AI Feynman: A physics-inspired method for symbolic regression

Silviu-Marian UdrescuMax Tegmark

article2019Science Advances1,359 citations

Develops AI Feynman, a recursive symbolic regression algorithm that combines neural network fitting with physical properties such as symmetry and separability to extract governing equations from data, raising discovery rates on difficult benchmarks from 15% to 90%.

Listen

Modern data-driven science faces a fundamental bottleneck in symbolic regression: automatically discovering exact, human-interpretable mathematical formulas from numerical data. Standard brute-force search over mathematical symbols is computationally impossible for complex expressions because search spaces grow exponentially. Meanwhile, popular genetic algorithm tools often get trapped in suboptimal approximations when formulas contain nested or non-additive structures.

The article develops and evaluates AI Feynman, a recursive symbolic regression algorithm that combines standard fitting techniques with neural networks to discover underlying physical properties—such as symmetries and separability—to break complex multi-variable problems into simpler, solvable sub-problems.

To evaluate the algorithm, the authors constructed a benchmark database of 100 physics equations from the Feynman Lectures on Physics and a separate, more challenging test set of 20 advanced physics formulas from standard graduate-level texts. Synthetic datasets with up to 100,000 points per equation were generated and evaluated under varying levels of data volume and injected noise. AI Feynman systematically applies dimensional analysis, polynomial fitting, and brute-force searches alongside neural network interpolators that test for translational symmetry, scaling, and additive or multiplicative separability.

The analysis produced several key findings. First, AI Feynman discovered 100% (100 of 100) of the basic Feynman benchmark equations, significantly outperforming the commercial genetic algorithm software Eureqa, which solved 71%. Second, on the 20 advanced test equations, AI Feynman improved the state-of-the-art discovery rate from 15% (3 of 20 solved by Eureqa) to 90% (18 of 20). Third, even when physical units were omitted and dimensional analysis was disabled, AI Feynman maintained a 93% success rate on the benchmark by relying directly on neural network property detection. Finally, the algorithm exhibited strong data efficiency and noise resilience: most simpler equations required as few as 10 to 100 data points and tolerated up to 1% injected noise, whereas complex multi-variable equations required larger sample sizes (up to 1,000,000 points) and lower noise thresholds to train the neural network effectively.

These results indicate that automated scientific discovery does not require intractable brute-force computation if algorithms exploit structural simplifications inherent in natural laws. By reducing multi-variable dependencies into modular sub-tasks, AI Feynman provides a deterministic path toward exact analytic solutions, reducing the risk of false convergence common in genetic programming. This capability accelerates scientific modeling and allows automated distillation of governing physical laws directly from experimental data.

For future development, the article recommends hybridizing AI Feynman with genetic algorithms to generate Pareto-optimal candidate expressions under noisy conditions. It also proposes extending the architecture to automatically discover differential equations by estimating derivatives, incorporating broader functional operators such as arbitrary exponentiation, and adopting optimized neural network architectures to lower fitting error.

The findings are supported with high confidence on clean, well-sampled synthetic data with known physical properties. However, readers should exercise caution when evaluating highly arbitrary mathematical formulas lacking physical symmetries or applications involving noisy experimental data with noise in the independent variables, where neural network training and property thresholding become more challenging.

Massachusetts Institute of Technology Theiss Research neural networks software symbolic regression AI Feynman symbolic regression dimensional analysis

arXiv: 1905.11481 SJ001/AI-Feynman

Paper: Theory-Guided Data Science: A New Paradigm for Scientific Discovery from Data, Anuj Karpatne et al. (2016). It provides the foundational conceptual framework for theory-guided data science and embedding physical properties into machine learning, which directly motivates the physics-inspired regression paradigm of AI Feynman.
Paper: Gene Expression Programming: A New Adaptive Algorithm for Solving Problems, Cândida Ferreira (2001). It introduces foundational evolutionary and expression-tree representations for symbolic regression, establishing the traditional baseline methods that AI Feynman seeks to dramatically improve upon.

