README.md

metadata

pretty_name: SRSD-Feynman (Hard)
annotations_creators:
  - expert
language_creators:
  - expert-generated
languages:
  - en
licenses:
  - mit
multilinguality:
  - monolingual
size_categories:
  - 100K<n<1M
source_datasets:
  - extended
task_categories:
  - symbolic-regression
task_ids: []

Dataset Card for SRSD-Feynman (Hard set)

Table of Contents

• Table of Contents
• Dataset Description
  • Dataset Summary
  • Supported Tasks and Leaderboards
  • Languages
• Dataset Structure
  • Data Instances
  • Data Fields
  • Data Splits
• Dataset Creation
  • Curation Rationale
  • Source Data
  • Annotations
  • Personal and Sensitive Information
• Considerations for Using the Data
  • Social Impact of Dataset
  • Discussion of Biases
  • Other Known Limitations
• Additional Information
  • Dataset Curators
  • Licensing Information
  • Citation Information
  • Contributions

Dataset Description

• Homepage:
• Repository: https://github.com/omron-sinicx/srsd-benchmark
• Paper: Rethinking Symbolic Regression Datasets and Benchmarks for Scientific Discovery
• Point of Contact: Yoshitaka Ushiku

Dataset Summary

Our SRSD (Feynman) datasets are designed to discuss the performance of Symbolic Regression for Scientific Discovery. We carefully reviewed the properties of each formula and its variables in the Feynman Symbolic Regression Database to design reasonably realistic sampling range of values so that our SRSD datasets can be used for evaluating the potential of SRSD such as whether or not an SR method con (re)discover physical laws from such datasets.

This is the Hard set of our SRSD-Feynman datasets, which consists of the following 50 different physics formulas:

