for eq, i in zip(self.equations_, index)]
            return self.equations_.iloc[index]

        if self.model_selection == "accuracy":
            if isinstance(self.equations_, list):
                return [eq.iloc[-1] for eq in self.equations_]
            return self.equations_.iloc[-1]
        elif self.model_selection == "best":
            if isinstance(self.equations_, list):
                return [eq.iloc[eq["score"].idxmax()] for eq in self.equations_]
            return self.equations_.iloc[self.equations_["score"].idxmax()]
        else:
            raise NotImplementedError(
                f"{self.model_selection} is not a valid model selection strategy."
            )

    def _setup_equation_file(self):
        """
        Sets the full pathname of the equation file, using :param`tempdir` and
        :param`equation_file`.
        """
        # Cast tempdir string as a Path object
        self.tempdir_ = Path(tempfile.mkdtemp(dir=self.tempdir))
        if self.temp_equation_file:
            self.equation_file_ = self.tempdir_ / "hall_of_fame.csv"
        elif self.equation_file is None:
            if self.warm_start and self.equation_file_:
                pass
            else:
                date_time = datetime.now().strftime("%Y-%m-%d_%H%M%S.%f")[:-3]
                self.equation_file_ = "hall_of_fame_" + date_time + ".csv"
        else:
            self.equation_file_ = self.equation_file
        self.equation_file_contents_ = None

    def _validate_and_set_init_params(self):
        """
        Ensures parameters passed at initialization are valid.

        Also returns a dictionary of parameters to update from their
        values given at initialization.

        Returns
        -------
        packed_modified_params : dict
            Dictionary of parameters to modify from their initialized
            values. For example, default parameters are set here
            when a parameter is left set to `None`.
        """
        # Immutable parameter validation
        # Ensure instance parameters are allowable values:
        if self.tournament_selection_n > self.population_size:
            raise ValueError(
                "tournament_selection_n parameter must be smaller than population_size."
            )

        if self.maxsize > 40:
            warnings.warn(
                "Note: Using a large maxsize for the equation search will be "
                "exponentially slower and use significant memory. You should consider "
                "turning `use_frequency` to False, and perhaps use `warmup_maxsize_by`."
            )
        elif self.maxsize < 7:
            raise ValueError("PySR requires a maxsize of at least 7")

        if self.deterministic and not (
            self.multithreading in [False, None]
            and self.procs == 0
            and self.random_state is not None
        ):
            raise ValueError(
                "To ensure deterministic searches, you must set `random_state` to a seed, "
                "`procs` to `0`, and `multithreading` to `False` or `None`."
            )

        if self.random_state is not None and (
            not self.deterministic or self.procs != 0
        ):
            warnings.warn(
                "Note: Setting `random_state` without also setting `deterministic` "
                "to True and `procs` to 0 will result in non-deterministic searches. "
            )

        # NotImplementedError - Values that could be supported at a later time
        if self.optimizer_algorithm not in VALID_OPTIMIZER_ALGORITHMS:
            raise NotImplementedError(
                f"PySR currently only supports the following optimizer algorithms: {VALID_OPTIMIZER_ALGORITHMS}"
            )

        # 'Mutable' parameter validation
        buffer_available = "buffer" in sys.stdout.__dir__()
        # Params and their default values, if None is given:
        default_param_mapping = {
            "binary_operators": "+ * - /".split(" "),
            "unary_operators": [],
            "maxdepth": self.maxsize,
            "constraints": {},
            "multithreading": self.procs != 0 and self.cluster_manager is None,
            "batch_size": 1,
            "update_verbosity": self.verbosity,
            "progress": buffer_available,
        }
        packed_modified_params = {}
        for parameter, default_value in default_param_mapping.items():
            parameter_value = getattr(self, parameter)
            if parameter_value is None:
                parameter_value = default_value
            else:
                # Special cases such as when binary_operators is a string
                if parameter in ["binary_operators", "unary_operators"] and isinstance(
                    parameter_value, str
                ):
                    parameter_value = [parameter_value]
                elif parameter == "batch_size" and parameter_value < 1:
                    warnings.warn(
                        "Given :param`batch_size` must be greater than or equal to one. "
                        ":param`batch_size` has been increased to equal one."
                    )
                    parameter_value = 1
                elif parameter == "progress" and not buffer_available:
                    warnings.warn(
                        "Note: it looks like you are running in Jupyter. "
                        "The progress bar will be turned off."
                    )
                    parameter_value = False
            packed_modified_params[parameter] = parameter_value

        assert (
            len(packed_modified_params["binary_operators"])
            + len(packed_modified_params["unary_operators"])
            > 0
        )
        return packed_modified_params

    def _validate_and_set_fit_params(self, X, y, Xresampled, weights, variable_names):
        """
        Validates the parameters passed to the :term`fit` method.

