SRSD: Rethinking Datasets of Symbolic Regression for Scientific Discovery
Abstract
Symbolic Regression (SR) is a task of recovering mathematical expressions from given data and has been attracting attention from the research community to discuss its potential for scientific discovery.
However, the community lacks datasets of symbolic regression for scientific discovery (SRSD) to discuss the potential of SR.
To address the critical issue, we revisit datasets of SRSD to discuss the potential of symbolic regression for scientific discovery.
Focused on a set of formulas used in the existing datasets based on Feynman Lectures on Physics, we recreate 120 datasets to discuss the performance of SRSD.
For each of the SRSD datasets, we carefully review the properties of the formula and its variables to design reasonably realistic sampling ranges of values so that our new SRSD datasets can be used for evaluating the potential of SRSD such as whether or not an SR method can (re)discover physical laws from such datasets.
We conduct experiments on our new SRSD datasets using five state-of-the-art SR methods in SRBench, and the results show that the new SRSD datasets are more challenging than the original ones.
Our datasets 123 and code repository 4 are publicly available.
Introduction
Recent advances in machine learning (ML), especially deep learning (DL), have led to the proposal of many methods that can reproduce the given data and make appropriate inferences on new inputs.
Such methods are, however, often black-box, which makes it difficult for humans to understand how they made predictions for given inputs.
This property will be more critical especially when non-ML experts apply ML to problems in their research domains such as physics and chemistry.
Symbolic regression (SR) is the task of producing a mathematical expression (symbolic expression) that fits a given dataset.
SR has been studied in the genetic programming (GP) community [1–6] and
DL-based SR has been attracting more attention from the ML/DL community [7–12].
Because of its interpretability, various scientific communities apply SR to advance research in their scientific fields e.g. , Physics [13–19], Applied Mechanics [20], Climatology [21], Materials [22–25], and
Chemistry [26].
Given that SR has been studied in various communities, La Cava et al.
[11] propose SRBench, a unified benchmark framework for symbolic regression methods.
In the benchmark study, they combine the Feynman Symbolic Regression Database (FSRD) [14] and the ODE-Strogatz repository [27] to compare a number of SR methods, using a large-scale heterogeneous computing cluster.
To discuss the potential of symbolic regression for scientific discovery (SRSD), there still remain some issues to be addressed: oversimplified datasets and lack of evaluation metric towards SRSD.
For symbolic regression tasks, existing datasets consist of values sampled from limited domains such as in range of 1 to 5, and there are no large-scale datasets with reasonably realistic values that capture the properties of the formula and its variables.
Thus, it is difficult to discuss the potential of symbolic regression for scientific discovery with such existing datasets.
For instance, the FSRD consists offormulas selected mostly from Feynman Lectures Series 6 [32–34] and are core benchmark datasets used in SRBench [11].
While the formulas indicate physical laws, variables and constants used in each dataset have no physical meanings and sampling processes are oversimplified since the datasets in the benchmark study are not designed to discover the physical laws from the observed data in the real world.
(See Section 3.1.)
To address these issues, we propose new SRSD datasets and conduct benchmark experiments using representative SR methods.
We carefully review and design annotation policies for the new datasets, considering the properties of the physics formulas.
Using the proposed SRSD datasets, we perform benchmark experiments with a set of symbolic regression baselines and find that even state of the art symbolic regression methods still need improvements to be used for scientific discovery.
Related Studies
In this section, we briefly introduce related studies focused on 1) symbolic regression for scientific discovery and 2) symbolic regression dataset and evaluation.
2.1 SRSD: Symbolic Regression for Scientific Discovery
A pioneer study on symbolic regression for scientific discovery is conducted by Schmidt and Lipson [35], who propose a data-driven scientific discovery method.
They collect data from standard experimental systems like those used in undergrad physics education: an air-track oscillator and a double pendulum.
Their proposed algorithm detects different types of laws from the data such as position manifolds, energy laws, and equations of motion and sum of forces laws.
Following the study, data-driven scientific discovery has been attracting attention from research communities and been applied to various domains such as Physics [13–19], Applied Mechanics [20],
Climatology [21], Materials [22–25], and Chemistry [26].
These studies leverage symbolic regression in different fields.
While general symbolic regression tasks use synthetic datasets with limited sampling domains for benchmarks, many of the SRSD studies collect data from the real world and discuss how we could leverage symbolic regression toward scientific discovery.
While SRSD tasks share the same input-output interface with general symbolic regression (SR) tasks ( i.e. , input: dataset, output: symbolic expression), we differentiate SRSD tasks in this study from general SR tasks by whether or not the datasets including true symbolic expressions are created with reasonably realistic assumptions for scientific discovery such as meaning of true symbolic expressions (whether or not they have physical meanings) and sampling domains for input variables.
2.2 Dataset and Evaluation
For symbolic regression methods, there exist several benchmark datasets and empirical studies.
The
Feynman Symbolic Regression Database [14] is one of the largest symbolic regression datasets, which consists of 100 physics-inspired equations based on Feynman Lectures on Physics [32–34].
By randomly sampling from small ranges of value, they generate the corresponding tabular datasets for the 100 equations.
Inspired by [1, 2, 4], Uy et al.
[5] suggest 10 different real-valued symbolic regression problems (functions) and create the corresponding dataset ( a.k.a.
Nguyen dataset).
The suggested functions consist of either 1 or 2 variables e.g. , f ( x ) = x 6 + x 5 + x 4 + x 3 + x 2 + x and f ( x, y )
=
sin( x ) + sin( y 2 ) .
They generate each dataset by randomly sampling 20 - 100 data points.
La Cava et al.
[11] design a symbolic regression benchmark, named SRBench, and conduct a comprehensive benchmark experiment, using existing symbolic regression datasets such as the
Feynman Symbolic Regression Database [14] and ODE-Strogatz repository [36].
