Data-driven discovery of governing differential equations across physical systems

2026-06-08

AI Summary

This work addresses the challenge of automatically discovering governing differential equations from data when the underlying physical mechanisms are unknown, proposing an analysis framework centered on discoverability. By constructing a two-dimensional phase diagram defined by structural and coefficient complexity, the problem space is systematically categorized. The authors introduce a unified Representation–Evaluation–Optimization (REO) framework that characterizes the core workflow of equation discovery, transcending the limitations of specific algorithms. This approach elucidates fundamental principles governing discoverability and clarifies the applicability boundaries of techniques such as sparse regression, symbolic regression, and machine learning across diverse scenarios. Consequently, it advances data-driven modeling beyond mere equation recovery toward a scientific discovery paradigm capable of theoretical refinement and novel concept generation.

Abstract

Differential equations play a critical role in scientific discovery because they provide a mathematical framework to describe the behaviour of physical phenomena. As a promising alternative to traditional first principles, data-driven differential equation discovery has attracted increasing attention for its ability to infer governing laws directly from experimental or simulated data, especially when the underlying physics is unclear. However, the field has expanded rapidly along diverse methodological directions, particularly with the emergence of AI-based approaches, and still lacks a clear organizing perspective. In this Review, we propose a problem-oriented perspective on data-driven differential equation discovery. We first introduce a two-dimensional phase diagram of equation discoverability, where discovery problems are organized according to structural complexity and coefficient complexity. This phase diagram shows how the field has moved from the discovery of sparse equations with simple coefficients toward more complex governing laws with richer structures and more flexible parameterizations. It also clarifies why different methodological families succeed or fail in different problem settings. We then present the representation-evaluation-optimization (REO) framework as a fundamental abstraction of the discovery process. By identifying the core problems of equation discovery that persist across algorithmic variations, REO shifts the discussion from individual algorithms to the fundamental principles that determine discoverability. We connect these perspectives to applications across physics and adjacent sciences, and argue that the next challenge is not merely recovering equations, but using them to revise existing theories, distil mechanisms and form new scientific concepts.

Problem

• differential equation discovery
• data-driven
• scientific discovery
• equation discoverability
• physical systems

Innovation

• data-driven discovery
• differential equations
• phase diagram of discoverability
• representation-evaluation-optimization (REO) framework
• scientific machine learning