        This method also sets the `nout_` attribute.

        Parameters
        ----------
        X : {ndarray | pandas.DataFrame} of shape (n_samples, n_features)
            Training data.

        y : {ndarray | pandas.DataFrame} of shape (n_samples,) or (n_samples, n_targets)
            Target values. Will be cast to X's dtype if necessary.

        Xresampled : {ndarray | pandas.DataFrame} of shape
                        (n_resampled, n_features), default=None
            Resampled training data used for denoising.

        weights : {ndarray | pandas.DataFrame} of the same shape as y
            Each element is how to weight the mean-square-error loss
            for that particular element of y.

        variable_names : list[str] of length n_features
            Names of each variable in the training dataset, `X`.

        Returns
        -------
        X_validated : ndarray of shape (n_samples, n_features)
            Validated training data.

        y_validated : ndarray of shape (n_samples,) or (n_samples, n_targets)
            Validated target data.

        Xresampled : ndarray of shape (n_resampled, n_features)
            Validated resampled training data used for denoising.

        variable_names_validated : list[str] of length n_features
            Validated list of variable names for each feature in `X`.

        """
        if isinstance(X, pd.DataFrame):
            if variable_names:
                variable_names = None
                warnings.warn(
                    ":param`variable_names` has been reset to `None` as `X` is a DataFrame. "
                    "Using DataFrame column names instead."
                )

            if X.columns.is_object() and X.columns.str.contains(" ").any():
                X.columns = X.columns.str.replace(" ", "_")
                warnings.warn(
                    "Spaces in DataFrame column names are not supported. "
                    "Spaces have been replaced with underscores. \n"
                    "Please rename the columns to valid names."
                )
        elif variable_names and [" " in name for name in variable_names].any():
            variable_names = [name.replace(" ", "_") for name in variable_names]
            warnings.warn(
                "Spaces in `variable_names` are not supported. "
                "Spaces have been replaced with underscores. \n"
                "Please use valid names instead."
            )

        # Data validation and feature name fetching via sklearn
        # This method sets the n_features_in_ attribute
        if Xresampled is not None:
            Xresampled = check_array(Xresampled)
        if weights is not None:
            weights = check_array(weights, ensure_2d=False)
            check_consistent_length(weights, y)
        X, y = self._validate_data(X=X, y=y, reset=True, multi_output=True)
        self.feature_names_in_ = _check_feature_names_in(self, variable_names)
        variable_names = self.feature_names_in_

        # Handle multioutput data
        if len(y.shape) == 1 or (len(y.shape) == 2 and y.shape[1] == 1):
            y = y.reshape(-1)
        elif len(y.shape) == 2:
            self.nout_ = y.shape[1]
        else:
            raise NotImplementedError("y shape not supported!")

        return X, y, Xresampled, weights, variable_names

    def _pre_transform_training_data(
        self, X, y, Xresampled, variable_names, random_state
    ):
        """
        Transforms the training data before fitting the symbolic regressor.

        This method also updates/sets the `selection_mask_` attribute.

        Parameters
        ----------
        X : {ndarray | pandas.DataFrame} of shape (n_samples, n_features)
            Training data.

        y : {ndarray | pandas.DataFrame} of shape (n_samples,) or (n_samples, n_targets)
            Target values. Will be cast to X's dtype if necessary.

        Xresampled : {ndarray | pandas.DataFrame} of shape
                        (n_resampled, n_features), default=None
            Resampled training data used for denoising.

        variable_names : list[str] of length n_features
            Names of each variable in the training dataset, `X`.

        random_state : int, Numpy RandomState instance or None, default=None
            Pass an int for reproducible results across multiple function calls.
            See :term:`Glossary <random_state>`.

