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SynPAT: A System for Generating Synthetic Physical Theories with Data

Jonathan Lenchner, Karan Srivastava, Joao Goncalves, Mark Squillante, Lior Horesh

TL;DR

SynPAT provides a framework to generate synthetic physical theories with data, enabling rigorous benchmarking of symbolic regression and theory-discovery methods on complete theories rather than isolated equations. It combines theory generation (dimensionally consistent axioms), consequence derivation via Gröbner bases, and data generation (including ODE-based dynamics) into a unified pipeline, producing 216 axiom systems with noiseless and noisy data. Empirical_results show that leveraging a background theory (AI Hilbert) generally improves consequence discovery over purely data-driven SR methods, while intentionally flawed theories reduce performance, highlighting the dependence of SR success on theory correctness. The work delivers open-source tooling and datasets to standardize and advance benchmarking in data-with-theory-driven discovery of physical laws.

Abstract

Machine-assisted methods for discovering physical laws from background theory and data have recently emerged, promising to advance our understanding of the physical world. However, training and benchmarking these systems remains challenging: real physical theories are limited in number. To address this need, we introduce SynPAT, a system for generating synthetic physical theories with accompanying data. SynPAT produces: (i) a consistent set of axioms forming a synthetic theory, (ii) a symbolic consequence of these axioms representing the discovery target, and (iii) noisy data approximating this consequence. Crucially, to mirror historically incorrect theories (e.g., Newtonian mechanics before Special Relativity), SynPAT can also generate theories whose axioms do not strictly entail, and in fact conflict with, the observed consequence, requiring a correction to the assumed axioms to bridge the gap. We detail SynPAT's methodology and benchmark several open-source symbolic regression systems on our generated theories and data.

SynPAT: A System for Generating Synthetic Physical Theories with Data

TL;DR

SynPAT provides a framework to generate synthetic physical theories with data, enabling rigorous benchmarking of symbolic regression and theory-discovery methods on complete theories rather than isolated equations. It combines theory generation (dimensionally consistent axioms), consequence derivation via Gröbner bases, and data generation (including ODE-based dynamics) into a unified pipeline, producing 216 axiom systems with noiseless and noisy data. Empirical_results show that leveraging a background theory (AI Hilbert) generally improves consequence discovery over purely data-driven SR methods, while intentionally flawed theories reduce performance, highlighting the dependence of SR success on theory correctness. The work delivers open-source tooling and datasets to standardize and advance benchmarking in data-with-theory-driven discovery of physical laws.

Abstract

Machine-assisted methods for discovering physical laws from background theory and data have recently emerged, promising to advance our understanding of the physical world. However, training and benchmarking these systems remains challenging: real physical theories are limited in number. To address this need, we introduce SynPAT, a system for generating synthetic physical theories with accompanying data. SynPAT produces: (i) a consistent set of axioms forming a synthetic theory, (ii) a symbolic consequence of these axioms representing the discovery target, and (iii) noisy data approximating this consequence. Crucially, to mirror historically incorrect theories (e.g., Newtonian mechanics before Special Relativity), SynPAT can also generate theories whose axioms do not strictly entail, and in fact conflict with, the observed consequence, requiring a correction to the assumed axioms to bridge the gap. We detail SynPAT's methodology and benchmark several open-source symbolic regression systems on our generated theories and data.
Paper Structure (30 sections, 15 equations, 2 figures, 8 tables, 4 algorithms)

This paper contains 30 sections, 15 equations, 2 figures, 8 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Data-driven Approaches
  4. Data-and-Theory Driven Approaches
  5. Benchmarking Data Sets
  6. Theory, Consequence, and Data Generation
  7. Theory Generation
  8. Consequence Generation and Consistency Checking
  9. Data Generation for Consequences
  10. Generating Theories with Mismatching Data
  11. Benchmarking Consequence Discovery
  12. Code and Generated Data Set Availability
  13. Conclusions and Future Work
  14. Theory Generation Algorithms
  15. Term Generation
  16. ...and 15 more sections

Figures (2)

  • Figure 1: The pipeline for generating a theory, then a consequence, and then data for the consequence.
  • Figure 2: The Pipeline for Generating Theories Together With Mismatching Data.