In SRBench, symbolic regression methods are assessed by 1) an error metric based on squared error between target and estimated values, and 2) solution rate that shows a percentage of the estimated symbolic regression models that match the true models (equations).
However, these datasets and evaluations are not necessarily designed to discuss symbolic regression for scientific discovery.
In Sections 3.1 and 4.1, we will further describe potential issues in such existing studies.
Datasets
In this section, we summarize issues we found in the existing symbolic regression datasets, and then propose new datasets to address them towards symbolic regression for scientific discovery (SRSD).
3.1 Issues in Existing Datasets
As introduced in Section 2.2, there are many symbolic regression datasets.
However, we consider that novel datasets are required to discuss SRSD for the following reasons:
No physical meaning: Many of the existing symbolic regression datasets [1, 2, 4, 5] are not necessarily physics-inspired, but instead randomly generated e.g. , f ( x ) = log( x ) , f ( x, y ) =
xy +sin(( x − 1)( y − 1)) .
To discuss the potential of symbolic regression for scientific discovery, we need to further elaborate datasets, considering how we would leverage symbolic regression in practice.
Oversimplified sampling process: While some of the datasets are physics-inspired such as the Feynman Symbolic Regression Database (FSRD) [14] and ODE-Strogatz repository [
36], their sampling strategies are very simplified.
Specifically, the strategies do not distinguish between constants and variables e.g. , speed of light 7 is treated as a variable and randomly sampled in range of 1 to 5.
Besides, most of the sampling domains are far from values we could observe in the real world e.g , II.4.23 in Table S1 (the vacuum permittivity values are sampled from range of 1 to 5).
When sampled ranges of the distributions are narrow, we cannot distinguish Lorentz transformation from Galilean transformation e.g. I.15.10 and I.16.6 in
Table S3, I.48.2 in Table S5, I.15.3t, I.15.3x, and I.34.14 in Table S7, or the black body radiation can be misestimated to Stephan-Boltzmann law or the Wien displacement law e.g. I.41.16 in
Table S8.
Duplicate equations: Due to the two issues above, many of the equations in existing datasets turn out to be duplicate.
e.g. , as shown in Table 1, F = µN n (I.12.1) and F = q 2 E (I.12.5) in the original Feynman Symbolic Regression Database are considered identical since both the equations are multiplicative and consists of two variables, and their sampling domains (Distributions in Table 1) are exactly the same.
For instance, approximately 25% of the symbolic regression problems in the original FSRD have 1 - 5 duplicates in that regard.
Incorrect/Inappropriate formulas: The Feynman Symbolic Regression Database [14] treat every variables as float whereas they should be integer to be physically meaningful.
For example, the number of phase difference in Bragg’s law should be integer but sampled as real number (I.30.5 in Table S1).
Furthermore, they don’t even give special treatment of angle variables (I.18.12, I.18.16, and I.26.2 in Table 1).
Physically some variables can be negative whereas the original Feynman Symbolic Regression Database [14] only samples positive values ( e.g. I.8.
and I.11.19 in Table S3).
We also avoid using arcsin / arccos in the equations since the use of arcsin / arccos in the Feynman Symbolic Regression Database [14] just to obtain angle variable is not experimentally meaningful (I.26.2 in Table 1, I.30.5 in Table S1, and B10 in Table S11).
Equations using arcsin and arccos in the original annotation are I.26.2 (Snell’s law), I.30. (Bragg’s law), and B10 (Relativistic aberration).
These are all describing physical phenomena related to two angles, and it is an unnatural deformation to describe only one of them with an inverse function.
Additionally, inverse function use implicitly limits the range of angles, but there is no such limitation in the actual physical phenomena.
3.2 Proposed SRSD Datasets
We address the issues in existing datasets above by proposing new SRSD datasets based on the equations used in the FSRD [14].
i.e. , Section 3.1 summarizes the differences between the FSRD and our SRSD datasets.
Our annotation policy is carefully designed to simulate typical physics experiments so that the SRSD datasets can engage studies on symbolic regression for scientific discovery in the research community.
3.2.1 Annotation policy
We thoroughly revised the sampling range for each variable from the annotations in the FSRD [14].
First, we reviewed the properties of each variable and treated physical constants ( e.g. , light speed, gravitational constant) as constants while such constants are treated as variables in the original FSRD datasets.
Next, variable ranges were defined to correspond to each typical physics experiment to confirm the physical phenomenon for each equation.
We also used [37] as a reference.
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.
Tablesand S1 – S11 show the detailed comparisons between the original FSRD and our proposed SRSD datasets.
3.2.2 Complexity-aware Dataset Categories
While the proposed datasets consist of 120 different problems, there will be non-trivial training cost required to train a symbolic regression model for all the problems individually [11] i.e. , there will be 120 separate training sessions to assess the symbolic regression approach.
To allow more flexibility in assessing symbolic regression models for scientific discovery, we define three clusters of the proposed datasets based on their complexity: Easy , Medium , and Hard sets, which consist of 30, 40, anddifferent problems respectively.
We define the complexity of problem, using the number of operations to represent the true equation tree and range of the sampling domains.
The former measures how many mathematical operations compose the true equation such as add , mul , pow , exp , and log operations.
The latter considers magnitude of sampling distributions ( Distributions column in Tables 1 and S1 – S11) and increases the complexity when sampling values from wide range of distributions.
We define the domain range as follows:
[Equation]
where S indicates a set of sampling domains ( distributions ) for a given symbolic regression problem.
As we will show in Section 5.3, these clusters represent problem difficulties at high level.
For instance, these subsets will help the research community to shortly tune and/or perform sanity-check new approaches on the Easy set (30 problems) instead of using the whole datasets (120 problems).