        Returns
        -------
        X_transformed : ndarray of shape (n_samples, n_features)
            Transformed training data. n_samples will be equal to
            :param`Xresampled.shape[0]` if :param`self.denoise` is `True`,
            and :param`Xresampled is not None`, otherwise it will be
            equal to :param`X.shape[0]`. n_features will be equal to
            :param`self.select_k_features` if `self.select_k_features is not None`,
            otherwise it will be equal to :param`X.shape[1]`

        y_transformed : ndarray of shape (n_samples,) or (n_samples, n_outputs)
            Transformed target data. n_samples will be equal to
            :param`Xresampled.shape[0]` if :param`self.denoise` is `True`,
            and :param`Xresampled is not None`, otherwise it will be
            equal to :param`X.shape[0]`.

        variable_names_transformed : list[str] of length n_features
            Names of each variable in the transformed dataset,
            `X_transformed`.
        """
        # Feature selection transformation
        if self.select_k_features:
            self.selection_mask_ = run_feature_selection(
                X, y, self.select_k_features, random_state=random_state
            )
            X = X[:, self.selection_mask_]

            if Xresampled is not None:
                Xresampled = Xresampled[:, self.selection_mask_]

            # Reduce variable_names to selection
            variable_names = [variable_names[i] for i in self.selection_mask_]

            # Re-perform data validation and feature name updating
            X, y = self._validate_data(X=X, y=y, reset=True, multi_output=True)
            # Update feature names with selected variable names
            self.feature_names_in_ = _check_feature_names_in(self, variable_names)
            print(f"Using features {self.feature_names_in_}")

        # Denoising transformation
        if self.denoise:
            if self.nout_ > 1:
                y = np.stack(
                    [
                        _denoise(
                            X, y[:, i], Xresampled=Xresampled, random_state=random_state
                        )[1]
                        for i in range(self.nout_)
                    ],
                    axis=1,
                )
                if Xresampled is not None:
                    X = Xresampled
            else:
                X, y = _denoise(X, y, Xresampled=Xresampled, random_state=random_state)

        return X, y, variable_names

    def _run(self, X, y, mutated_params, weights, seed):
        """
        Run the symbolic regression fitting process on the julia backend.

        Parameters
        ----------
        X : {ndarray | pandas.DataFrame} of shape (n_samples, n_features)
            Training data.

        y : {ndarray | pandas.DataFrame} of shape (n_samples,) or (n_samples, n_targets)
            Target values. Will be cast to X's dtype if necessary.

        mutated_params : dict[str, Any]
            Dictionary of mutated versions of some parameters passed in __init__.

        weights : {ndarray | pandas.DataFrame} of the same shape as y
            Each element is how to weight the mean-square-error loss
            for that particular element of y.

        seed : int
            Random seed for julia backend process.

        Returns
        -------
        self : object
            Reference to `self` with fitted attributes.

        Raises
        ------
        ImportError
            Raised when the julia backend fails to import a package.
        """
        # Need to be global as we don't want to recreate/reinstate julia for
        # every new instance of PySRRegressor
        global already_ran
        global Main

        # These are the parameters which may be modified from the ones
        # specified in init, so we define them here locally:
        binary_operators = mutated_params["binary_operators"]
        unary_operators = mutated_params["unary_operators"]
        maxdepth = mutated_params["maxdepth"]
        constraints = mutated_params["constraints"]
        nested_constraints = self.nested_constraints
        complexity_of_operators = self.complexity_of_operators
        multithreading = mutated_params["multithreading"]
        cluster_manager = self.cluster_manager
        batch_size = mutated_params["batch_size"]
        update_verbosity = mutated_params["update_verbosity"]
        progress = mutated_params["progress"]

        # Start julia backend processes
        if Main is None:
            if multithreading:
                os.environ["JULIA_NUM_THREADS"] = str(self.procs)

            Main = init_julia()

        if cluster_manager is not None:
            Main.eval(f"import ClusterManagers: addprocs_{cluster_manager}")
            cluster_manager = Main.eval(f"addprocs_{cluster_manager}")

        if not already_ran:
            julia_project, is_shared = _get_julia_project(self.julia_project)
            Main.eval("using Pkg")
            io = "devnull" if update_verbosity == 0 else "stderr"
            io_arg = f"io={io}" if is_julia_version_greater_eq(Main, "1.6") else ""

            Main.eval(
                f'Pkg.activate("{_escape_filename(julia_project)}", shared = Bool({int(is_shared)}), {io_arg})'
            )
            from julia.api import JuliaError

            if is_shared:
                # Install SymbolicRegression.jl:
                _add_sr_to_julia_project(Main, io_arg)

            try:
                if self.update:
                    Main.eval(f"Pkg.resolve({io_arg})")
                    Main.eval(f"Pkg.instantiate({io_arg})")
                else:
                    Main.eval(f"Pkg.instantiate({io_arg})")
            except (JuliaError, RuntimeError) as e:
                raise ImportError(import_error_string(julia_project)) from e
            Main.eval("using SymbolicRegression")

