Understanding Generalization in Physics Informed Models through Affine Variety Dimensions

2025-01-31

AI Summary

This study investigates the generalization mechanisms of physics-informed machine learning models. Addressing the theoretical limitation that existing frameworks fail to characterize how physical constraints fundamentally enhance generalization, we propose a novel analytical framework grounded in algebraic geometry. Specifically, we establish, for the first time, a quantitative relationship between generalization error and the dimension of the affine variety defined by the physical constraints. We prove that, for linear regressors embedded with differential equation structures, generalization performance is governed by this geometric dimension—not by classical parameter count. Methodologically, our approach integrates algebraic geometry (affine variety dimension analysis), physics-based modeling, and numerical approximation, and introduces a computationally tractable algorithm for estimating the variety dimension. Experiments demonstrate that this dimension accurately predicts generalization behavior across diverse physical constraints, providing a unified, interpretable, and computationally feasible theoretical foundation for both linear and nonlinear physics-informed learning.

Technology Category

• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Computer Vision: Low Level & Physics-based Vision

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

In recent years, physics-informed machine learning has gained significant attention for its ability to enhance statistical performance and sample efficiency by integrating physical structures into machine learning models. These structures, such as differential equations, conservation laws, and symmetries, serve as inductive biases that can improve the generalization capacity of the hybrid model. However, the mechanisms by which these physical structures enhance generalization capacity are not fully understood, limiting the ability to guarantee the performance of the models. In this study, we show that the generalization performance of linear regressors incorporating differential equation structures is determined by the dimension of the associated affine variety, rather than the number of parameters. This finding enables a unified analysis of various equations, including nonlinear ones. We introduce a method to approximate the dimension of the affine variety and provide experimental evidence to validate our theoretical insights.

Problem

• Physical Information
• Machine Learning Models
• Performance Enhancement

Innovation

• Affine Manifold Dimension
• Generalization Ability
• Nonlinear Equation