Feynman Symbolic Regression Database

• FSReD is a curated database of symbolic regression tasks derived from canonical physics, featuring 100 primary equations and bonus formulas from classical texts.
• It organizes machine-readable data with CSV files for input–output pairs, SI unit vectors, LaTeX and sympy expressions, and detailed JSON metadata.
• FSReD enables benchmarking via exact symbolic recovery, numerical accuracy, noise robustness, and complexity–accuracy Pareto analysis to drive scientific discovery.

The Feynman Symbolic Regression Database (FSReD) is a standardized, meticulously curated repository of symbolic regression tasks derived from canonical physical laws, designed as a benchmark for evaluating and developing algorithms in data-driven scientific discovery. Established originally in conjunction with the AI Feynman project, FSReD comprises exact, non-trivial analytic formulas sampled from the three volumes of the Feynman Lectures on Physics and several classical physics texts. It provides machine-readable datasets associating symbolic expressions, units, metadata, and large-scale numerical input/output samples, enabling rigorous quantitative assessment and reproducibility for symbolic regression methods across the domains of physics, machine learning, and scientific computing (Udrescu et al., 2019).

1. Construction, Contents, and Structure

FSReD was assembled by extracting 100 “mystery” equations from the Feynman Lectures on Physics, supplemented by 20 “bonus” formulas from Goldstein, Jackson, Weinberg, and Schwartz. The selection emphasizes algebraic relationships, excluding explicit derivatives or integrals, with equations involving up to nine independent variables and incorporating a vocabulary of elementary operations and functions: addition, subtraction, multiplication, division, square root, exponentials, logarithms, trigonometric, hyperbolic, and inverse trigonometric functions. Each equation is parsed into a symbolic expression language (SEL) using reverse-Polish notation to ensure machine-level uniqueness (e.g., “mv*1vv*cc*/-R/” for $m v/\sqrt{1-v^2/c^2}$).

Variable definitions for each entry include physical meaning and a SI unit vector (five exponents over $$[\mathrm{m},\mathrm{s},\mathrm{kg},\mathrm{K},\mathrm{V}]$$). For example, mass ($m, m_1, m_2$) maps to (0, 0, 1, 0, 0), velocity ($v, u, w, c$) to (1, –1, 0, 0, 0), and force ($F$) to (1, –2, 1, 0, 0) (Udrescu et al., 2019).

Data and metadata are organized as follows:

• /data/: CSV files encoding sampled input–output tables covering the input domain (usually $[1,5]^{n}$) with $10^5$–$10^6$ points per equation, avoiding singular domains.
• /units/: per-variable SI exponent vectors in CSV format.
• /eq/: ground-truth symbolic formulas in LaTeX and sympy.
• /metadata/: JSON index tracking ID, input dimensionality, invoked physics-inspired decomposition methods, minimum required data points, solution times, and recommended noise tolerance.

Each equation is referenced by its Feynman volume-section (e.g., "I.6.20") and documented in index tables with associated metadata.

2. Data Generation, Preprocessing, and Units

The database numerically instantiates formulas by sampling each input variable uniformly and independently within a user-specified physical range, adjusted to avoid mathematical singularities. For each equation, $10^5$ input–output pairs $(x_1,\dots,x_n, y)$ are generated per default, supporting large-scale statistical testing and cross-validation.

Physical units are encoded per-variable as integer exponent vectors, enabling dimensional analysis and the detection of consistent unit relations. Each sampling range per variable is stored in the JSON metadata, supporting both uniform and log-uniform strategies.

Ground-truth symbolic expressions are provided redundantly via raw LaTeX and sympy-parseable strings, ensuring compatibility with major symbolic regression, algebra, and numerical libraries (Udrescu et al., 2019).

3. Usage Protocols and Benchmarking Methodology

FSReD's primary purpose is as a challenge suite for benchmarking symbolic regression algorithms in terms of:

• Exact symbolic recovery: A model must reconstruct the ground-truth formula $$[\mathrm{m},\mathrm{s},\mathrm{kg},\mathrm{K},\mathrm{V}]$$0, with symbolic equivalence established via computer algebra (e.g., $$[\mathrm{m},\mathrm{s},\mathrm{kg},\mathrm{K},\mathrm{V}]$$1).
• Numerical accuracy: Assessed via root-mean-square error (RMSE) on a held-out subset of data,

$$[\mathrm{m},\mathrm{s},\mathrm{kg},\mathrm{K},\mathrm{V}]$$2

• Noise robustness: Gaussian noise is added to $$[\mathrm{m},\mathrm{s},\mathrm{kg},\mathrm{K},\mathrm{V}]$$3 with standard deviation $$[\mathrm{m},\mathrm{s},\mathrm{kg},\mathrm{K},\mathrm{V}]$$4 up to a prescribed noise limit per equation.
• Complexity–accuracy Pareto analysis: Candidate solutions are ranked using description length versus error tradeoffs, inspired by MDL (minimum description length) principles, yielding Pareto frontiers over model space (Udrescu et al., 2020).

The canonical FSReD protocol recommends an 80/20 train/test split, with cross-validation for low-data regimes ($$[\mathrm{m},\mathrm{s},\mathrm{kg},\mathrm{K},\mathrm{V}]$$5), and mandates that no derivatives or integrals be used in candidate expressions.