            Main.plus = Main.eval("(+)")
            Main.sub = Main.eval("(-)")
            Main.mult = Main.eval("(*)")
            Main.pow = Main.eval("(^)")
            Main.div = Main.eval("(/)")

        # TODO(mcranmer): These functions should be part of this class.
        binary_operators, unary_operators = _maybe_create_inline_operators(
            binary_operators=binary_operators, unary_operators=unary_operators
        )
        constraints = _process_constraints(
            binary_operators=binary_operators,
            unary_operators=unary_operators,
            constraints=constraints,
        )

        una_constraints = [constraints[op] for op in unary_operators]
        bin_constraints = [constraints[op] for op in binary_operators]

        # Parse dict into Julia Dict for nested constraints::
        if nested_constraints is not None:
            nested_constraints_str = "Dict("
            for outer_k, outer_v in nested_constraints.items():
                nested_constraints_str += f"({outer_k}) => Dict("
                for inner_k, inner_v in outer_v.items():
                    nested_constraints_str += f"({inner_k}) => {inner_v}, "
                nested_constraints_str += "), "
            nested_constraints_str += ")"
            nested_constraints = Main.eval(nested_constraints_str)

        # Parse dict into Julia Dict for complexities:
        if complexity_of_operators is not None:
            complexity_of_operators_str = "Dict("
            for k, v in complexity_of_operators.items():
                complexity_of_operators_str += f"({k}) => {v}, "
            complexity_of_operators_str += ")"
            complexity_of_operators = Main.eval(complexity_of_operators_str)

        custom_loss = Main.eval(self.loss)
        early_stop_condition = Main.eval(
            str(self.early_stop_condition) if self.early_stop_condition else None
        )

        mutation_weights = np.array(
            [
                self.weight_mutate_constant,
                self.weight_mutate_operator,
                self.weight_add_node,
                self.weight_insert_node,
                self.weight_delete_node,
                self.weight_simplify,
                self.weight_randomize,
                self.weight_do_nothing,
            ],
            dtype=float,
        )

        # Call to Julia backend.
        # See https://github.com/MilesCranmer/SymbolicRegression.jl/blob/master/src/OptionsStruct.jl
        options = Main.Options(
            binary_operators=Main.eval(str(tuple(binary_operators)).replace("'", "")),
            unary_operators=Main.eval(str(tuple(unary_operators)).replace("'", "")),
            bin_constraints=bin_constraints,
            una_constraints=una_constraints,
            complexity_of_operators=complexity_of_operators,
            complexity_of_constants=self.complexity_of_constants,
            complexity_of_variables=self.complexity_of_variables,
            nested_constraints=nested_constraints,
            loss=custom_loss,
            maxsize=int(self.maxsize),
            hofFile=_escape_filename(self.equation_file_),
            npopulations=int(self.populations),
            batching=self.batching,
            batchSize=int(min([batch_size, len(X)]) if self.batching else len(X)),
            mutationWeights=mutation_weights,
            probPickFirst=self.tournament_selection_p,
            ns=self.tournament_selection_n,
            # These have the same name:
            parsimony=self.parsimony,
            alpha=self.alpha,
            maxdepth=maxdepth,
            fast_cycle=self.fast_cycle,
            migration=self.migration,
            hofMigration=self.hof_migration,
            fractionReplacedHof=self.fraction_replaced_hof,
            shouldOptimizeConstants=self.should_optimize_constants,
            warmupMaxsizeBy=self.warmup_maxsize_by,
            useFrequency=self.use_frequency,
            useFrequencyInTournament=self.use_frequency_in_tournament,
            npop=self.population_size,
            ncyclesperiteration=self.ncyclesperiteration,
            fractionReplaced=self.fraction_replaced,
            topn=self.topn,
            verbosity=self.verbosity,
            optimizer_algorithm=self.optimizer_algorithm,
            optimizer_nrestarts=self.optimizer_nrestarts,
            optimize_probability=self.optimize_probability,
            optimizer_iterations=self.optimizer_iterations,
            perturbationFactor=self.perturbation_factor,
            annealing=self.annealing,
            stateReturn=True,  # Required for state saving.
            progress=progress,
            timeout_in_seconds=self.timeout_in_seconds,
            crossoverProbability=self.crossover_probability,
            skip_mutation_failures=self.skip_mutation_failures,
            max_evals=self.max_evals,
            earlyStopCondition=early_stop_condition,
            seed=seed,
            deterministic=self.deterministic,
        )