Figure 1 shows the three different distribution maps of our proposed datasets.
Easy , Medium , and
Hard sets consist of 30, 40, and 50 individual symbolic regression problems, respectively.  Benchmark
Besides the conventional metrics, we propose a new metric to discuss the performance of symbolic regression for scientific discovery in Section 4.1.
Following the set of metrics, we design an evaluation framework of symbolic regression for scientific discovery.
[Table]
[Table] [Table]
I.14.
U =
k spring x
[Table]
U
Elastic energy
V, F
V,
[Table]
, P
[Table] [Table]
[Table]
k spring Spring constant
V,
[Table] [Table]
[Table]
x
Position
[Table] [Table]
[Table]
I.18.12 τ = rF sin θ τ
Torque
[Table] [Table]
Distance
[Table]
Force
[Table]
Angle
[Table]
I.18.
[Table]
= mrv sin θ
L
Angular momentum
[Table]
Mass
[Table]
Distance
[Table]
Velocity
[Table]
Angle
[Table]
I.25.
V = qC
V
Voltage
[Table]
q
Electric charge
[Table]
C
Electrostatic Capacitance
[Table]
I.26.
[Table]
n
Relative refractive index
V,
[Table]
V,
[Table]
4.1 Metrics
In general, it would be difficult to define “accuracy” of symbolic regression models since we will compare its estimated equation to the ground truth equation and need criteria to determine whether or not it is “correct”.
La Cava et al.
[11] suggested a reasonable definition of symbolic solution, which is designed to capture symbolic regression models that differ from the true model by a constant or scalar.
Using R 2 score (Eq. (2)), they also defined as accuracy the percentage of symbolic regression problems that a model meets R 2 > τ , where τ is a threshold e.g. , τ = 0 .
999 in [11].
[Equation]
where N indicates the number of test samples ( i.e. , the number of rows in the test dataset), and ¯ y is a mean of target outputs produced by f true .
f pred and f true are a trained SR model and a true model, respectively.
4.2 Model Selection
For real datasets (assuming observed datasets), only tabular data are available for training and validation.
(In practice, a test dataset does not include the true equation).
For benchmark purposes, true equations are provided as test data besides test tabular data.
For each problem, we use the validation tabular dataset and choose the best trained SR model f ∗ pred from F , a set of the trained models by a given method respect to Eq.
[Equation]
where X i indicates the i -th row of the validation tabular dataset X .
We use the geometrical distance between predicted values against a validation tabular dataset to choose the best model obtained through hyperparameter tuning.
Using the best model per method, we compute R 2 score to assess the method.
Experiments 5.1 Baseline Methods
For baselines, we use the five best symbolic regression methods in SRBench [11].
Specifically, we choose gplearn [3], AFP [38], AFP-FE [35], AI Feynman [15], and DSR [7], referring to the rankings of solution rate for the FSRD datasets in their study.
1. gplearn [3]: a genetic programming based symbolic regression method published as a Python package gplearn .
AFP [38]: Age-fitness pareto optimization.
AFP-FE [35]: AFP optimization with fitness estimates.
AI Feynman [15]: an iterative approach to generate symbolic regression to seek to fit data to formulas that are Pareto-optimal.
DSR
For the details of the baseline models, we refer readers to the corresponding papers [3, 7, 15, 35, 38].
5.2 Runtime Constraints
The implementations of the baseline methods in Section 5.1 do not use any GPUs.
We run 600 high performance computing (HPC) jobs in total, using “C.small” and “C.large” computing nodes, which [Table] [Table] [Table]
have 5 - 20 assigned physical CPU cores, 30 - 120 GB RAM
[Table] and 720 GB local storage available in AI Bridging Cloud Infrastructure (ABCI). 8 Due to the properties of our HPC resource, we have some runtime constraints:
Since each HPC job is designed to run for up to 24 hours due to the limited resource, we run a job with a pair of a target tabular dataset and a symbolic regression method.
Given a pair of a dataset and a method, each of our HPC jobs runs up to 100 separate training sessions with different hyperparameter values.
5.3 Results
In this section, we discuss the experimental results of our baseline methods, using the proposed SRSD datasets.
Tables 2 and 3 show the performance of the symbolic regression baseline methods in terms of R {+
+} -driven accuracy
( R 2 > 0 . 999 ) and solution rate respectively, and both the metrics are used in SRBench [11].
According to the metrics, DSR significantly outperforms all the other baselines we considered.
The DSR results also indicate difficulty levels of the three categories of our SRSD datasets, which looks aligned with our complexity-aware dataset categorization (Section 3.2.2).
Table 4 compares the solution rates of the five common baselines for the FSRD and our SRSD datasets.
We can confirm that the overall solution rates for our SRSD are significantly degraded compared to those for the FSRD reported in SRBench [11].
The results indicate that our SRSD datasets are more challenging than the FSRD in terms of solution rate.
Limitations and Discussion 6.1 Implicit Functions
Symbolic regression generally has a limitation in inferring implicit functions, as the model infers a trivial constant function if there are no restrictions on variables.
For example, f ( x, y ) = 0 is [-inferredas 
-]inferred as 0 = 0 ∀ x, y .
This problem can be solved by applying the constraint that an inferred function should depend on at least two variables e.g. , inferring f ( x, y )
= 0 with ∂f∂x ̸
= 0 and ∂f∂y ̸ = 0 , or by converting the function to an explicit form e.g. , y = g ( x ) .
We converted some functions in the datasets into explicit forms and avoided the inverse trigonometric functions as described in Section 3.1.
6.2 Dummy Variables and Noise Injection
When applying machine learning to real-world problems, it is often true that 1) not all the observed features (variables in symbolic regression) are necessary to solve the problems, and 2) the observed values contain some noise.
