Data-driven Progressive Discovery of Physical Laws

2026-03-13

AI Summary

Traditional symbolic regression methods often produce overly complex expressions lacking physical interpretability and generalizability, as they neglect the incremental nature of scientific discovery—from simplicity to complexity. This work proposes a Chain-based Symbolic Regression (CoSR) framework that, for the first time, integrates the principle of progressive scientific discovery into symbolic regression. CoSR constructs hierarchical knowledge chains composed of physically meaningful units, progressively combining them to emulate the historical process of uncovering physical laws—such as the derivation of Newton’s law of universal gravitation from Kepler’s third law. By synergistically blending data-driven learning with physics-informed modeling, the approach successfully refines classical scaling laws in complex systems including turbulent convection, viscous pipe flow, and laser–metal interactions, and further uncovers novel scaling relationships in aerodynamic coefficient prediction for aircraft.

Technology Category

• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Machine Learning: Neuro-Symbolic Learning
• Computer Vision: Visual Reasoning & Symbolic Representations

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Symbolic regression is a powerful tool for knowledge discovery, enabling the extraction of interpretable mathematical expressions directly from data. However, conventional symbolic discovery typically follows an end-to-end, "one-step" process, which often generates lengthy and physically meaningless expressions when dealing with real physical systems, leading to poor model generalization. This limitation fundamentally stems from its deviation from the basic path of scientific discovery: physical laws do not exist in a single form but follow a hierarchical and progressive pattern from simplicity to complexity. Motivated by this principle, we propose Chain of Symbolic Regression (CoSR), a novel framework that models the discovery of physical laws as a chain of symbolic knowledge. This knowledge chain is formed by progressively combining multiple knowledge units with clear physical meanings along a specific logic, ultimately enabling the precise discovery of the underlying physical laws from data. CoSR fully recapitulates the progressive discovery path from Kepler's third law to the law of universal gravitation in classical mechanics, and is applied to three types of problems: turbulent Rayleigh-Benard convection, viscous flows in a circular pipe, and laser-metal interaction, demonstrating its ability to improve classical scaling theories. Finally, CoSR showcases its capability to discover new knowledge in the complex engineering problem of aerodynamic coefficients scaling for different aircraft.

Problem

• symbolic regression
• physical laws
• scientific discovery
• model generalization
• hierarchical discovery

Innovation

• symbolic regression
• progressive discovery
• physical laws
• knowledge chain
• interpretable AI