ID	Formula
I.6.20	$f=\exp (-\frac{{\theta }^2}{2{\sigma }^2})\mathrm{/}\sqrt{2\pi {\sigma }^2}f\; =\; \backslash exp\backslash left(-\backslash frac\{\backslash theta^2\}\{2\backslash sigma^2\}\backslash right)/\backslash sqrt\{2\backslash pi\backslash sigma^2\}$
I.6.20a	$f=\exp (-\frac{{\theta }^2}{2})\mathrm{/}\sqrt{2\pi }f\; =\; \backslash exp\backslash left(-\backslash frac\{\backslash theta^2\}\{2\}\backslash right)/\backslash sqrt\{2\backslash pi\}$
I.6.20b	$f=\exp (-\frac{{(\theta -{\theta }_1)}^2}{2{\sigma }^2})\mathrm{/}\sqrt{2\pi \sigma }f\; =\; \backslash exp\backslash left(-\backslash frac\{\backslash left(\backslash theta-\backslash theta\_1\backslash right)^2\}\{2\backslash sigma^2\}\backslash right)/\backslash sqrt\{2\backslash pi\backslash sigma\}$
I.9.18	$F=\frac{G{m}_1{m}_2}{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}F\; =\; \backslash frac\{G\; m\_1\; m\_2\}\{(x\_2-x\_1)^2+(y\_2-y\_1)^2+(z\_2-z\_1)^2\}$
I.15.3t	$t_1=\frac{t-ux\mathrm{/}{c}^2}{\sqrt{1-u^2\mathrm{/}{c}^2}}t\_1\; =\; \backslash frac\{t-u\; x/c^2\}\{\backslash sqrt\{1-u^2/c^2\}\}$
I.15.3x	$x_1=\frac{x-ut}{\sqrt{1-u^2\mathrm{/}{c}^2}}x\_1\; =\; \backslash frac\{x\; -\; u\; t\}\{\backslash sqrt\{1\; -\; u^2/c^2\}\}$
I.29.16	$x=\sqrt{x_1^2+x_2^2+2x_1{x}_2\cos ({\theta }_1-{\theta }_2)}x\; =\; \backslash sqrt\{x\_1^2+x\_2^2\; +\; 2\; x\_1\; x\_2\; \backslash cos\backslash left(\backslash theta\_1-\backslash theta\_2\backslash right)\}$
I.30.3	$I=I_0\frac{{\sin }^2\left(n\theta \mathrm{/}2\right)}{{\sin }^2\left(\theta \mathrm{/}2\right)}I\; =\; I\_0\; \backslash frac\{\backslash sin^2\backslash left(n\; \backslash theta/2\backslash right)\}\{\backslash sin^2\backslash left(\backslash theta/2\backslash right)\}$
I.32.17	$P=\left(\frac{1}{2}ϵc{E}^2\right)\left(\frac{8\pi r^2}{3}\right)\left(\frac{{\omega }^4}{{({\omega }^2-{\omega }_0^2)}^2}\right)P\; =\; \backslash left(\backslash frac\{1\}\{2\}\; \backslash epsilon\; c\; E^2\backslash right)\; \backslash left(\backslash frac\{8\; \backslash pi\; r^2\}\{3\}\backslash right)\; \backslash left(\backslash frac\{\backslash omega^4\}\{\backslash left(\backslash omega^2-\backslash omega\_0^2\backslash right)^2\}\backslash right)$
I.34.14	$\omega =\frac{1+v\mathrm{/}c}{\sqrt{1-v^2\mathrm{/}{c}^2}}{\omega }_0\backslash omega\; =\; \backslash frac\{1+v/c\}\{\backslash sqrt\{1-v^2/c^2\}\}\; \backslash omega\_0$
I.37.4	$I_{12}=I_1+I_2+2\sqrt{I_1{I}_2}\cos \delta I\_\{12\}\; =\; I\_1+I\_2+2\; \backslash sqrt\{I\_1\; I\_2\}\; \backslash cos\backslash delta$
I.39.22	$P=\frac{nkT}{V}P\; =\; \backslash frac\{n\; k\; T\}\{V\}$
I.40.1	$n=n_0\exp (-mgx\mathrm{/}kT)n\; =\; n\_0\; \backslash exp\backslash left(-m\; g\; x/\; k\; T\backslash right)$
I.41.16	$L_{\text{rad}}=\frac{h}{2\pi }\frac{{\omega }^3}{{\pi }^2{c}^2(\exp (h\omega \mathrm{/}2\pi kT)-1)}L\_\backslash text\{rad\}\; =\; \backslash frac\{h\}\{2\; \backslash pi\}\; \backslash frac\{\backslash omega^3\}\{\backslash pi^2\; c^2\; (\backslash exp(h\; \backslash omega/2\; \backslash pi\; k\; T)-1)\}$
I.44.4	$Q=nkT\ln (\frac{V_2}{V_1})Q\; =\; n\; k\; T\; \backslash ln(\backslash frac\{V\_2\}\{V\_1\})$
I.50.26	$x=K(\cos \omega t+ϵ{\cos }^2\omega t)x\; =\; K\; \backslash left(\backslash cos\backslash omega\; t\; +\; \backslash epsilon\; \backslash cos^2\; \backslash omega\; t\backslash right)$
II.6.15a	$E=\frac{p}{4\pi ϵ}\frac{3z}{r^5}\sqrt{x^2+y^2}E\; =\; \backslash frac\{p\}\{4\; \backslash pi\; \backslash epsilon\}\; \backslash frac\{3\; z\}\{r^5\}\; \backslash sqrt\{x^2+y^2\}$
II.6.15b	$E=\frac{p}{4\pi ϵ}\frac{3\cos \theta \sin \theta }{r^3}E\; =\; \backslash frac\{p\}\{4\; \backslash pi\; \backslash epsilon\}\; \backslash frac\{3\; \backslash cos\backslash theta\; \backslash sin\backslash theta\}\{r^3\}$