Paper: KAN: Kolmogorov-Arnold Networks, Ziming Liu et al. (2025). It extends AI Feynman's goal of interpretable, physics-inspired scientific discovery by introducing Kolmogorov-Arnold Networks as an interpretable alternative to MLPs for discovering symbolic and mathematical formulas.
Paper: Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next, Salvatore Cuomo et al. (2022). It surveys the subsequent landscape of scientific machine learning and physics-informed models, providing broader context on how physics-guided neural architectures have evolved for scientific discovery.

Cover for AI Feynman: A physics-inspired method for symbolic regression

Abstract

A core challenge for both physics and artificial intellicence (AI) is symbolic regression: finding a symbolic expression that matches data from an unknown function. Although this problem is likely to be NP-hard in principle, functions of practical interest often exhibit symmetries, separability, compositionality and other simplifying properties. In this spirit, we develop a recursive multidimensional symbolic regression algorithm that combines neural network fitting with a suite of physics-inspired techniques. We apply it to 100 equations from the Feynman Lectures on Physics, and it discovers all of them, while previous publicly available software cracks only 71; for a more difficult test set, we improve the state of the art success rate from 15% to 90%.

Topics

neural networks software symbolic regression test samples

Keywords

AI Feynman symbolic regression dimensional analysis genetic programming neural networks sparse identification evolutionary computation brute-force search

I Introduction
II Methods
II.1 Overall Algorithm
II.2 Dimensional Analysis
II.3 Polynomial Fit
II.4 Brute Force
II.5 Neural-network-based tests & transformations
II.5.1 Neural network training
II.5.2 Translational symmetry and generalizations
II.5.3 Separability
II.5.4 Setting variables equal
II.6 Extra Transformations
III Results
III.1 The Feynman Symbolic Regression Database
III.2 Method comparison
III.3 Dependence on data size
III.4 Dependence on noise level
III.5 Bonus mysteries
IV Conclusions
IV.1 Key findings
IV.2 Opportunities for further work
References

Knowls

Knowl 1 — AI Feynman Recursive Symbolic Regression Architecture
model/method
AI Feynman is a recursive, multidimensional symbolic regression algorithm designed to discover an exact analytical function $y = f(x_1, \dots, x_n)$ from a dataset of numerical inputs and outputs. Rather than attempting a global search across an exponentially large space of mathematical formulas, AI Feynman decomposes regression tasks by exploiting physical and mathematical properties: physical units, low-degree polynomial structures, compositionality, smoothness, continuous symmetries, and separability.

The algorithm executes a sequential pipeline of specialized modules on a given dataset:
1. Dimensional Analysis: Utilizes the physical units of input and output variables to construct dimensionless variables, reducing the total variable count from $n$ to the nullity $n'$ of the unit matrix.
2. Polynomial Fitting: Tests whether the mystery data can be fitted by a multivariate polynomial of degree $d \le 4$ by solving a linear system of equations. If the root-mean-square error satisfies $\epsilon \le 10^{-4}$ , the polynomial is returned as the exact solution.
3. Brute-Force Symbolic Search: Enumerates syntactically valid reverse Polish notation (RPN) expressions of increasing length over an alphabet of elementary mathematical operations and constants, selecting candidates that minimize a Minimum Description Length (MDL) criterion.
4. Neural Network Interpolation: Trains a fully connected feed-forward neural network to act as a smooth, high-dimensional interpolator $\hat{f}(x_1, \dots, x_n)$ over the data domain.
5. Symmetry Detection: Uses the neural network to test for translational, scaling, additive, or multiplicative symmetries between variable pairs. If a symmetry is found, the pair is replaced by a single composite variable, and the reduced problem is recursively fed back to a fresh instantiation of the algorithm.
6. Separability Detection: Tests whether $\hat{f}$ can be factored into additively or multiplicatively separable components involving disjoint subsets of variables. If separable, sub-datasets corresponding to each component are generated and solved recursively.
7. Variable Equating and Transformations: Tests the effect of setting pairs of variables equal or applying elementary functional transformations (such as logarithms, square roots, squares, trigonometric functions, or inverses) to dependent and independent variables before re-running polynomial fitting or brute-force search.
Whenever a reduction or decomposition step succeeds, the algorithm spawns sub-problems with strictly fewer variables, guaranteeing monotonic dimensional reduction until all sub-expressions are solved.
Knowl 2 — Automated Dimensional Analysis for Dimensionality Reduction
algorithm
Dimensional analysis reduces the number of independent variables in a symbolic regression task by enforcing unit consistency across fundamental physical dimensions.

