What question did this study set out to answer?

This framework aims to enhance automated discovery of physical laws from data using a structured approach.

May 17, 2026Open Access

Physics Intermediate Representation (PIR): A Structured Framework for Automated Physical Law Discovery from Data — Version 3.3

Key Points

This framework aims to enhance automated discovery of physical laws from data using a structured approach.
Combined physics-aware symbolic search with soft dimensional consistency filtering.
Evaluated on a 44-equation subset of the Feynman Symbolic Regression Database using a held-out numerical-equivalence protocol.
Targeted improvements identified for future versions to enhance recovery of certain equations.
Achieved median relative error of < 5% on a disjoint 500-point test set.
Successfully recovered 27.3% of the tasks with 11 out of 44 equations fully recovered.
Highlighted failures in recovering certain single-variable and complex multi-variable equations.

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. 3 (May 2026) — External-benchmark evaluation upgrade. KEY NEW RESULT (Section 9. 4): PIR evaluated on a 44-equation Tier A subset of the Feynman Symbolic Regression Database (Udrescu result will be published in v3. 3. 1 or v3. 4 regardless of outcome. UNCHANGED FROM v3. 2: - PIR-Bench-20 results (Table 1, mean DCS 0. 840) - gplearn head-to-head (Section 9. 3, PIR 100% / gplearn 50% mean DR; 14W/6T/0L) - All design constants (alpha=0. 7, beta=0. 3, gamma=0. 2, Langevin T=500) DEFERRED TO v3. 4 OR LATER: - Tier B and Tier C Feynman evaluations- Formal SRBench submission (sklearn-compatible PIR algorithm) - PySR head-to-head (Python 3. 13 / PythonCall. jl upstream issue) - Full PIR component ablation Bundle contents: - PIRArchitectureᵥ3₃. pdf — main paper (14 pages) - PIRArchitectureᵥ3₃. tex — LaTeX source- feynmanₗoader. py — manifest-driven Feynman dataset generator- verifyfeynmanₜierA. py — held-out structural verifier (SRBench-style) - aggregatefeynmanₜierA. py — per-task results aggregator- tierₐᵥerified. md /. csv /. json — verified per-task results Reproducibility: the Feynman Tier A evaluation reruns end-to-end from the PIR repository using the included scripts. The Feynman manifest (FeynmanEquations. csv) is sourced from Udrescu & Tegmark's MIT page and not redistributed.

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