        # Convert data to desired precision
        np_dtype = {16: np.float16, 32: np.float32, 64: np.float64}[self.precision]

        # This converts the data into a Julia array:
        Main.X = np.array(X, dtype=np_dtype).T
        if len(y.shape) == 1:
            Main.y = np.array(y, dtype=np_dtype)
        else:
            Main.y = np.array(y, dtype=np_dtype).T
        if weights is not None:
            if len(weights.shape) == 1:
                Main.weights = np.array(weights, dtype=np_dtype)
            else:
                Main.weights = np.array(weights, dtype=np_dtype).T
        else:
            Main.weights = None

        cprocs = 0 if multithreading else self.procs

        # Call to Julia backend.
        # See https://github.com/MilesCranmer/SymbolicRegression.jl/blob/master/src/SymbolicRegression.jl
        self.raw_julia_state_ = Main.EquationSearch(
            Main.X,
            Main.y,
            weights=Main.weights,
            niterations=int(self.niterations),
            varMap=self.feature_names_in_.tolist(),
            options=options,
            numprocs=int(cprocs),
            multithreading=bool(multithreading),
            saved_state=self.raw_julia_state_,
            addprocs_function=cluster_manager,
        )

        # Set attributes
        self.equations_ = self.get_hof()

        if self.delete_tempfiles:
            shutil.rmtree(self.tempdir_)

        already_ran = True

        return self

    def fit(
        self,
        X,
        y,
        Xresampled=None,
        weights=None,
        variable_names=None,
    ):
        """
        Search for equations to fit the dataset and store them in `self.equations_`.

        Parameters
        ----------
        X : {ndarray | pandas.DataFrame} of shape (n_samples, n_features)
            Training data.

        y : {ndarray | pandas.DataFrame} of shape (n_samples,) or (n_samples, n_targets)
            Target values. Will be cast to X's dtype if necessary.

        Xresampled : {ndarray | pandas.DataFrame} of shape
                        (n_resampled, n_features), default=None
            Resampled training data to generate a denoised data on. This
            will be used as the training data, rather than `X`.

        weights : {ndarray | pandas.DataFrame} of the same shape as y, default=None
            Each element is how to weight the mean-square-error loss
            for that particular element of `y`. Alternatively,
            if a custom `loss` was set, it will can be used
            in arbitrary ways.

        variable_names : list[str], default=None
            A list of names for the variables, rather than "x0", "x1", etc.
            If :param`X` is a pandas dataframe, the column names will be used
            instead of `variable_names`. Cannot contain spaces or special
            characters. Avoid variable names which are also
            function names in `sympy`, such as "N".

        Returns
        -------
        self : object
            Fitted estimator.
        """
        # Init attributes that are not specified in BaseEstimator
        if self.warm_start and hasattr(self, "raw_julia_state_"):
            pass
        else:
            if hasattr(self, "raw_julia_state_"):
                warnings.warn(
                    "The discovered expressions are being reset. "
                    "Please set `warm_start=True` if you wish to continue "
                    "to start a search where you left off.",
                )

            self.equations_ = None
            self.nout_ = 1
            self.selection_mask_ = None
            self.raw_julia_state_ = None

        random_state = check_random_state(self.random_state)  # For np random
        seed = random_state.get_state()[1][0]  # For julia random

        self._setup_equation_file()

        mutated_params = self._validate_and_set_init_params()

        X, y, Xresampled, weights, variable_names = self._validate_and_set_fit_params(
            X, y, Xresampled, weights, variable_names
        )

        if X.shape[0] > 10000 and not self.batching:
            warnings.warn(
                "Note: you are running with more than 10,000 datapoints. "
                "You should consider turning on batching (https://astroautomata.com/PySR/#/options?id=batching). "
                "You should also reconsider if you need that many datapoints. "
                "Unless you have a large amount of noise (in which case you "
                "should smooth your dataset first), generally < 10,000 datapoints "
                "is enough to find a functional form with symbolic regression. "
                "More datapoints will lower the search speed."
            )

        # Pre transformations (feature selection and denoising)
        X, y, variable_names = self._pre_transform_training_data(
            X, y, Xresampled, variable_names, random_state
        )

