What question did this study set out to answer?

This research aims to enhance the automation of physical law discovery using a structured framework called PIR.

May 10, 2026Open Access

Physics Intermediate Representation (PIR): A Structured Framework for Automated Physical Law Discovery from Data — Version 3.2

Key Points

This research aims to enhance the automation of physical law discovery using a structured framework called PIR.
Automated discovery framework combining symbolic search and soft filtering techniques.
Evaluation against gplearn using identical datasets and success criteria across 20 tasks.
Version 3.2 updates include refined consistency and documented reproducibility scripts.
PIR achieved 100% mean discovery rate (DR) versus gplearn's 50% DR.
PIR scored 0.838 on dimensional consistency (DCS) compared to gplearn's 0.672.
PIR outperformed gplearn on 14 out of 20 evaluated tasks.

Abstract

PIR is a research framework for automated physical law discovery combining physics-aware symbolic search, soft dimensional consistency filtering, residual-driven refinement, sparse model selection, and a hybrid loss (MSE + Optimal Transport, alpha=0. 7, beta=0. 3). It produces typed scientific output objects (Equation, PIRDifferentialEquation, PIRHamiltonian, PIRLagrangian, PIRSystem) decoupling data-level discovery from knowledge-level reasoning. Version 3. 2 (May 2026) additions over v3. 1: 1. Section 9. 3 — Head-to-head comparison with gplearn 0. 4. 3 on the full PIR-Bench-20 under an identical evaluation protocol (same datasets, same seeds, same success criterion, same DCS formula). Result: PIR 100% mean DR / DCS 0. 838 vs gplearn 50% / 0. 672. PIR wins on 14/20 tasks, tied on 6/20, gplearn wins on 0/20. Eight catastrophic gplearn failures (0% DR) on structurally demanding tasks (Lagrangians, FK, Jacobian, pendulum, inverse-square). Six perfect ties confirm the benchmark contains genuinely solvable tasks. This retires the standard SR baseline gap identified in v3. 1 critique. 2. Section 11 Conclusion — Stale "noise ≤ 0. 05" parenthetical removed; now consistent with Abstract, Table 1, Section 10. 4, and Appendix A (all of which correctly reported sigma in 0. 01, 0. 05, 0. 10, 0. 20). 3. PySR head-to-head and component ablation are documented as deferred to v3. 3 (Python 3. 13 / PythonCall. jl C-API incompatibility for PySR; benchmark-runner cache requires invalidation logic for component ablation). The Langevin-step ablation already published in v3. 1 is unaffected. All Table 1 numbers, feature comparisons, and design constants are unchanged from v3. 1. Bundle contents: - PIRArchitectureᵥ3₂. pdf — main paper (13 pages) - PIRArchitectureᵥ3₂. tex — LaTeX source- comparisonₜable. md,. csv,. json — gplearn comparison artifacts- rungplearnₕeadtoheadᵥ2. py — reproducibility script- aggregateₚysrcomparison. py — aggregation script Reproducibility: the gplearn comparison reruns end-to-end via the two included Python scripts on the public PIR codebase. Per-run JSON artifacts and provenance trail are preserved.

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