II.11.17	$n=n_0(1+\frac{p_{0}E\cos \theta }{kT})n\; =\; n\_0\; \backslash left(1\; +\; \backslash frac\{p\_0\; E\; \backslash cos\backslash theta\}\{k\; T\}\backslash right)$
II.11.20	$P=\frac{n_0{p}_0^{2}E}{3kT}P\; =\; \backslash frac\{n\_0\; p\_0^2\; E\}\{3\; k\; T\}$
II.11.27	$P=\frac{N\alpha }{1-(n\alpha \mathrm{/}3)}ϵEP\; =\; \backslash frac\{N\; \backslash alpha\}\{1-(n\; \backslash alpha/3)\}\; \backslash epsilon\; E$
II.11.28	$\kappa =1+\frac{N\alpha }{1-(N\alpha \mathrm{/}3)}\backslash kappa\; =\; 1\; +\; \backslash frac\{N\; \backslash alpha\}\{1-(N\; \backslash alpha/3)\}$
II.13.23	$\rho =\frac{{\rho }_0}{\sqrt{1-v^2\mathrm{/}{c}^2}}\backslash rho\; =\; \backslash frac\{\backslash rho\_0\}\{\backslash sqrt\{1-v^2/c^2\}\}$
II.13.34	$j=\frac{{\rho }_{0}v}{\sqrt{1-v^2\mathrm{/}{c}^2}}j\; =\; \backslash frac\{\backslash rho\_0\; v\}\{\backslash sqrt\{1-v^2/c^2\}\}$
II.24.17	$k=\sqrt{{\omega }^2\mathrm{/}{c}^2-{\pi }^2\mathrm{/}{a}^2}k\; =\; \backslash sqrt\{\backslash omega^2\; /\; c^2\; -\; \backslash pi^2/a^2\}$
II.35.18	$a=\frac{N}{\exp (\mu B\mathrm{/}kT)+\exp (-\mu B\mathrm{/}kT)}a\; =\; \backslash frac\{N\}\{\backslash exp(\backslash mu\; B/k\; T)+\backslash exp(-\backslash mu\; B/k\; T)\}$
II.35.21	$M=N\mu \tanh \frac{\mu B}{kT}M\; =\; N\; \backslash mu\; \backslash tanh\backslash frac\{\backslash mu\; B\}\{k\; T\}$
II.36.38	$x=\frac{\mu H}{kT}+\frac{\mu \lambda }{ϵc^{2}kT}Mx\; =\; \backslash frac\{\backslash mu\; H\}\{k\; T\}+\backslash frac\{\backslash mu\; \backslash lambda\}\{\backslash epsilon\; c^2\; k\; T\}\; M$
III.4.33	$E=\frac{h\omega }{2\pi (\exp \left(h\omega \mathrm{/}2\pi kT\right)-1)}E\; =\; \backslash frac\{h\; \backslash omega\}\{2\; \backslash pi\; \backslash left(\backslash exp\backslash left(h\; \backslash omega/2\; \backslash pi\; k\; T\backslash right)\; -\; 1\backslash right)\}$
III.9.52	$P_{\text{I}\to \text{II}}={\left(\frac{2\pi \mu Et}{h}\right)}^2\frac{{\sin }^2\left((\omega -{\omega }_0)t\mathrm{/}2\right)}{(\omega -{\omega }_0)t\mathrm{/}2{)}^2}P\_\{\backslash text\{I\}\; \backslash rightarrow\; \backslash text\{II\}\}\; =\; \backslash left(\backslash frac\{2\; \backslash pi\; \backslash mu\; E\; t\}\{h\}\backslash right)^2\; \backslash frac\{\backslash sin^2\backslash left(\backslash left(\backslash omega-\backslash omega\_0\backslash right)\; t/2\backslash right)\}\{\backslash left(\backslash omega-\backslash omega\_0\backslash right)\; t\; /\; 2)^2\}$
III.10.19	$E=\mu \sqrt{B_x^2+B_y^2+B_z^2}E\; =\; \backslash mu\; \backslash sqrt\{B\_x^2+B\_y^2+B\_z^2\}$
III.21.20	$J=-\rho \frac{q}{m}AJ\; =\; -\backslash rho\; \backslash frac\{q\}\{m\}\; A$
B1	$A={\left(\frac{Z_1{Z}_2\alpha hc}{4E{\sin }^2\left(\theta \mathrm{/}2\right)}\right)}^{2}A\; =\; \backslash left(\backslash frac\{Z\_1\; Z\_2\; \backslash alpha\; h\; c\}\{4\; E\; \backslash sin^2\backslash left(\backslash theta/2\backslash right)\}\backslash right)^2$
B2	$k=\frac{m{k}_G}{L^2}(1+\sqrt{1+\frac{2E{L}^2}{m{k}_G^2}}\cos ({\theta }_1-{\theta }_2))k\; =\; \backslash frac\{m\; k\_G\}\{L^2\}\; \backslash left(1+\backslash sqrt\{1+\backslash frac\{2\; E\; L^2\}\{m\; k\_G^2\}\}\; \backslash cos\backslash left(\backslash theta\_1-\backslash theta\_2\backslash right)\backslash right)$
B3	$r=\frac{d(1-{\alpha }^2)}{1+\alpha \cos ({\theta }_1-{\theta }_2)}r\; =\; \backslash frac\{d\; (1-\backslash alpha^2)\}\{1+\backslash alpha\; \backslash cos(\backslash theta\_1-\backslash theta\_2)\}$
B4	$v=\sqrt{\frac{2}{m}(E-U-\frac{L^2}{2m{r}^2})}v\; =\; \backslash sqrt\{\backslash frac\{2\}\{m\}\; \backslash left(E-U-\backslash frac\{L^2\}\{2\; m\; r^2\}\backslash right)\}$