Let physical units be represented as 5-dimensional integer vectors $\mathbf{u} = (u_{\text{m}}, u_{\text{s}}, u_{\text{kg}}, u_{\text{T}}, u_{\text{V}})^T \in \mathbb{Z}^5$ , corresponding to the powers of length (meter), time (second), mass (kilogram), temperature (kelvin), and voltage (volt). For a mystery relationship $y = f(x_1, \dots, x_n)$ , let $\mathbf{M} \in \mathbb{Z}^{5 \times n}$ denote the matrix whose $j$ -th column is the unit vector $\mathbf{u}_j$ of variable $x_j$ , and let $\mathbf{b} \in \mathbb{Z}^5$ be the unit vector of the target output $y$ .

The automated dimensional analysis procedure operates as follows:
Input: Data table D = {(x_1, ..., x_n, y)}, Unit matrix M in Z^{5 x n}, Target unit vector b in Z^5
Output: Dimensionless data table D' = {(x'_1, ..., x'_{n'}, y')}, Transformation relations
Find a particular solution p in Q^n satisfying M p = b
Compute basis matrix U in Z^{n x n'} for the null space of M such that M U = 0, where n' is the nullity of M
Select basis U and shift p -> p + U a (for a in Q^{n'}) to maximize the number of zero entries in p and U
for each sample (x_1, ..., x_n, y) in D do:
    y* = prod_{i=1}^n x_i^{p_i}
    y' = y / y*
    for j = 1 to n' do:
        x'_j = prod_{i=1}^n x_i^{U_{ij}}
    add (x'_1, ..., x'_{n'}, y') to D'
return D', p, U
The resulting regression task $y' = f'(x'_1, \dots, x'_{n'})$ operates on $n'$ dimensionless variables. If $n' = 0$ , the problem reduces to solving for a dimensionless constant.
Knowl 3 — Neural Network-Based Continuous Function Interpolator
model/method
To evaluate a mystery function $f(x_1, \dots, x_n)$ at arbitrary coordinates for symmetry and separability testing, AI Feynman trains a deep neural network interpolator on the dataset.
- Architecture: A feed-forward, fully connected neural network with 6 hidden layers and softplus activation functions. The first three hidden layers contain 128 neurons each, and the subsequent three hidden layers contain 64 neurons each.
- Dataset Partition: $100{,}000$ points are sampled over the input domain, partitioned into an $80\%$ training set ( $80{,}000$ points) and a $20\%$ validation set ( $20{,}000$ points).
- Optimization and Schedules: Trained for 100 epochs using the Adam optimizer with a weight decay of $10^{-2}$ and mini-batch size of 2048. Training employs the 1cycle learning rate and momentum policy: a maximum learning rate of $0.005$ with a ratio of 20 between maximum and minimum learning rates, allocating the final $10\%$ of iterations to a cooldown phase. Momentum $\beta_1$ is cycled between $0.85$ and $0.95$ , while $\beta_2 = 0.99$ .
- Performance: Across target equations, the achieved root-mean-square validation error $\epsilon_{\text{NN}}$ typically ranges between $10^{-3} f_{\text{rms}}$ and $10^{-5} f_{\text{rms}}$ , where $f_{\text{rms}} = \left( \frac{1}{N_d} \sum_{k=1}^{N_d} y_k^2 \right)^{1/2}$ .
Knowl 4 — Neural Network-Based Symmetry Detection and Variable Elimination
algorithm
AI Feynman detects continuous symmetries between pairs of input variables $(x_i, x_j)$ by querying the trained neural network interpolator $\text{net}(\mathbf{x})$ . If a symmetry condition holds within a tolerance threshold $\epsilon_{\text{sym}} = 7 \epsilon_{\text{NN}}$ (where $\epsilon_{\text{NN}}$ is the neural network validation error), the two variables are replaced by a single composite variable.