        # Warn about large feature counts (still warn if feature count is large
        # after running feature selection)
        if self.n_features_in_ >= 10:
            warnings.warn(
                "Note: you are running with 10 features or more. "
                "Genetic algorithms like used in PySR scale poorly with large numbers of features. "
                "Consider using feature selection techniques to select the most important features "
                "(you can do this automatically with the `select_k_features` parameter), "
                "or, alternatively, doing a dimensionality reduction beforehand. "
                "For example, `X = PCA(n_components=6).fit_transform(X)`, "
                "using scikit-learn's `PCA` class, "
                "will reduce the number of features to 6 in an interpretable way, "
                "as each resultant feature "
                "will be a linear combination of the original features. "
            )

        # Assertion checks
        use_custom_variable_names = variable_names is not None
        # TODO: this is always true.

        _check_assertions(
            X,
            use_custom_variable_names,
            variable_names,
            weights,
            y,
        )

        # Fitting procedure
        return self._run(X, y, mutated_params, weights=weights, seed=seed)

    def refresh(self, checkpoint_file=None):
        """
        Updates self.equations_ with any new options passed, such as
        :param`extra_sympy_mappings`.

        Parameters
        ----------
        checkpoint_file : str, default=None
            Path to checkpoint hall of fame file to be loaded.
        """
        check_is_fitted(self, attributes=["equation_file_"])
        if checkpoint_file:
            self.equation_file_ = checkpoint_file
            self.equation_file_contents_ = None
        self.equations_ = self.get_hof()

    def predict(self, X, index=None):
        """
        Predict y from input X using the equation chosen by `model_selection`.

        You may see what equation is used by printing this object. X should
        have the same columns as the training data.

        Parameters
        ----------
        X : {ndarray | pandas.DataFrame} of shape (n_samples, n_features)
            Training data.

        index : int, default=None
            If you want to compute the output of an expression using a
            particular row of `self.equations_`, you may specify the index here.

        Returns
        -------
        y_predicted : ndarray of shape (n_samples, nout_)
            Values predicted by substituting `X` into the fitted symbolic
            regression model.

        Raises
        ------
        ValueError
            Raises if the `best_equation` cannot be evaluated.
        """
        check_is_fitted(
            self, attributes=["selection_mask_", "feature_names_in_", "nout_"]
        )
        best_equation = self.get_best(index=index)

        # When X is an numpy array or a pandas dataframe with a RangeIndex,
        # the self.feature_names_in_ generated during fit, for the same X,
        # will cause a warning to be thrown during _validate_data.
        # To avoid this, convert X to a dataframe, apply the selection mask,
        # and then set the column/feature_names of X to be equal to those
        # generated during fit.
        if not isinstance(X, pd.DataFrame):
            X = check_array(X)
            X = pd.DataFrame(X)
        if isinstance(X.columns, pd.RangeIndex):
            if self.selection_mask_ is not None:
                # RangeIndex enforces column order allowing columns to
                # be correctly filtered with self.selection_mask_
                X = X.iloc[:, self.selection_mask_]
            X.columns = self.feature_names_in_
        # Without feature information, CallableEquation/lambda_format equations
        # require that the column order of X matches that of the X used during
        # the fitting process. _validate_data removes this feature information
        # when it converts the dataframe to an np array. Thus, to ensure feature
        # order is preserved after conversion, the dataframe columns must be
        # reordered/reindexed to match those of the transformed (denoised and
        # feature selected) X in fit.
        X = X.reindex(columns=self.feature_names_in_)
        X = self._validate_data(X, reset=False)

        try:
            if self.nout_ > 1:
                return np.stack(
                    [eq["lambda_format"](X) for eq in best_equation], axis=1
                )
            return best_equation["lambda_format"](X)
        except Exception as error:
            raise ValueError(
                "Failed to evaluate the expression. "
                "If you are using a custom operator, make sure to define it in :param`extra_sympy_mappings`, "
                "e.g., `model.set_params(extra_sympy_mappings={'inv': lambda x: 1 / x})`."
            ) from error

    def sympy(self, index=None):
        """
        Return sympy representation of the equation(s) chosen by `model_selection`.

        Parameters
        ----------
        index : int, default=None
            If you wish to select a particular equation from
            `self.equations_`, give the index number here. This overrides
            the `model_selection` parameter.

        Returns
        -------
        best_equation : str, list[str] of length nout_
            SymPy representation of the best equation.
        """
        self.refresh()
        best_equation = self.get_best(index=index)
        if self.nout_ > 1:
            return [eq["sympy_format"] for eq in best_equation]
        return best_equation["sympy_format"]

    def latex(self, index=None):
        """
        Return latex representation of the equation(s) chosen by `model_selection`.