While we follow [11] and show experimental results for our SRSD datasets with noise-injected target variables in the supplementary material, these aspects are not thoroughly discussed in this study, such discussions can be a separate paper built on this work and further engage studies of symbolic regression for scientific discovery.
6.3 Interpretability Evaluation
Though symbolic regression methods are popular for intrpretability in their behaviors/outputs, there is a lack of appropriate metrics to evaluate these methods, taking into account the property.
One of the most common approaches would be to measure the prediction error or correlation between the predicted values and the target values in the test data, as in standard regression problems.
However, low prediction errors could be achieved even by complex models that differ from the original law.
Some studies [11, 15] use complexity of the predicted expression as an evaluation metric (the simpler the better).
However, it is based on a big assumption that a simpler expression may be more likely to be a hidden law in the data (scientific discovery such as physics law), which may not be true for SRSD.
SRBench [11] present the percentage of agreement between the target and the estimated equations, using solution rate they defined.
But in such cases, both 1) equations that do not match at all and 2) that differ by only one term 9 are equally treated as incorrect.
As a result, it is considered as a coarse-resolution evaluation method for accuracy in SRSD, which still needs more discussion towards real-world SRSD applications.
Conclusion
In this work, we pointed out issues of existing datasets and benchmarks of symbolic regression for scientific discovery (SRSD).
To address the issues, we proposed 120 new SRSD datasets based on a set of physics formulas in FSRD [14] and conducted benchmark experiments using the proposed
SRSD datasets.
The results show the new SRSD datasets are significantly more challenging than the original FSRD datasets in terms of solution rate.
We also discussed the limitations in this study and pointed out lack of evaluation metrics suitable for SRSD.
Matsubara et al.
[39] further discuss the issue and propose a new evaluation metric for SRSD, performing benchmark experiments with an additional baseline.
To encourage the studies of SRSD, we publish our datasets 1, 2, 3 and code repository 4 with MIT License.
Acknowledgments
We thank the anonymous reviewers for their comments.
In this study, we used the computing resource of AI Bridging Cloud Infrastructure (ABCI) provided by National Institute of Advanced Industrial
Science and Technology (AIST).
This work is supported by JST-Mirai Program Grant Number
JPMJMI21G2, Japan. A Our SRSD Datasets: Additional Information
This section provides additional information regarding our SRSD datasets.
We created the datasets to discuss the performance of symbolic regression for scientific discovery (SRSD).
We refer readers to
Section 3 for details of the datasets.
Tables S1 – S11 comprehensively summarize the differences between FSRD and our SRSD datasets.
Note that the table of Easy set (part 1) is provided as Tablein Section 3.1.
As described in Section 3.2.2, we categorized each of the 120 SRSD datasets into one of Easy, Medium [Table] and Hard sets.
We published the three sets of the SRSD datasets with MIT License at Hugging Face Dataset repositories.
The dataset documentations are publicly available as Hugging Face Dataset cards.
1, 2, 3 These repositories are version-controlled with Git 10 so that users can track the log of the changes.
We bear all responsibility in case of violation of rights.
[Table]
V, F
U (1 , 5)
U log (10 −
[Table]
) d
Distance
V, F
V, F, P U (1 ,
[Table]
U log (
[Table]
[Table]
I.47.
c = (cid:113)
γPρ c
Velocity of sound
V, F
V, F, P
N/A
N/A γ
Heat capacity ratio
V, F
V, F, P U (1 , 5)
U
[Table]
[Table]
) P
Atmospheric pressure
V, F
V, F,
[Table]
U
[Table]
ρ
Density of air
[Table]
[Table]
II [Table] 42 J = κ ( T 2 − T 1 ) Ad
J Energy difference
[Table] κ Thermal conductivity [Table]
2 Temperature
[Table]
1 Temperature
[Table]
A
Area
[Table] d
Length
[Table]
II.
[Table]
h = W 4 πrh
Heat flux
[Table]
W Work
[Table]
r
Distance
[Table]
ϕ =
q 4 πϵr ϕ
Electric potential
V, F V, F
N/A
N/A q
Electric charge
V, F V, F
U (
[Table] U log ( [Table] [Table]
ϵ
Vacuum permittivity
V, F
C, F, P U ( [Table] [Table] r
Distance
V, F
V, F, P U (1 ,
[Table] U log ( [Table]
[Table]
u = ϵE 2u
Energy
V, F V, F
N/A
N/A ϵ
[Table]
V, F
C, F, P U ( [Table]
[Table]
, F
[Table]
=
[Table]
[Table] [Table]
N/A
N/A
[Table]
, F
[Table]
, F
[Table]
U log
[Table]
= [Table] 4 πϵc [Table]
Ir
B
[Table]
/A
[Table]
/A
[Table]
[Table]
P U
[Table]
[Table]
I
[Table]
V
[Table]
F
[Table]
F
U
[Table]
U log
[Table]
U log
[Table]
=
[Table]
µB
[Table]
Energy from
[Table]
V, F
V
[Table]
10 https://git-scm.com/
[Table]
, F
V, F, P U (1 , 5)
U log (10 − 30 , 10 − 28 )
II.34.29b U =
2 πgµB Jzh
U Energy
V, F
V, F, P
N/A
N/A g g-factor
V, F V, F
U (1 , 5)
U ( − 1 , 1) µ Bohr magneton
V, F
C, F, P U (1 , 5) 9 . 2740100783 × 10 − 24 B Magnetic field strength
V, F V, F
U (1 , 5)
U log (10 − 3 , 10 − 1 )
J z Element of angular momentum
V, F V, F
U (1 , 5)
U log (10 − 26 , 10 − 22 )
h
Planck constant
V, F
C, F, P U (1 , 5) 6 .