The algorithm tests for translational symmetry ( $x_j - x_i$ ), scaling symmetry ( $x_j / x_i$ ), additive combination ( $x_i + x_j$ ), and multiplicative combination ( $x_i x_j$ ):
Input: Dataset D = {(x, y)}, Trained neural network net, Validation error NN_error, Shift parameter a = 1
Output: Reduced dataset D_trans, Transformed indices (i, j)
for i = 1 to len(x) - 1 do:
    for j = i + 1 to len(x) do:
        x_shifted = copy(x)
        x_shifted[i] = x_shifted[i] + a
        x_shifted[j] = x_shifted[j] + a
        error = RMSE(net(x), net(x_shifted)) / RMSE(net(x))
        if error < 7 * NN_error then:
            x_new = copy(x)
            x_new[i] = x_new[j] - x_new[i]
            x_new = delete_coordinate(x_new, j)
            return x_new, i, j
return No_Symmetry_Found
If translational symmetry is identified, $f$ depends on $x_i$ and $x_j$ only through the difference $x'_i = x_j - x_i$ , eliminating one independent variable. Analogous substitution is performed for scaling symmetry ( $x'_i = x_j / x_i$ ), sum ( $x'_i = x_i + x_j$ ), and product ( $x'_i = x_i x_j$ ).
Knowl 5 — Neural Network-Based Functional Separability Decomposition
algorithm
A multivariate function $f(x_1, \dots, x_n)$ is separable if it can be partitioned into expressions involving disjoint subsets of variables. AI Feynman tests for multiplicative and additive separability using the neural network interpolator $\text{net}(\mathbf{x})$ .

For a two-variable multiplicative relation $f(x_1, x_2) = g(x_1)h(x_2)$ , constants $c_1, c_2$ are chosen as the sample means $\bar{x}_1, \bar{x}_2$ . The non-separability metric is: $\Delta_{\text{sep}}(x_1, x_2) = \frac{1}{f_{\text{rms}}} \left| f(x_1, x_2) - \frac{f(x_1, c_2) f(c_1, x_2)}{f(c_1, c_2)} \right|$ If the root-mean-square of $\Delta_{\text{sep}}$ is less than $\epsilon_{\text{sep}} = 10 \epsilon_{\text{NN}}$ , the problem is split into two univariate sub-problems $y' = f(x_1, c_2)$ and $y'' = f(c_1, x_2)/f(c_1, c_2)$ . Once $y'$ is solved symbolically, $y''$ is redefined as $y'' = y / (y' c_{\text{num}})$ where $c_{\text{num}}$ absorbs numerical prefactors, yielding $y = y' y'' / c_{\text{num}}$ .

For general additive separability across arbitrary variable partitions:
Input: Dataset D = {(x, y)}, Trained neural network net, Validation error NN_error
Output: Decomposed datasets D_1, D_2, Partition indices idx_1, idx_2
x_mean = mean_vector(x)
for k = 1 to len(x) - 1 do:
    for idx_1 in combinations(range(1, len(x) + 1), k) do:
        idx_2 = complement(idx_1, range(1, len(x) + 1))
        x_part1 = copy(x); set_coordinates(x_part1, idx_2, x_mean[idx_2])
        x_part2 = copy(x); set_coordinates(x_part2, idx_1, x_mean[idx_1])
        f_pred = net(x_part1) + net(x_part2) - net(x_mean)
        error = RMSE(net(x), f_pred) / RMSE(net(x))
        if error < 10 * NN_error then:
            D_1 = project_coordinates(x_part1, idx_1)
            D_2 = project_coordinates(x_part2, idx_2)
            return D_1, D_2, idx_1, idx_2
return No_Separability_Found
Knowl 6 — Minimum Description Length Objective for Symbolic String Selection
equation
In the brute-force symbolic search module, expressions are generated in reverse Polish notation in increasing order of complexity. To avoid overfitting noisy data and select the simplest expression matching the data, candidate functions with root-mean-square fitting error $\epsilon < \epsilon_b$ are selected by minimizing the total Description Length $\text{DL}$ :