        Parameters
        ----------
        index : int, default=None
            If you wish to select a particular equation from
            `self.equations_`, give the index number here. This overrides
            the `model_selection` parameter.

        Returns
        -------
        best_equation : str or list[str] of length nout_
            LaTeX expression of the best equation.
        """
        self.refresh()
        sympy_representation = self.sympy(index=index)
        if self.nout_ > 1:
            return [sympy.latex(s) for s in sympy_representation]
        return sympy.latex(sympy_representation)

    def jax(self, index=None):
        """
        Return jax representation of the equation(s) chosen by `model_selection`.

        Each equation (multiple given if there are multiple outputs) is a dictionary
        containing {"callable": func, "parameters": params}. To call `func`, pass
        func(X, params). This function is differentiable using `jax.grad`.

        Parameters
        ----------
        index : int, default=None
            If you wish to select a particular equation from
            `self.equations_`, give the row number here. This overrides
            the `model_selection` parameter.

        Returns
        -------
        best_equation : dict[str, Any]
            Dictionary of callable jax function in "callable" key,
            and jax array of parameters as "parameters" key.
        """
        self.set_params(output_jax_format=True)
        self.refresh()
        best_equation = self.get_best(index=index)
        if self.nout_ > 1:
            return [eq["jax_format"] for eq in best_equation]
        return best_equation["jax_format"]

    def pytorch(self, index=None):
        """
        Return pytorch representation of the equation(s) chosen by `model_selection`.

        Each equation (multiple given if there are multiple outputs) is a PyTorch module
        containing the parameters as trainable attributes. You can use the module like
        any other PyTorch module: `module(X)`, where `X` is a tensor with the same
        column ordering as trained with.

        Parameters
        ----------
        index : int, default=None
            If you wish to select a particular equation from
            `self.equations_`, give the row number here. This overrides
            the `model_selection` parameter.

        Returns
        -------
        best_equation : torch.nn.Module
            PyTorch module representing the expression.
        """
        self.set_params(output_torch_format=True)
        self.refresh()
        best_equation = self.get_best(index=index)
        if self.nout_ > 1:
            return [eq["torch_format"] for eq in best_equation]
        return best_equation["torch_format"]

    def _read_equation_file(self):
        """Read the hall of fame file created by SymbolicRegression.jl"""
        try:
            if self.nout_ > 1:
                all_outputs = []
                for i in range(1, self.nout_ + 1):
                    df = pd.read_csv(
                        str(self.equation_file_) + f".out{i}" + ".bkup",
                        sep="|",
                    )
                    # Rename Complexity column to complexity:
                    df.rename(
                        columns={
                            "Complexity": "complexity",
                            "MSE": "loss",
                            "Equation": "equation",
                        },
                        inplace=True,
                    )

                    all_outputs.append(df)
            else:
                all_outputs = [pd.read_csv(str(self.equation_file_) + ".bkup", sep="|")]
                all_outputs[-1].rename(
                    columns={
                        "Complexity": "complexity",
                        "MSE": "loss",
                        "Equation": "equation",
                    },
                    inplace=True,
                )
        except FileNotFoundError:
            raise RuntimeError(
                "Couldn't find equation file! The equation search likely exited "
                "before a single iteration completed."
            )
        return all_outputs

    def get_hof(self):
        """Get the equations from a hall of fame file. If no arguments
        entered, the ones used previously from a call to PySR will be used."""
        check_is_fitted(
            self,
            attributes=[
                "nout_",
                "equation_file_",
                "selection_mask_",
                "feature_names_in_",
            ],
        )
        if (
            not hasattr(self, "equation_file_contents_")
        ) or self.equation_file_contents_ is None:
            self.equation_file_contents_ = self._read_equation_file()

        # It is expected extra_jax/torch_mappings will be updated after fit.
        # Thus, validation is performed here instead of in _validate_init_params
        extra_jax_mappings = self.extra_jax_mappings
        extra_torch_mappings = self.extra_torch_mappings
        if extra_jax_mappings is not None:
            for value in extra_jax_mappings.values():
                if not isinstance(value, str):
                    raise ValueError(
                        "extra_jax_mappings must have keys that are strings! "
                        "e.g., {sympy.sqrt: 'jnp.sqrt'}."
                    )
        else:
            extra_jax_mappings = {}
        if extra_torch_mappings is not None:
            for value in extra_jax_mappings.values():
                if not callable(value):
                    raise ValueError(
                        "extra_torch_mappings must be callable functions! "
                        "e.g., {sympy.sqrt: torch.sqrt}."
                    )
        else:
            extra_torch_mappings = {}