B5	$t=\frac{2\pi d^{3\mathrm{/}2}}{\sqrt{G(m_1+m_2)}}t\; =\; \backslash frac\{2\; \backslash pi\; d^\{3/2\}\}\{\backslash sqrt\{G(m\_1+m\_2)\}\}$
B6	$\alpha =\sqrt{1+\frac{2{ϵ}^{2}E{L}^2}{m(Z_1{Z}_2{q}^2{)}^2}}\backslash alpha\; =\; \backslash sqrt\{1+\backslash frac\{2\; \backslash epsilon^2\; E\; L^2\}\{m\; (Z\_1\; Z\_2\; q^2)^2\}\}$
B7	$H=\sqrt{\frac{8\pi G\rho }{3}-\frac{k_{\text{f}}{c}^2}{a_{\text{f}}^2}}H\; =\; \backslash sqrt\{\backslash frac\{8\; \backslash pi\; G\; \backslash rho\}\{3\}-\backslash frac\{k\_\backslash text\{f\}\; c^2\}\{a\_\backslash text\{f\}^2\}\}$
B9	$P=-\frac{32}{5}\frac{G^4}{c^5}\frac{(m_1{m}_2{)}^2(m_1+m_2)}{r^5}P\; =\; -\backslash frac\{32\}\{5\}\; \backslash frac\{G^4\}\{c^5\}\; \backslash frac\{(m\_1\; m\_2)^2\; (m\_1+m\_2)\}\{r^5\}$
B10	$\cos {\theta }_1=\frac{\cos {\theta }_2-v\mathrm{/}c}{(1-v\mathrm{/}c)\cos {\theta }_2}\backslash cos\backslash theta\_1\; =\; \backslash frac\{\backslash cos\backslash theta\_2-v/c\}\{(1-v/c)\; \backslash cos\backslash theta\_2\}$
B11	$I=I_0{\left(\frac{\sin (\alpha \mathrm{/}2)}{\alpha \mathrm{/}2}\frac{\sin (N\delta \mathrm{/}2)}{\sin (\delta \mathrm{/}2)}\right)}^{2}I\; =\; I\_0\; \backslash left(\backslash frac\{\backslash sin(\backslash alpha/2)\}\{\backslash alpha/2\}\; \backslash frac\{\backslash sin(N\; \backslash delta/2)\}\{\backslash sin(\backslash delta/2)\}\backslash right)^2$
B12	$F=\frac{q}{4\pi ϵy^2}(4\pi ϵV_{\text{e}}d-\frac{qd{y}^3}{(y^2-d^2{)}^2})F\; =\; \backslash frac\{q\}\{4\; \backslash pi\; \backslash epsilon\; y^2\}\; \backslash left(4\; \backslash pi\; \backslash epsilon\; V\_\backslash text\{e\}\; d\; -\; \backslash frac\{q\; d\; y^3\}\{(y^2-d^2)^2\}\backslash right)$
B13	$V_{\text{e}}=\frac{q}{4\pi ϵ\sqrt{r^2+d^2-2dr\cos \alpha }}V\_\backslash text\{e\}\; =\; \backslash frac\{q\}\{4\; \backslash pi\; \backslash epsilon\; \backslash sqrt\{r^2+d^2-2\; d\; r\; \backslash cos\backslash alpha\}\}$
B14	$V_{\text{e}}=E_{\text{f}}\cos \theta (\frac{\alpha -1}{\alpha +2}\frac{d^3}{r^2}-r)V\_\backslash text\{e\}\; =\; E\_\backslash text\{f\}\; \backslash cos\backslash theta\; \backslash left(\backslash frac\{\backslash alpha-1\}\{\backslash alpha+2\}\; \backslash frac\{d^3\}\{r^2\}-r\backslash right)$
B15	${\omega }_0=\frac{\sqrt{1-\frac{v^2}{c^2}}}{1+\frac{v}{c}\cos \theta }\omega \backslash omega\_0\; =\; \backslash frac\{\backslash sqrt\{1-\backslash frac\{v^2\}\{c^2\}\}\}\{1+\backslash frac\{v\}\{c\}\; \backslash cos\backslash theta\}\; \backslash omega$
B16	$E=q{V}_{\text{e}}+\sqrt{(p-qA{)}^2{c}^2+m^2{c}^4}E\; =\; q\; V\_\backslash text\{e\}\; +\; \backslash sqrt\{(p-q\; A)^2\; c^2+m^2\; c^4\}$
B17	$E=\frac{1}{2m}(p^2+m^2{\omega }^2{x}^2(1+\alpha \frac{x}{y}))E\; =\; \backslash frac\{1\}\{2\; m\}\; \backslash left(p^2+m^2\; \backslash omega^2\; x^2\; \backslash left(1+\backslash alpha\; \backslash frac\{x\}\{y\}\backslash right)\backslash right)$
B19	$p_{\text{f}}=-\frac{1}{8\pi G}(\frac{c^4{k}_{\text{f}}}{a_{\text{f}}^2}+c^2{H}^2(1-2\alpha ))p\_\backslash text\{f\}\; =\; -\backslash frac\{1\}\{8\; \backslash pi\; G\}\; \backslash left(\backslash frac\{c^4\; k\_\backslash text\{f\}\}\{a\_\backslash text\{f\}^2\}+c^2\; H^2\; \backslash left(1-2\; \backslash alpha\backslash right)\backslash right)$
B20	$A=\frac{{\alpha }^2{h}^2}{4\pi m^2{c}^2}{\left(\frac{{\omega }_0}{\omega }\right)}^2(\frac{{\omega }_0}{\omega }+\frac{\omega }{{\omega }_0}-{\sin }^2\theta )A\; =\; \backslash frac\{\backslash alpha^2\; h^2\}\{4\; \backslash pi\; m^2\; c^2\}\; \backslash left(\backslash frac\{\backslash omega\_0\}\{\backslash omega\}\backslash right)^2\; \backslash left(\backslash frac\{\backslash omega\_0\}\{\backslash omega\}+\backslash frac\{\backslash omega\}\{\backslash omega\_0\}-\backslash sin^2\backslash theta\backslash right)$