$\text{DL} \equiv \log_2 N + \lambda \log_2 \left[ \max\left(1, \frac{\epsilon}{\epsilon_d}\right) \right]$

where:
- $N \in \mathbb{N}$ is the lexicographical integer rank of the candidate symbolic string in the generation list, representing the number of bits needed to store the formula structure.
- $\epsilon = \left( \frac{1}{N_d} \sum_{i=1}^{N_d} (y_i - f(\mathbf{x}_i))^2 \right)^{1/2}$ is the root-mean-square error on the dataset.
- $\epsilon_d = 10^{-15}$ is the baseline machine precision floor.
- $\lambda = N_d^{1/2}$ is a hyperparameter scaling the penalty for residual error relative to expression complexity, with $N_d$ being the number of data points in the mystery dataset.
- $\epsilon_b$ is the acceptance error threshold, set to $10^{-5}$ for original datasets and $10\epsilon_{\text{NN}}$ for sub-datasets generated from neural network interpolations.
Knowl 7 — The Feynman Symbolic Regression Database (FSReD)
experimental setup
The Feynman Symbolic Regression Database (FSReD) is a standardized benchmark database created to evaluate symbolic regression algorithms on equations from physics.
- Core Dataset (100 Equations): 100 non-differential, non-integral closed-form equations selected from Volumes I, II, and III of The Feynman Lectures on Physics, spanning classical mechanics, electromagnetism, thermodynamics, wave optics, and quantum mechanics. The number of independent variables ranges between 1 and 9.
- Bonus Test Set (20 Equations): 20 highly complex equations selected from standard graduate physics textbooks: Goldstein's Classical Mechanics, Jackson's Classical Electrodynamics, Weinberg's Gravitation and Cosmology, and Schwartz's Quantum Field Theory and the Standard Model.
- Data Structure: For each equation, the database provides:
  1. A data table of $10^5$ numerical rows $\{x_1, \dots, x_n, y\}$ , where input variables are sampled uniformly at random in $[1, 5]$ (adjusted when necessary to avoid negative square roots or division by zero).
  2. A unit table specifying the physical unit vector $\mathbf{u} \in \mathbb{Z}^5$ (powers of $\text{m}, \text{s}, \text{kg}, \text{K}, \text{V}$ ) for all input and output variables.
  3. The ground-truth analytical formula $f(x_1, \dots, x_n)$ .
- Evaluation Criterion: A candidate symbolic solution $f'$ is classified as correct if and only if algebraic simplification of the difference $f' - f$ (e.g., via SymPy's simplify or Mathematica's Simplify) evaluates identically to the constant $0$ .

Knowl 8 — Symbolic Regression Benchmark Results: AI Feynman vs. Eureqa

empirical result

AI Feynman was benchmarked against the commercial genetic-programming-based symbolic regression software Eureqa on three datasets with a runtime budget of 2 hours of CPU time per mystery:

Benchmark Dataset	Total Equations	AI Feynman Success (%)	Eureqa Success (%)
Feynman Lectures Core Benchmark	100	100% (100/100)	71% (71/100)
Advanced Physics Bonus Benchmark	20	90% (18/20)	15% (3/20)
McDermott et al. Synthetic Benchmark	45	66.7% (30/45)	48.9% (22/45)

Key empirical findings:

On the 100 core Feynman equations, AI Feynman discovered $100\%$ of the analytical formulas, whereas Eureqa discovered $71\%$ .
On the 20 difficult bonus physics equations, AI Feynman solved $18/20$ ( $90\%$ ), while Eureqa solved $3/20$ ( $15\%$ ). The performance advantage of AI Feynman is greatest for complex multidimensional expressions, where recursive symmetry reduction and separability factorization prevent the combinatorial search collapse that traps genetic algorithms in local optima.
On arbitrary synthetic functions from McDermott et al. that lack physical symmetries or separability, AI Feynman still outperformed Eureqa ( $66.7\%$ vs. $48.9\%$ ).