626 × 10 −
II.38.
F = Y
A ∆ ll
F Force
V, F V, F
N/A
N/A Y Young’s modulus
V, F
V, F, P U (1 , 5)
U log (10 − 1 , 10 1 )
A Area
V, F
V, F, P U (1 , 5)
U log (10 − 4 , 10 − 2 ) δl Displacement
V, F V, F
U (1 , 5)
U log (10 − 3 , 10 − 1 ) l
Length
V, F
V, F, P U (1 , 5)
U log (10 − 2 , 10 0 )
II.38.
µ =
Y 2(1+ σ ) µ Rigidity modulus
V, F
V, F, P
N/A
N/A Y Young’s modulus
V, F
V, F, P U (1 , 5)
U log (10 − 1 , 10 1 ) σ Poisson coefficient
V, F
V, F, P U (1 , 5)
U log (10 − 2 , 10 0 )
III.7.
ω = 4 πµBh ω Precession frequency
V, F V, F
N/A
N/A µ Magnetic moment
V, F V, F
U (1 , 5)
U log (10 − 11 , 10 − 9 )
B Magnetic flux density
V, F V, F
U (1 , 5)
U log (10 − 3 , 10 − 1 ) h
Planck constant
V, F
C, F, P U (1 , 5) 6 .
626 × 10 −
III.12.
J = mh 2 π
J
Variable
V, F V, F
N/A
N/A m Spin state
V, F
V, I,NN U (1 , 5)
U log (10 0 , 10 2 ) h
Planck constant
V, F
C, F, P U (1 , 5) 6 .
626 × 10 −
III.15.
k
=
πNb s k
Wavenumber
V, F V, F
N/A N/A s
Parameter of state
V, F
V, I
U (1 , 5)
U log (10 0 , 10 2 ) N Number of atoms
V, F
V, I,P U (1 , 5)
U log (10 0 , 10 2 ) b
Lattice constant
V, F
V, F, P U (1 , 5)
U log (10 − 10 , 10 − 8 )
[Table]
vector
V, F V, F
U (1 , 5)
U log (10 − 1 , 10 1 )
y 2 Element of a vector
V, F V, F
U (1 , 5)
U log (10 − 1 , 10 1 ) x 3 Element of a vector
V, F V, F
U (1 , 5)
U log (10 − 1 , 10 1 )
y 3 Element of a vector
V, F V, F
U (1 , 5)
U log (10 − 1 , 10 1 )
I.12.
F = q 1 q 2 4 πϵr
F
Electrostatic force
V, F V, F
N/A
N/A q 1 Electric charge
V, F
V, F
U (1 , 5) U log (10 − 3 , 10 − 1 ) q 2 Electric charge
V, F
V, F
U (1 , 5) U log (10 − 3 , 10 − 1 ) r
Distance
V, F
V, F, P
U (1 , 5)
U log (10 − 2 , 10 0 )
ϵ
Vacuum permittivity
V, F
C, F, P
U (1 , 5) 8 . 854 × 10 −
I.12.
F = q ( E + Bv sin( θ ))
F
Force
V, F V, F
N/A
N/A q
Electric charge
V, F V, F
U (1 , 5)
U log (10 − 1 , 10 1 )
E
Electric field
V, F V, F
U (1 , 5)
U log (10 − 1 , 10 1 )
B
Magnetic field strength
V, F
V, F, P
U (1 , 5)
U log (10 − 1 , 10 1 ) v
Velocity
V, F
V, F, P
U (1 , 5)
U log (10 − 1 , 10 1 ) θ
Angle
V, F
V, F, NN U (1 , 5)
U (0 , 2 π )
I.13.
K = 12 m ( v 2 + u 2 + w 2 )
K
Kinetic energy
V, F
V, F, P
N/A
N/A m Mass
V, F
V, F, P
U (1 , 5)
U log (10 − 2 , 10 0 ) v
Element of velocity
V, F
V, F, P
U (1 , 5)
U log (10 − 1 , 10 1 ) u
Element of velocity
V, F
V, F, P
U (1 , 5)
U log (10 − 1 , 10 1 ) w
Element of velocity
V, F
V, F, P
U (1 , 5)
U log (10 − 1 , 10 1 )
I.13.
U =
Gm 1 m 2 (cid:16) 1 r 2 − 1 r 1 (cid:17)
U
Potential energy
V, F
V, F, P
N/A
N/A G
Gravitational constant
V, F
C, F, P
U (1 , 5) 6 .
674 × 10 −m 1 Mass (The Earth)
V, F
V, F, P
U (1 , 5)
U log (10 − 2 , 10 0 ) m 2 Mass
V, F
V, F, P
U (1 , 5)
U log (10 − 2 , 10 0 ) r 2 Distance
V, F
V, F, P
U (1 , 5)
U log (10 − 2 , 10 0 ) r 1 Distance
V, F
V, F, P
U (1 , 5)
U log (10 − 2 , 10 0 )
I.15.
p
= m 0 v √ 1 − v 2 /cp
Relativistic mass
V, F
V, F, P
N/A
N/A m 0 Rest Mass
V, F
V, F, P
U (1 , 5)
U log (10 − 2 , 10 0 ) v
Velocity
V, F V, F
U (1 , 2)
U log (10 5 , 10 7 ) c
Speed of light
V, F
C, F, P U (3 , 10) 2 .
