Datasets:

Task Categories: symbolic-regression
Languages: en
Multilinguality: monolingual
Size Categories: 100K<n<1M
Licenses: mit
Language Creators: expert-generated
Annotations Creators: expert
Source Datasets: extended
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Dataset Card for SRSD-Feynman (Hard set)

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(θ22σ2)/2πσ2f = \exp\left(-\frac{\theta^2}{2\sigma^2}\right)/\sqrt{2\pi\sigma^2}
I.6.20a f=exp(θ22)/2πf = \exp\left(-\frac{\theta^2}{2}\right)/\sqrt{2\pi}
I.6.20b f=exp((θθ1)22σ2)/2πσf = \exp\left(-\frac{\left(\theta-\theta_1\right)^2}{2\sigma^2}\right)/\sqrt{2\pi\sigma}
I.9.18 F=Gm1m2(x2x1)2+(y2y1)2+(z2z1)2F = \frac{G m_1 m_2}{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}
I.15.3t t1=tux/c21u2/c2t_1 = \frac{t-u x/c^2}{\sqrt{1-u^2/c^2}}
I.15.3x x1=xut1u2/c2x_1 = \frac{x - u t}{\sqrt{1 - u^2/c^2}}
I.29.16 x=x12+x22+2x1x2cos(θ1θ2)x = \sqrt{x_1^2+x_2^2 + 2 x_1 x_2 \cos\left(\theta_1-\theta_2\right)}
I.30.3 I=I02(nθ/2)2(θ/2)I = I_0 \frac{\sin^2\left(n \theta/2\right)}{\sin^2\left(\theta/2\right)}
I.32.17 P=(12ϵcE2)(8πr23)(ω4(ω2ω02)2)P = \left(\frac{1}{2} \epsilon c E^2\right) \left(\frac{8 \pi r^2}{3}\right) \left(\frac{\omega^4}{\left(\omega^2-\omega_0^2\right)^2}\right)
I.34.14 ω=1+v/c1v2/c2ω0\omega = \frac{1+v/c}{\sqrt{1-v^2/c^2}} \omega_0
I.37.4 I12=I1+I2+2I1I2cosδI_{12} = I_1+I_2+2 \sqrt{I_1 I_2} \cos\delta
I.39.22 P=nkTVP = \frac{n k T}{V}
I.40.1 n=n0exp(mgx/kT)n = n_0 \exp\left(-m g x/ k T\right)
I.41.16 Lrad=h2πω3π2c2(exp(hω/2πkT)1)L_\text{rad} = \frac{h}{2 \pi} \frac{\omega^3}{\pi^2 c^2 (\exp(h \omega/2 \pi k T)-1)}
I.44.4 Q=nkTln(V2V1)Q = n k T \ln(\frac{V_2}{V_1})
I.50.26 x=K(cosωt+ϵ2ωt)x = K \left(\cos\omega t + \epsilon \cos^2 \omega t\right)
II.6.15a E=p4πϵ3zr5x2+y2E = \frac{p}{4 \pi \epsilon} \frac{3 z}{r^5} \sqrt{x^2+y^2}
II.6.15b E=p4πϵ3cosθsinθr3E = \frac{p}{4 \pi \epsilon} \frac{3 \cos\theta \sin\theta}{r^3}
II.11.17 n=n0(1+p0EcosθkT)n = n_0 \left(1 + \frac{p_0 E \cos\theta}{k T}\right)
II.11.20 P=n0p02E3kTP = \frac{n_0 p_0^2 E}{3 k T}
II.11.27 P=Nα1(nα/3)ϵEP = \frac{N \alpha}{1-(n \alpha/3)} \epsilon E
II.11.28 κ=1+Nα1(Nα/3)\kappa = 1 + \frac{N \alpha}{1-(N \alpha/3)}
II.13.23 ρ=ρ01v2/c2\rho = \frac{\rho_0}{\sqrt{1-v^2/c^2}}
II.13.34 j=ρ0v1v2/c2j = \frac{\rho_0 v}{\sqrt{1-v^2/c^2}}
II.24.17 k=ω2/c2π2/a2k = \sqrt{\omega^2 / c^2 - \pi^2/a^2}
II.35.18 a=Nexp(μB/kT)+exp(μB/kT)a = \frac{N}{\exp(\mu B/k T)+\exp(-\mu B/k T)}
II.35.21 M=NμtanhμBkTM = N \mu \tanh\frac{\mu B}{k T}