Knowl 9 — Data Efficiency, Noise Robustness, and Ablation of Dimensional Analysis
empirical result
Systematic empirical evaluations of AI Feynman's performance under data constraints, observational noise, and module ablation revealed:
- Data Efficiency: For equations solvable purely through polynomial fitting or brute-force search, AI Feynman requires as few as $10$ to $100$ data points to identify the exact analytical expression without overfitting. Equations requiring neural network interpolation for symmetry or separability discovery require $10^2$ to $10^6$ data points to train the network to an rms validation accuracy below $10^{-3} f_{\text{rms}}$ .
- Noise Robustness: When independent Gaussian random noise with standard deviation $\epsilon y_{\text{rms}}$ is added to the dependent variable $y$ , most Feynman equations are solved exactly at relative noise levels up to $\epsilon = 10^{-4}$ , and nearly half remain solvable at $\epsilon = 10^{-2}$ ( $1\%$ relative noise).
- Dimensional Analysis Ablation: When the automated dimensional analysis module is completely disabled, AI Feynman still solves $93\%$ ( $93/100$ ) of the Feynman equations. Without dimensional analysis, the algorithm heavily utilizes the neural network module to detect translational symmetries and multiplicative separability to factor out dimensionless groups and constant parameters one variable at a time (e.g., deploying the neural network module six times to solve Newton's gravitational law).
Knowl 10 — Failure Modes and Algorithmic Limitations of AI Feynman
limitation
AI Feynman failed on 2 of the 20 advanced physics bonus equations:
1. Radiated Gravitational Wave Power: $P = -\frac{32}{5} \frac{G^4}{c^5} \frac{(m_1 m_2)^2 (m_1 + m_2)}{r^5}$ . After reduction by dimensional analysis to $y = -\frac{32}{5} \frac{a^2(1+a)}{b^5}$ , the $b^5$ factor in the denominator produced an extreme dynamic range across the dataset domain $[1, 5]$ . This large dynamic range degraded neural network fitting accuracy, causing the network validation error to exceed the separability tolerance threshold $\epsilon_{\text{sep}}$ . The problem was relegated to brute-force search, which required an estimated 2 years of runtime for the string length.
2. Jackson Electrodynamics Problem 2.11: $F = \frac{q}{4\pi\epsilon y^2} \left[ 4\pi\epsilon V_e d - \frac{q d y^3}{(y^2 - d^2)^2} \right]$ . Reduction via dimensional analysis resulted in $a - \frac{1}{4\pi} \frac{a}{b} (1 - a^2)^2$ . The representation of the $4\pi$ numerical constant required seven RPN symbols (PPPP+++), and the squared binomial required repeated multiplication symbols, generating a string whose brute-force evaluation time exceeded $100$ times the age of the universe.
3. Domain Specificity: The algorithm's superior search efficiency relies heavily on physical symmetries, separability, and dimensional homogeneity; performance decreases on arbitrary compositions of elementary functions lacking these structural properties.

Coverage note — Individual formulas, numerical runtimes, and exact method sequences for all 100 individual Feynman equations (Tables 4 and 5) were summarized into overall empirical results rather than listed one by one to avoid excessive and repetitive tabulation.