Supported Tasks and Leaderboards

Symbolic Regression

Dataset Structure

Data Instances

Tabular data + Ground-truth equation per equation

Tabular data: (num_samples, num_variables+1), where the last (rightmost) column indicate output of the target function for given variables. Note that the number of variables (num_variables) varies from equation to equation.

Ground-truth equation: pickled symbolic representation (equation with symbols in sympy) of the target function.

Data Fields

For each dataset, we have

1. train split (txt file, whitespace as a delimiter)
2. val split (txt file, whitespace as a delimiter)
3. test split (txt file, whitespace as a delimiter)
4. true equation (pickle file for sympy object)

Data Splits

• train: 8,000 samples per equation
• val: 1,000 samples per equation
• test: 1,000 samples per equation

Dataset Creation

Curation Rationale

We chose target equations based on the Feynman Symbolic Regression Database.

Annotations

Annotation process

We significantly revised the sampling range for each variable from the annotations in the Feynman Symbolic Regression Database. First, we checked the properties of each variable and treat physical constants (e.g., light speed, gravitational constant) as constants. Next, variable ranges were defined to correspond to each typical physics experiment to confirm the physical phenomenon for each equation. In cases where a specific experiment is difficult to be assumed, ranges were set within which the corresponding physical phenomenon can be seen. Generally, the ranges are set to be sampled on log scales within their orders as 10^2 in order to take both large and small changes in value as the order changes. Variables such as angles, for which a linear distribution is expected are set to be sampled uniformly. In addition, variables that take a specific sign were set to be sampled within that range.

Who are the annotators?

The main annotators are

• Naoya Chiba (@nchiba)
• Ryo Igarashi (@rigarash)

Personal and Sensitive Information

N/A

Considerations for Using the Data

Social Impact of Dataset

We annotated this dataset, assuming typical physical experiments. The dataset will engage research on symbolic regression for scientific discovery (SRSD) and help researchers discuss the potential of symbolic regression methods towards data-driven scientific discovery.

Discussion of Biases

Our choices of target equations are based on the Feynman Symbolic Regression Database, which are focused on a field of Physics.

Other Known Limitations

Some variables used in our datasets indicate some numbers (counts), which should be treated as integer. Due to the capacity of 32-bit integer, however, we treated some of such variables as float e.g., number of molecules (10^{23} - 10^{25})

Additional Information

Dataset Curators

The main curators are

• Naoya Chiba (@nchiba)
• Ryo Igarashi (@rigarash)

Licensing Information

MIT License

Citation Information

[Preprint]

@article{matsubara2022rethinking,
  title={Rethinking Symbolic Regression Datasets and Benchmarks for Scientific Discovery},
  author={Matsubara, Yoshitomo and Chiba, Naoya and Igarashi, Ryo and Tatsunori, Taniai and Ushiku, Yoshitaka},
  journal={arXiv preprint arXiv:2206.10540},
  year={2022}
}

Contributions

Authors:

• Yoshitomo Matsubara (@yoshitomo-matsubara)
• Naoya Chiba (@nchiba)
• Ryo Igarashi (@rigarash)
• Tatsunori Taniai
• Yoshitaka Ushiku (@yushiku)