4. Applications and Evolution in Symbolic Regression Research

FSReD rapidly established itself as the de facto standard for assessing symbolic regression systems in physics, including both conventional genetic programming and neural methods. It has been used to:

• Quantitatively compare solution rates and time-to-recovery for standard and GPU-accelerated GP frameworks (e.g., Beagle (Haut et al., 10 Mar 2026)), deep learning–guided pipelines (KAN-SR (Bühler et al., 12 Sep 2025)), and semantic LLM-driven evolutionary agents (IdeaSearchFitter (Song et al., 9 Oct 2025)).
• Drive the development of physics-informed enhancements such as on-the-fly dimensional analysis, modularity detection, and feature engineering by domain-specific variable precomputation (Bruneton, 24 Mar 2025).
• Enable robust validation of noise-tolerant symbolic regression, variable selection in the presence of irrelevant (“dummy”) features, and meaningful model complexity assessment (Matsubara et al., 2022).
• Serve as an analytic baseline in hybrid strategies for deducing symbolic structure from high-precision numerical samples (e.g., for Feynman integral regression (Barrera et al., 23 Jul 2025)).

FSReD's comprehensive metadata structure and public availability facilitate reproducibility, algorithmic benchmarking, and method ablation studies.

5. Extensions: SRSD-Feynman, Dummy Variables, and Advanced Metrics

FSReD has been expanded by several groups, most notably as part of the “SRSD-Feynman” extension (Matsubara et al., 2022):

• The core set was recalibrated to 120 base problems, with variable sampling ranges adjusted to realistic physical SI units and distributions.
• Each core problem is mirrored by variants with $$[\mathrm{m},\mathrm{s},\mathrm{kg},\mathrm{K},\mathrm{V}]$$6–$$[\mathrm{m},\mathrm{s},\mathrm{kg},\mathrm{K},\mathrm{V}]$$7 dummy (irrelevant) inputs sampled over wide log-uniform ranges, explicitly challenging a method’s feature selection and variable significance capacity.
• Data is provided in modern Hugging Face datasets format (Apache Arrow + JSON), with formal specification of formula complexity (tree node count), domain ranges, and constant values.
• Standard splits are train (8000), validation (1000), and test (1000) per problem.
• Normalized Edit Distance (NED) is introduced as a structural metric, defined as the normalized tree edit distance between predicted and true symbolic expressions after canonicalization and simplification, providing a continuous, human-correlated analog to binary solution rates and $$[\mathrm{m},\mathrm{s},\mathrm{kg},\mathrm{K},\mathrm{V}]$$8 (Matsubara et al., 2022).

6. Algorithmic Performance and Impact

FSReD supports rigorous quantitative comparison across diverse symbolic regression algorithms. Key findings include:

• Early genetic programming systems reached 71/100 exact formula recovery; AI Feynman solved all 100 in the baseline and improved recovery from 15% to 90% in the extended “hard” set (Udrescu et al., 2019).
• GPU-accelerated frameworks (Beagle) demonstrate scalable improvements in throughput: up to 84/100 problems solved in 30 min wall-clock time using correlation-based fitness, 10–15% ahead of leading CPU-only approaches (Haut et al., 10 Mar 2026).
• KAN-SR leverages Kolmogorov–Arnold decomposition and symmetry-based divide-and-conquer, attaining 93% success on “easy” problems, 60% on “medium”, and breaking double-digit recovery on “hard” classes, significantly outperforming all prior methods (Bühler et al., 12 Sep 2025).
• LLM-driven semantic evolutionary agents (IdeaSearchFitter) achieve >80% exact recovery even under high noise (up to 0.1 times the signal), exceeding classic and neural symbolic regression methods (Song et al., 9 Oct 2025).
• QDSR, integrating quality-diversity grids, on-the-fly dimensional analysis, and precomputed physics-derived features, reaches 91.6% noiseless exact recovery and 79.4% at 10% noise—prevailing over all preceding SR paradigms (Bruneton, 24 Mar 2025).
• Enhanced feature sets and unit-aware variable engineering yield a beneficial trade-off: increased vocabulary expressiveness shortens minimal formula depth, dramatically shrinking the effective search space (Bruneton, 24 Mar 2025).

7. Access, Version Control, and Future Directions

FSReD v1.0 is distributed as a 6.5 GB multi-directory archive, including data, units, expressions, and metadata. Download is available from https://space.mit.edu/home/tegmark/aifeynman.html with matching MD5 checksums and version history (Udrescu et al., 2019). SRSD-Feynman extensions are published at https://github.com/omron-sinicx/srsd-benchmark and Hugging Face repositories.

A recommended FSReD-style schema for future entries includes canonical LaTeX/sympy expressions, complete unit metadata, SI-appropriate sampling regimes, support for analytic, neural, and integral regression domains, and extensible cross-referencing for multi-loop or non-polylogarithmic physical function classes (Barrera et al., 23 Jul 2025). Emerging directions involve integrating lattice-reduction methods for analytic regression, modularity-based Pareto optimization, and further expansion of coverage to quantum and chaotic dynamical regimes.

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FSReD, through its meticulous curation, physics-aware variable annotation, and expansive coverage, has become the consensus benchmark underpinning state-of-the-art research in symbolic regression for scientific discovery, providing an indispensable infrastructure for algorithm evaluation, comparison, and the principled design of new methods grounded in domain knowledge (Udrescu et al., 2019, Udrescu et al., 2020, Matsubara et al., 2022, Bruneton, 24 Mar 2025, Bühler et al., 12 Sep 2025, Song et al., 9 Oct 2025, Haut et al., 10 Mar 2026, Barrera et al., 23 Jul 2025).