References

1.A. Koyré, The Astronomical Revolution: Copernicus-Kepler-Borelli (Routledge, 2013).
2.N. M. Amil, N. Bredeche, C. Gagné, S. Gelly, M. Schoenauer, O.Teytaud, European Conference on Genetic Programming (Springer, 2009), pp. 327–338.
3.S. K. Pal, P. P. Wang, Genetic algorithms for pattern recognition (CRC press, 2017).
4.J. D. Lohn, W. F. Kraus, D. S. Linden, IEEE Antenna & Propagation Society Mtg. 3, 814 (2002).
5.D. S. Linden, Proceedings 2002 NASA/DoD Conference on Evolvable Hardware (IEEE, 2002), pp. 147–151.
6.H. Yu, N. Yu, The Pennsylvania State University, (University Park, 2003) pp. 1–9.
7.S. Panthong, S. Jantarang, CCECE 2003-Canadian Conference on Electrical and Computer Engineering. Toward a Caring and Humane Technology (Cat. No. 03CH37436) (IEEE, 2003), vol. 3, pp. 1597–1600.
8.B. Oh, Y. Na, J. Yang, S. Park, J. Nang, J. Kim, Genetic algorithm-based dynamic vehicle route search using car-to-car communication. Adv. Electr. Comput. En. 10, 81–86 (2010).
9.A. Ram, R. Arkin, G. Boone, M. Pearce, Using genetic algorithms to learn reactive control parameters for autonomous robotic navigation. Adapt. Behav. 2, 277–305 (1994).
10.B. Delman, Genetic algorithms in cryptography. Rochester Institute of Technology (2004).
11.S. Kim, P. Lu, S. Mukherjee, M. Gilbert, L. Jing, V. Ceperic, M. Soljacic, arXiv preprint arXiv:1912.04825 (2019).
12.R. J. Bauer, Genetic algorithms and investment strategies, (John Wiley & Sons, ed. 1, 1994), vol. 19, p. 320.
13.R. Venkatesan, V. Kumar, A genetic algorithms approach to forecasting of wireless subscribers. Int. J. Forecast. 18, 625–646 (2002).
14.W. L. Cava, T. R. Singh, J. Taggart, S. Suri, J. Moore, International Conference on Learning Representations (2019), https://openreview.net/forum?id=Hke-JhA9Y7.
15.S. McAleer, F. Agostinelli, A. Shmakov, P. Baldi, International Conference on Learning Representations (2019); https://openreview.net/forum?id=Hyfn2jCcKm.
16.J. R. Koza, J. R. Koza, Genetic Programming: On the Programming of Computers by Means of Natural Selection (MIT Press, 1992), vol. 1.
17.M. D. Schmidt, R. R. Vallabhajosyula, J. W. Jenkins, J. E. Hood, A. S. Soni, J. P. Wikswo, H. Lipson, Automated refinement and inference of analytical models for metabolic networks. Phys. Biol. 8, 055011 (2011).
18.R. K. McRee, Proceedings of the 12th Annual Conference Companion on Genetic and Evolutionary Computation (ACM, New York, NY, USA, 2010), GECCO ‘10, pp. 1983–1990; http://doi.acm.org/10.1145/1830761.1830841.
19.S. Stijven, W. Minnebo, K. Vladislavleva, Proceedings of the 13th Annual Conference Companion on Genetic and Evolutionary Computation (ACM, New York, NY, USA, 2011), GECCO ‘11, pp. 623–630; http://doi.acm.org/10.1145/2001858.2002059.
20.W. Kong, C. Liaw, A. Mehta, D. Sivakumar, International Conference on Learning Representations (2019); https://openreview.net/forum?id=rkluJ2R9KQ.
21.T. McConaghy, Genetic Programming Theory and Practice IX (Springer, 2011), pp. 235–260.
22.I. Arnaldo, U.-M. O’Reilly, K. Veeramachaneni, Proceedings of the 2015 Annual Conference on Genetic and Evolutionary Computation (ACM, 2015), pp. 983–990.
23.S. L. Brunton, J. L. Proctor, J. N. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl. Acad. Sci. U.S.A. 113, 3932–3937 (2016).
24.M. Quade, M. Abel, J. Nathanutz, S. L. Brunton, Sparse identification of nonlinear dynamics for rapid model recovery. Chaos 28, 063116 (2018).
25.D. P. Searson, D. E. Leahy, M. J. Willis, Proceedings of the International multiconference of engineers and computer scientists (Citeseer, 2010), vol. 1, pp. 77–80.
26.R. Praksova, Eureqa: Software review. Genet. Program. Evol. M. 12, 173–178 (2011).
27.M. Schmidt, H. Lipson, Distilling free-form natural laws from experimental data. Science 324, 81–85 (2009).
28.H. Mhaskar, Q. Liao, T. Poggio, Technical report, Center for Brains, Minds and Machines (CBMM), arXiv (2016).
29.H. W. Lin, M. Tegmark, D. Rolnick, Why does deep and cheap learning work so well? J. Stat. Phys. 168, 1223–1247 (2017).
30.T. Wu, M. Tegmark, Toward an artificial intelligence physicist for unsupervised learning. Phys. Rev. E. 100, 033311 (2019).
31.L. N. Smith, N. Topin, Super-convergence: Very fast training of residual networks using large learning rates (2018); https://openreview.net/forum?id=H1A5ztj3b.
32.L. N. Smith, A disciplined approach to neural network hyper-parameters: Part 1 – learning rate, batch size, momentum, and weight decay. arXiv:1803.09820 (2018).
33.J. Howard et al., Fastai, https://github.com/fastai/fastai (2018).
34.R. Feynman, R. Leighton, M. Sands, The Feynman Lectures on Physics: The New Millennium Edition: Mainly Mechanics, Radiation, and Heat, vol. 1 (Basic Books, 1963); https://books.google.com/books?id=d76DBQAAQBAJ.
35.R. Feynman, R. Leighton, M. Sands, The Feynman Lectures on Physics, vol. 2 in The Feynman Lectures on Physics (Pearson/Addison-Wesley, 1963b); https://books.google.com/books?id=AbruAAAAMAAJ.
36.R. Feynman, R. Leighton, M. Sands, The Feynman Lectures on Physics, vol. 3 in The Feynman Lectures on Physics (Pearson/Addison-Wesley, 1963); https://books.google.com/books?id=_6XvAAAAMAAJ.
37.H. Goldstein, C. Poole, J. Safko, Classical Mechanics (Addison Wesley, 2002); https://books.google.com/books?id=tJCuQgAACAAJ.
38.J. D. Jackson, Classical electrodynamics (Wiley, New York, NY, ed. 3, 1999); http://cdsweb.cern.ch/record/490457.
39.S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (New York: Wiley, 1972).
40.M. Schwartz, Quantum Field Theory and the Standard Model, Quantum Field Theory and the Standard Model (Cambridge Univ. Press, 2014); https://books.google.com/books?id=HbdEAgAAQBAJ.
41.J. McDermott, D. R. White, S. Luke, L. Manzoni, M. Castelli, L. Vanneschi, W. Jaskowski, K. Krawiec, R. Harper, K. De Jong, Proceedings of the 14th Annual Conference on Genetic and Evolutionary Computation (ACM, 2012), pp. 791–798.

