# Discovering physical laws with parallel symbolic enumeration

arXiv: https://arxiv.org/abs/2407.04405 · HTML: https://mlsanity.com/paper/2407.04405/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jul 2024.

## Built on by (most cited first, 9 of 9)

- [The Agentic Researcher: A Practical Guide to AI-Assisted Research in Mathematics and Machine Learning](https://mlsanity.com/paper/2603.15914/) (2026)
- [Data-driven discovery of governing differential equations across physical systems](https://mlsanity.com/paper/2606.09638/) (2026)
- [EGG-SR: Embedding Symbolic Equivalence into Symbolic Regression via Equality Graph](https://mlsanity.com/paper/2511.05849/) (2025)
- [Interpretable Analytic Calabi-Yau Metrics via Symbolic Distillation](https://mlsanity.com/paper/2602.07834/) (2026)
- [Discovering Latent Response Laws in Forced Physical Systems](https://mlsanity.com/paper/2607.09801/) (2026)
- [One-shot learning for the complex dynamical behaviors of weakly nonlinear forced oscillators](https://mlsanity.com/paper/2604.15181/) (2026)
- [EditSR: Enhancing Neural Symbolic Regression via Edit-based Rectification](https://mlsanity.com/paper/2606.07915/) (2026)
- [AutoSR: Automatic Symbolic Regression by Searching Research States](https://mlsanity.com/paper/2608.16876/) (2026)
- [Hypothesize, Evaluate, Refine: A Scientific Agent for PDE Discovery with Unknown Spatial Coefficient Fields](https://mlsanity.com/paper/2608.27475/) (2026)

## Abstract (arXiv)

Symbolic regression plays a crucial role in modern scientific research thanks to its capability of discovering concise and interpretable mathematical expressions from data. A key challenge lies in the search for parsimonious and generalizable mathematical formulas, in an infinite search space, while intending to fit the training data. Existing algorithms have faced a critical bottleneck of accuracy and efficiency over a decade when handling problems of complexity, which essentially hinders the pace of applying symbolic regression for scientific exploration across interdisciplinary domains. To this end, we introduce parallel symbolic enumeration (PSE) to efficiently distill generic mathematical expressions from limited data. Experiments show that PSE achieves higher accuracy and faster computation compared to the state-of-the-art baseline algorithms across over 200 synthetic and experimental problem sets (e.g., improving the recovery accuracy by up to 99% and reducing runtime by an order of magnitude). PSE represents an advance in accurate and efficient data-driven discovery of symbolic, interpretable models (e.g., underlying physical laws), and improves the scalability of symbolic learning.