[Paper Review] Deep symbolic regression for physics guided by units constraints: toward the automated discovery of physical laws

Wassim Tenachi, Rodrigo Ibata+1 more|arXiv (Cornell University)|Mar 6, 2023

Evolutionary Algorithms and Applications52 references4 citations

TL;DR

This paper introduces Φ-SO, a deep reinforcement learning-based symbolic regression framework that enforces physical unit consistency from the ground up, drastically reducing the search space and improving performance in discovering physical laws. It achieves state-of-the-art results on noisy physics benchmarks and successfully recovers complex analytical expressions in astrophysics, including those elusive to prior methods.

ABSTRACT

Frozen release of github.com/WassimTenachi/PhySO for: - The ApJ paper "Deep symbolic regression for physics guided by units constraints: toward the automated discovery of physical laws" - The NeurIPS ML4PS proceeding "Physical Symbolic Optimization" Main feature: Symbolic Regression guided by Dimensional Analysis. Version: v1.0.0

Motivation & Objective

• To address the challenge of automated discovery of physical laws from data by integrating physical unit constraints into symbolic regression.
• To reduce the combinatorial explosion of candidate equations by embedding dimensional analysis as a grammatical constraint during generation.
• To improve generalization and robustness in symbolic regression, especially under noisy data conditions, by enforcing physical consistency a priori.
• To enable the recovery of analytically complex expressions in astrophysics that are beyond the reach of existing symbolic regression methods.
• To provide a scalable, interpretable, and physically sound alternative to black-box deep learning models in data-rich physics and astrophysics applications.

Proposed method

• The framework uses a recurrent neural network (RNN) to generate symbolic expressions while tracking the cumulative physical units of each sub-expression.
• It applies deep reinforcement learning to train the RNN to maximize reward based on both accuracy and unit consistency.
• Physical units are enforced via a custom embedding layer that tracks dimensions (e.g., length, time, mass) at each step of expression construction.
• The method incorporates a 'units prior' that prunes invalid expressions during search, reducing the search space from 268 to just 6 in a toy example.
• It leverages pre-processing techniques such as additive separability (Udrescu & Tegmark, 2020) to further improve performance on complex functions.
• The algorithm is trained end-to-end using policy gradient methods, with rewards shaped to favor both low error and correct dimensional balance.

Experimental results

Research questions

• RQ1 Can enforcing physical unit consistency during symbolic expression generation significantly improve the efficiency and accuracy of symbolic regression in physics?
• RQ2 How does the integration of dimensional analysis as a grammatical constraint affect performance on noisy and complex physical datasets?
• RQ3 Can the method discover non-trivial physical laws—such as those in special relativity or cosmology—that are missed by standard symbolic regression baselines?
• RQ4 To what extent can the framework recover analytically complex expressions in astrophysics, such as radial action in galactic dynamics, where prior methods fail?
• RQ5 How robust is the method under high levels of data noise, and can it generalize across diverse physical domains?

Key findings

• Φ-SO achieves state-of-the-art performance on a benchmark of 120 equations from the Feynman Lectures and physics textbooks, outperforming 17 baselines, especially under noise levels exceeding 0.1%.
• The method maintains consistent performance even at 10% noise, demonstrating robustness where other approaches fail.
• It successfully recovers the relativistic energy equation E = γmc², a case that the standard Petersen et al. (2021a) method fails to solve.
• On the Hubble diagram of Type Ia supernovae, Φ-SO recovers the standard ΛCDM form of the Hubble parameter H(z), but also identifies simpler alternative models that fit the data better in isolation.
• For a complex galactic dynamics problem involving radial action in an isochrone potential, Φ-SO initially fails but recovers the correct expression after applying additive separability pre-processing.
• The integration of dimensional analysis as a built-in constraint reduces the search space by orders of magnitude, enabling efficient exploration of physically plausible equations.