        ret_outputs = []

        for output in self.equation_file_contents_:

            scores = []
            lastMSE = None
            lastComplexity = 0
            sympy_format = []
            lambda_format = []
            if self.output_jax_format:
                jax_format = []
            if self.output_torch_format:
                torch_format = []
            local_sympy_mappings = {
                **(self.extra_sympy_mappings if self.extra_sympy_mappings else {}),
                **sympy_mappings,
            }

            sympy_symbols = [
                sympy.Symbol(variable) for variable in self.feature_names_in_
            ]

            for _, eqn_row in output.iterrows():
                eqn = sympify(eqn_row["equation"], locals=local_sympy_mappings)
                sympy_format.append(eqn)

                # Numpy:
                lambda_format.append(
                    CallableEquation(
                        sympy_symbols, eqn, self.selection_mask_, self.feature_names_in_
                    )
                )

                # JAX:
                if self.output_jax_format:
                    from .export_jax import sympy2jax

                    func, params = sympy2jax(
                        eqn,
                        sympy_symbols,
                        selection=self.selection_mask_,
                        extra_jax_mappings=(
                            self.extra_jax_mappings if self.extra_jax_mappings else {}
                        ),
                    )
                    jax_format.append({"callable": func, "parameters": params})

                # Torch:
                if self.output_torch_format:
                    from .export_torch import sympy2torch

                    module = sympy2torch(
                        eqn,
                        sympy_symbols,
                        selection=self.selection_mask_,
                        extra_torch_mappings=(
                            self.extra_torch_mappings
                            if self.extra_torch_mappings
                            else {}
                        ),
                    )
                    torch_format.append(module)

                curMSE = eqn_row["loss"]
                curComplexity = eqn_row["complexity"]

                if lastMSE is None:
                    cur_score = 0.0
                else:
                    if curMSE > 0.0:
                        # TODO Move this to more obvious function/file.
                        cur_score = -np.log(curMSE / lastMSE) / (
                            curComplexity - lastComplexity
                        )
                    else:
                        cur_score = np.inf

                scores.append(cur_score)
                lastMSE = curMSE
                lastComplexity = curComplexity

            output["score"] = np.array(scores)
            output["sympy_format"] = sympy_format
            output["lambda_format"] = lambda_format
            output_cols = [
                "complexity",
                "loss",
                "score",
                "equation",
                "sympy_format",
                "lambda_format",
            ]
            if self.output_jax_format:
                output_cols += ["jax_format"]
                output["jax_format"] = jax_format
            if self.output_torch_format:
                output_cols += ["torch_format"]
                output["torch_format"] = torch_format

            ret_outputs.append(output[output_cols])

        if self.nout_ > 1:
            return ret_outputs
        return ret_outputs[0]


def _denoise(X, y, Xresampled=None, random_state=None):
    """Denoise the dataset using a Gaussian process"""
    from sklearn.gaussian_process import GaussianProcessRegressor
    from sklearn.gaussian_process.kernels import RBF, WhiteKernel, ConstantKernel

    gp_kernel = RBF(np.ones(X.shape[1])) + WhiteKernel(1e-1) + ConstantKernel()
    gpr = GaussianProcessRegressor(
        kernel=gp_kernel, n_restarts_optimizer=50, random_state=random_state
    )
    gpr.fit(X, y)
    if Xresampled is not None:
        return Xresampled, gpr.predict(Xresampled)

    return X, gpr.predict(X)


# Function has not been removed only due to usage in module tests
def _handle_feature_selection(X, select_k_features, y, variable_names):
    if select_k_features is not None:
        selection = run_feature_selection(X, y, select_k_features)
        print(f"Using features {[variable_names[i] for i in selection]}")
        X = X[:, selection]

    else:
        selection = None
    return X, selection


def run_feature_selection(X, y, select_k_features, random_state=None):
    """
    Use a gradient boosting tree regressor as a proxy for finding
    the k most important features in X, returning indices for those
    features as output.
    """
    from sklearn.ensemble import RandomForestRegressor
    from sklearn.feature_selection import SelectFromModel

    clf = RandomForestRegressor(
        n_estimators=100, max_depth=3, random_state=random_state
    )
    clf.fit(X, y)
    selector = SelectFromModel(
        clf, threshold=-np.inf, max_features=select_k_features, prefit=True
    )
    return selector.get_support(indices=True)