998 ×
v 1 = u + v 1+ uv/cv 1 Velocity
V, F V, F
N/A
N/A u
Velocity
V, F V, F
U (1 , 5)
U log (10 6 , 10 8 ) v
Velocity
V, F V, F
U (1 , 5)
U log (10 6 , 10 8 ) c
Speed of light
V, F
C, F, P
U (1 , 5) 2 . 998 ×
I.18.
r = m 1 r 1+ m 2 r 2 m 1+ mr
Center of gravity
V, F V, F
N/A
N/A m 1 Mass
V, F
V, F, P
U (1 , 5)
U log (10 − 1 , 10 1 ) r 1 Position
V, F V, F
U (1 , 5)
U log (10 − 1 , 10 1 ) m 2 Mass
V, F
V, F, P
U (1 , 5)
U log (10 − 1 , 10 1 ) r 2 Position
V, F V, F
U (1 , 5)
U log (10 − 1 , 10 1 )
[Table]
v
Velocity
[Table]
[Table] [Table]
v/
[Table]
W =
[Table]
W
[Table]
U = 32 PV
U Internal energy
[Table]
, F
V, F, P
[Table]
[Table] [Table]
PVγ
[Table]
[Table] [Table]
U
[Table]
V, F V, F
N/A
N/A Q
[Table]
charge
V, F V, F
U
[Table] [Table]
) U log
[Table]
ϵ
[Table]
permittivity
V, F
C, F, P
U
[Table]
a
Radius
[Table]
[Table] [Table]
, P
[Table]
x = qE m ( ω
[Table]
2) x
Position
[Table]
F
[Table]
, F
N/A
[Table] [Table]
q
[Table]
field
[Table]
=
[Table] [Table]
= qvrµ
Magnetic moment
V, F V, F
N/A
N/A q
Electric charge
V, F V, F
U (1 , 5) U log (10 − 11 , 10 − 9 ) v
Velocity
V, F V, F
U (1 , 5)
U log (10 5 , 10 7 ) r
Radius
V, F
V, F, P
U (1 , 5) U log (10 − 11 , 10 − 9 )
II.34.2a
I = qv
2 πr
I
Electric Current
V, F V, F
N/A
N/A q
Electric charge
V, F V, F
U (1 , 5) U log (10 − 11 , 10 − 9 ) v
Velocity
V, F V, F
U (1 , 5)
U log (10 5 , 10 7 ) r
Radius
V, F
V, F, P
U (1 , 5) U log (10 − 11 , 10 − 9 )
II.34.29a µ = qh 4 πm µ
Bohr magneton
V, F V, F
N/A
N/A q
Electric charge
V, F V, F
U (1 , 5) U log (10 − 11 , 10 − 9 ) h
Planck constant
V, F
C, F, P
U (1 , 5) 6 .
626 × 10 −m Mass
V, F
V, F, P
U (1 , 5) U log (10 − 30 , 10 − 28 )
II
E = µ (1 + χ ) B
E Energy of magnetic field
V, F V, F
N/A
N/A µ
Magnetic moment
V, F
V, F
U (1 , 5) U log (10 − 25 , 10 − 23 ) χ Volume magnetic susceptibility
V, F V, F
U (1 , 5)
U log (10 4 , 10 6 ) B Magnetic field strength
V, F V, F
U (1 , 5)
U log (10 − 3 , 10 − 1 )
[Table]
I =
I 0 (exp( q ∆ V/κT ) − 1)
I
Electric Current
V, F V, F
N/A N/A
I
Electric current
V, F
V, F
U (1 , 5) U log (10 − 3 , 10 − 1 ) q
Electric charge
V, F
V, F, P U (1 , 2)
U log (10 − 22 , 10 − 20 ) ∆ V Voltage
V, F V, F
U (1 , 2)
U log (10 − 1 , 10 1 ) κ
Boltzmann constant
V, F
C, F, P U (1 , 2) 1 .
381 × 10 −
T
Temperature
V, F
V, F, P U (1 , 2)
U log (10 1 , 10 3 )
III.15.
E = 2 A (1 − cos( kd ))
E
Energy
V, F
V, F, P
N/A
N/A A
Amplitude
V, F
V, F, P U (1 , 5)
U log (10 − 18 , 10 − 16 ) k
Propagation coefficient
V, F
V, F, P U (1 , 5)
U log (10 − 1 , 10 1 ) d
Lattice constant
V, F
V, F, P U (1 , 5)
U log (10 − 10 , 10 − 8 )
III.15.
m = h 2 8 π 2 Abm
Effective mass
V, F
V, F, P
N/A
N/A h
Planck constant
V, F
C, F, P U (1 , 5) 6 .
626 × 10 −
A
Amplitude
V, F
V, F, P U (1 , 5)
U log (10 − 18 , 10 − 16 )
b
Lattice constant
V, F
V, F, P U (1 , 5)
U log (10 − 10 , 10 − 8 )
III.17.
f = β (1 + α cos θ ) f
Distribution
V, F V, F
N/A
N/A β
Variable
V, F
V, F, P U (1 , 5)
U log (10 − 18 , 10 − 16 )
α
Variable
V, F V, F
U (1 , 5) U log (10 − 18 , 10 − 16 )
θ
Angle
V, F
V, F, NN U (1 , 5)
U (0 , 2 π )
III.19.51 E = − mq 4 2(4 πϵ )2( h/ (2 π ))2 n
E
Energy
V, F
V, F, P
N/A
N/A m
Mass
V, F
V, F, P U (1 , 5)
U log (10 − 30 , 10 − 28 ) q
Electric charge
V, F V, F
U (1 , 5) U log (10 − 11 , 10 − 9 )
ϵ
Vacuum permittivity
V, F
C, F, P U (1 , 5) 8 .
854 × 10 −h
Planck constant
V, F
C, F, P U (1 , 5) 6 .
626 × 10 −n
Number of protons
V, F
V, I,P
U (1 , 5)
U log (10 0 , 10 2 )
B
U =
E
+ Emc 2 (1 − cos θ )
U
Variable
V, F
V, F, P
N/A
N/A E
Electromagnetic energy
V, F
V, F, P U (1 , 3) U log (10 − 24 , 10 − 22 ) m
Electron mass
V, F
C, F, P U (1 , 3) 9 .