II.36.38 x=μHkT+μλϵc2kTMx = \frac{\mu H}{k T}+\frac{\mu \lambda}{\epsilon c^2 k T} M
III.4.33 E=hω2π(exp(hω/2πkT)1)E = \frac{h \omega}{2 \pi \left(\exp\left(h \omega/2 \pi k T\right) - 1\right)}
III.9.52 PIII=(2πμEth)22((ωω0)t/2)(ωω0)t/2)2P_{\text{I} \rightarrow \text{II}} = \left(\frac{2 \pi \mu E t}{h}\right)^2 \frac{\sin^2\left(\left(\omega-\omega_0\right) t/2\right)}{\left(\omega-\omega_0\right) t / 2)^2}
III.10.19 E=μBx2+By2+Bz2E = \mu \sqrt{B_x^2+B_y^2+B_z^2}
III.21.20 J=ρqmAJ = -\rho \frac{q}{m} A
B1 A=(Z1Z2αhc4E2(θ/2))2A = \left(\frac{Z_1 Z_2 \alpha h c}{4 E \sin^2\left(\theta/2\right)}\right)^2
B2 k=mkGL2(1+1+2EL2mkG2cos(θ1θ2))k = \frac{m k_G}{L^2} \left(1+\sqrt{1+\frac{2 E L^2}{m k_G^2}} \cos\left(\theta_1-\theta_2\right)\right)
B3 r=d(1α2)1+αcos(θ1θ2)r = \frac{d (1-\alpha^2)}{1+\alpha \cos(\theta_1-\theta_2)}
B4 v=2m(EUL22mr2)v = \sqrt{\frac{2}{m} \left(E-U-\frac{L^2}{2 m r^2}\right)}
B5 t=2πd3/2G(m1+m2)t = \frac{2 \pi d^{3/2}}{\sqrt{G(m_1+m_2)}}
B6 α=1+2ϵ2EL2m(Z1Z2q2)2\alpha = \sqrt{1+\frac{2 \epsilon^2 E L^2}{m (Z_1 Z_2 q^2)^2}}
B7 H=8πGρ3kfc2af2H = \sqrt{\frac{8 \pi G \rho}{3}-\frac{k_\text{f} c^2}{a_\text{f}^2}}
B9 P=325G4c5(m1m2)2(m1+m2)r5P = -\frac{32}{5} \frac{G^4}{c^5} \frac{(m_1 m_2)^2 (m_1+m_2)}{r^5}
B10 cosθ1=cosθ2v/c(1v/c)cosθ2\cos\theta_1 = \frac{\cos\theta_2-v/c}{(1-v/c) \cos\theta_2}
B11 I=I0(sin(α/2)α/2sin(Nδ/2)sin(δ/2))2I = I_0 \left(\frac{\sin(\alpha/2)}{\alpha/2} \frac{\sin(N \delta/2)}{\sin(\delta/2)}\right)^2
B12 F=q4πϵy2(4πϵVedqdy3(y2d2)2)F = \frac{q}{4 \pi \epsilon y^2} \left(4 \pi \epsilon V_\text{e} d - \frac{q d y^3}{(y^2-d^2)^2}\right)
B13 Ve=q4πϵr2+d22drcosαV_\text{e} = \frac{q}{4 \pi \epsilon \sqrt{r^2+d^2-2 d r \cos\alpha}}
B14 Ve=Efcosθ(α1α+2d3r2r)V_\text{e} = E_\text{f} \cos\theta \left(\frac{\alpha-1}{\alpha+2} \frac{d^3}{r^2}-r\right)
B15 ω0=1v2c21+vccosθω\omega_0 = \frac{\sqrt{1-\frac{v^2}{c^2}}}{1+\frac{v}{c} \cos\theta} \omega
B16 E=qVe+(pqA)2c2+m2c4E = q V_\text{e} + \sqrt{(p-q A)^2 c^2+m^2 c^4}
B17 E=12m(p2+m2ω2x2(1+αxy))E = \frac{1}{2 m} \left(p^2+m^2 \omega^2 x^2 \left(1+\alpha \frac{x}{y}\right)\right)
B19 pf=18πG(c4kfaf2+c2H2(12α))p_\text{f} = -\frac{1}{8 \pi G} \left(\frac{c^4 k_\text{f}}{a_\text{f}^2}+c^2 H^2 \left(1-2 \alpha\right)\right)
B20 A=α2h24πm2c2(ω0ω)2(ω0ω+ωω02θ)A = \frac{\alpha^2 h^2}{4 \pi m^2 c^2} \left(\frac{\omega_0}{\omega}\right)^2 \left(\frac{\omega_0}{\omega}+\frac{\omega}{\omega_0}-\sin^2\theta\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)
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