Citation

MLA

Udrescu, S.-M., and M. Tegmark. “AI Feynman: A Physics-Inspired Method for Symbolic Regression”. Science Advances, 6:eaay2631, April 15, 2020, 2019, http://arxiv.org/abs/1905.11481v2.

APA

Udrescu, S.-M., & Tegmark, M. (2019). AI Feynman: a Physics-Inspired Method for Symbolic Regression. Science Advances, 6:eaay2631, April 15, 2020. http://arxiv.org/abs/1905.11481v2

Chicago

Udrescu, S.-M., and M. Tegmark. 2019. “AI Feynman: A Physics-Inspired Method for Symbolic Regression”. Science Advances, 6:eaay2631, April 15, 2020. http://arxiv.org/abs/1905.11481v2.

Harvard

Udrescu, S.-M. and Tegmark, M. (2019) “AI Feynman: a Physics-Inspired Method for Symbolic Regression”, Science Advances, 6:eaay2631, April 15, 2020 [Preprint]. Available at: http://arxiv.org/abs/1905.11481v2.

Vancouver

1. Udrescu S-M, Tegmark M (2019) AI Feynman: a Physics-Inspired Method for Symbolic Regression. Science Advances, 6:eaay2631, April 15, 2020

BibTeX

@article{udrescu2019feynman,
  title = {AI Feynman: a Physics-Inspired Method for Symbolic Regression},
  author = {Udrescu, Silviu-Marian and Tegmark, Max},
  year = {2019},
  journal = {Science Advances, 6:eaay2631, April 15, 2020},
  url = {http://arxiv.org/abs/1905.11481v2},
  eprint = {1905.11481}
}

Metadata:arXiv

Source Code

This paper has an official code repository available. Click below to access the source code.

View Repository

Access the Paper

This paper is available from its original source. Click below to access the PDF.

Open PDF

AI Feynman: A physics-inspired method for symbolic regression

Abstract

Topics

Keywords

Table of Contents

Knowls

Knowl 1 — AI Feynman Recursive Symbolic Regression Architecture

Knowl 2 — Automated Dimensional Analysis for Dimensionality Reduction

Knowl 3 — Neural Network-Based Continuous Function Interpolator

Knowl 4 — Neural Network-Based Symmetry Detection and Variable Elimination

Knowl 5 — Neural Network-Based Functional Separability Decomposition

Knowl 6 — Minimum Description Length Objective for Symbolic String Selection

Knowl 7 — The Feynman Symbolic Regression Database (FSReD)