109 × 10 −c
Speed of light
V, F
C, F, P U (1 , 3) 2 .
998 × 10 8 θ
Incidence angle
V, F V, F
U (1 , 3)
U ( − π,π )
B ρ = 38 πG (cid:18) c
2 k f a 2 f + H 2 (cid:19)
ρ
Variable
V, F V, F
N/A
N/A G
Gravitational constant
V, F
C, F, P U (
[Table] [Table]
[Equation]
[Table] [Table]
[Equation]
[Table]
[Table]
[Table]
L rad =
h
2 π ωπ
[Table]
c 2 (exp( hω/ 2 πkT ) − 1)
[Table] [Table]
= nkT ln( V 2 V 1 )
Q
[Table] [Table]
F
V, F, P U (
[Table]
U log
[Table]
) v
Velocity
V, F
V, F, P U
[Table]
U log
[Table]
c
Speed
[Table]
light
V, F
C,
[Table] [Table]
U
[Table]
k = (cid:112) ω 2 /c 2 − π 2 /ak
Wavenumber
V, F
[Table]
, F, P
N/A
N/A ω
Angular velocity
[Table] [Table]
U (
[Table]
c
[Table]
II.35.
a =
N exp(
[Equation] [Equation]
+
[Equation] [Equation]
) a
Number of atoms with the equivalent magnetic moment
V, F
V, I ⋆ , P
N/
[Table] [Table]
N
Numberofatomsperunitvolume
[Table]
, F
[Table]
µ
[Table]
B
Magnetic flux density
[Table]
U
[Table]
k
Boltzmann constant
V, F
C, F, P U
[Table]
−
T
Temperature
[Table]
, F
V, F
[Table]
P U
[Table]
U log
[Table]
II.35.
M =
Nµ tanh (cid:16) µBkT (cid:17)
M
Number of magnetized atoms
V, F
V, I ⋆ , P
N/A N/A
N
Number of atom
V, F
V, I
[Table]
x =
[Equation]
+
[Equation]
2 kT M x
[Table]
F
[Table]
E =
hω
[Equation] [Equation]
[Equation] [Equation]
) − 1)
E
[Table]
V, F
V, F, P
N/A
N/A h
[Table]
V, F
C, F, P U (1 , 5)
[Table]
[Equation]
[Table]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
E − U − L 2 2 mr 2 (cid:17) v Velocity
V, F V, F, P
N/A
N/A m Mass
(The Earth)
V, F V, F, P U (1 , 3) U log (10 23 , 10 25 ) E Energy
V, F V, F, P U (8 , 12) U log (10 25 , 10 27 )
U Potential energy
V, F V, F, P U (1 , 3) U log (10 25 , 10 27 )
L Angular momentum
V, F
V, F U (1 , 3)
U log (10 8 , 10 10 ) r Distance
V, F V, F, P U (1 , 3)
U log (10 8 , 10 10 )
B t = 2 πd 3 / 2 √ G ( m 1+ m 2) t Orbital period
V, F V, F, P
N/A
N/A d Semimajor axis of elliptical orbit
V, F V, F, P U (1 , 3)
U log (10 8 , 10 10 ) G Gravitational constant
V, F C, F, P U (1 , 3) 6 .
674 × 10 −m 1 Mass (The Earth)
V, F V, F, P U (1 , 3) U log (10 23 , 10 25 ) m 2 Mass
(The Earth)
V, F V, F, P U (1 , 3) U log (10 23 , 10 25 )
B α = (cid:114) 1 + 2 ϵ 2 EL 2 m ( Z 1 Z 2 q 2)
α Orbital eccentricity
V, F V, F, P
N/A
N/A ϵ Energy
V, F
V, F U (1 , 3) U log (10 − 18 , 10 − 16 )
E Energy
V, F V, F, P U (1 , 3) U log (10 − 18 , 10 − 16 )
L Distance
V, F V, F, P U (1 , 3) U log (10 − 10 , 10 − 8 ) m Mass
V, F V, F, P U (1 , 3) U log (10 − 30 , 10 − 28 )
Z 1 Atomic number
V, F
V, I,P U (1 , 3)
U log (10 0 , 10 1 ) Z 2 Atomic number
V, F
V, I,P U (1 , 3)
U log (10 0 , 10 1 ) q Electric charge
V, F
V, F U (1 , 3) U log (10 − 11 , 10 − 9 )
B
[Equation]
H Hubble’s constant
V, F V,
[Equation]
N
[Equation]
A
N/A G Gravitational constant
V, F C, F, P U (1 , 3) 6 . 674 × 10 −
[Table]
Density of the Universe
V, F V, F, P U (1 , 3) U log (10 − 27 , 10 − 25 )
k
f Spacetime curvature
V, F
V, I
U (1 , 2)
U ( − 1 , 1) c Speed of light
V, F C, F, P U (1 , 2) 2 .
998 × 10 8 a f Radius
V, F V, F, P U (1 , 3)
U log (10 8 , 10 10 )
B
[Equation]
P Gravitational wave energy
V, F V, F
N/A
N/A G Gravitational constant
V, F C, F, P U (1 , 2) 6 .
674 × 10 −c Speed of light
V, F C, F, P U (1 , 2) 2 .
998 × 10 8 m 1 Mass
V, F V, F, P U (1 , 5) U log (10 23 , 10 25 ) m 2 Mass
V, F V, F, P U (1 , 5) U log (10 23 , 10 25 ) r Distance
V, F V, F, P U (1 , 2)
U log (10 8 , 10 10 )
[Table]
Voltage
[Table]
Distance
[Table]
B
[Equation]
[Table]
B
[Equation]
[Table]
B
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
Frequency of electromagnetic waves
V, F V, F
N/A
N/A v
Velocity
V, F V, F, P U
[Table]
= qV
e +
[Equation]
[Table]
[Equation]
[Table]
F
V, F U
[Table]
) U log
[Table]
1 ) y
Distance
[Table]
F
[Table]
F
[Table]
U
[Table]
) U log
[Table] [Table]
[Equation]
Pressure
V, F V, F
N/A
N/A G
Gravitational constant
V, F C, F, P U (
[Table] [Table]
c
[Table]
light
[Table] [Table]
8 k f
Variable
[Table]
a f
Distance
[Table]
Variable
[Table]
α
Variable
[Table]
F
[Table]
U
[Table]
B
[Equation]
A
Differential cross section
V, F V, F
N/A
N/A α
Fine structure constant
V, F C, F, P U (1 , 5)
[Table] [Table]
−h
Planck constant
V, F C, F, P U (1 , 5)
[Table]
m
Electron mass
V, F C, F
[Table]
P U
[Table]
[Table] [Table]
c
Speed of light
[Table]
F
[Table]
, F
[Table]
U
[Table] [Table]
8 ω
Frequency
[Table]
Frequency
[Table]
F
[Table] [Table]
B Hyperparameters for Five Existing SR Baselines
Table S12 shows the hyperparameter space for the five existing symbolic regression baselines.
The hyperparameters of gplearn [3] 11 , AFP [38], and AFP-FE [35] 12 are optimized by Optuna [40], a hyperparameter optimization framework.
[Table]
100 trials with random combinations of the following hyperparameter spaces: population_size : U (10 2 , 10 3 ) , generations : U (10 , 10 2 ) , stopping_criteria : U (10 − 10 , 10 − 2 ) , warm_start : {True, False}, const_range : {None, ( − 1 . 0 , 1 . 0) , ( − 10 , 10) , ( − 10 2 , 10 2 ) , ( − 10 3 , 10 3 ) , ( − 10 4 , 10 4 ) }, max_samples : U (0 . 9 , 1 . 0) ,
parsimony_coefficient : U (10 − 3 , 10 − 2 )
AFP 100 trials with random combinations of the following hyperparameter spaces: popsize : U (100 , 1000) , g : U (250 , 2500) , stop_threshold : U (10 − 10 , 10 − 2 ) , op_list : {[’n’, ’v’, ’+’, ’-’, ’*’, ’/’, ’exp’, ’log’, ’2’, ’3’, ’sqrt’], [’n’, ’v’, ’+’, ’-’, ’*’, ’/’, ’exp’, ’log’, ’2’, ’3’, ’sqrt’, ’sin’, ’cos’]}
AFP-FE 100 trials with random combinations of the following hyperparameter spaces: popsize : U (100 , 1000) , g : U (250 , 2500) , stop_threshold : U (10 − 10 , 10 − 2 ) , op_list : {[’n’, ’v’, ’+’, ’-’, ’*’, ’/’, ’exp’, ’log’, ’2’, ’3’, ’sqrt’], [’n’, ’v’, ’+’, ’-’, ’*’, ’/’, ’exp’, ’log’, ’2’, ’3’, ’sqrt’, ’sin’, ’cos’]}
AI Feynman { bftt : 60, epoch : 300, op : ’7ops.txt’, poly_deg : 3}, { bftt : 60, epoch : 300, op : ’10ops.txt’, poly_deg : 3}, { bftt : 60, epoch : 300, op : ’14ops.txt’, poly_deg : 3}, { bftt : 60, epoch : 300, op : ’19ops.txt’, poly_deg : 3}, { bftt : 120, epoch : 300, op : ’14ops.txt’, poly_deg : 4}, { bftt : 120, epoch : 300, op : ’19ops.txt’, poly_deg : 4}, { bftt : 60, epoch : 500, op : ’7ops.txt’, poly_deg : 3}, { bftt : 60, epoch : 500, op : ’10ops.txt’, poly_deg : 3}, { bftt : 60, epoch : 500, op : ’14ops.txt’, poly_deg : 3}, { bftt : 60, epoch : 500, op : ’19ops.txt’, poly_deg : 3}
DSR { seed : 1, function_set : [’add’, ’sub’, ’mul’, ’div’, ’sin’, ’cos’, ’exp’, ’log’}, { seed : 2, function_set : [’add’, ’sub’, ’mul’, ’div’, ’sin’, ’cos’, ’exp’, ’log’}, { seed : 3, function_set : [’add’, ’sub’, ’mul’, ’div’, ’sin’, ’cos’, ’exp’, ’log’}, { seed : 4, function_set : [’add’, ’sub’, ’mul’, ’div’, ’sin’, ’cos’, ’exp’, ’log’}, { seed : 5, function_set : [’add’, ’sub’, ’mul’, ’div’, ’sin’, ’cos’, ’exp’, ’log’}, { seed : 1, function_set : [’add’, ’sub’, ’mul’, ’div’, ’sin’, ’cos’, ’exp’, ’log’, ’const’]}, { seed : 2, function_set : [’add’, ’sub’, ’mul’, ’div’, ’sin’, ’cos’, ’exp’, ’log’, ’const’]}, { seed : 3, function_set : [’add’, ’sub’, ’mul’, ’div’, ’sin’, ’cos’, ’exp’, ’log’, ’const’]}, { seed : 4, function_set : [’add’, ’sub’, ’mul’, ’div’, ’sin’, ’cos’, ’exp’, ’log’, ’const’]}, { seed : 5, function_set : [’add’, ’sub’, ’mul’, ’div’, ’sin’, ’cos’, ’exp’, ’log’, ’const’]}