Physics-informed learning under mixing: How physical knowledge speeds up learning

2025-09-29

AI Summary

In physics-informed machine learning, conventional Sobolev minimax rates are constrained by data dependence structures, limiting achievable learning rates under non-i.i.d. settings. Method: We propose a physics-regularized empirical risk minimization framework that explicitly incorporates accurate physical priors into the loss function, enabling improved learning rates under dependent data. Contribution/Results: We theoretically establish that when the embedded physical model exactly matches the underlying governing law, the excess risk converges at the optimal i.i.d. rate—accelerating beyond standard Sobolev rates—without compromising sample efficiency. This is the first result to quantify the convergence-rate acceleration induced by physical priors in non-i.i.d. regimes. Our analysis provides a novel generalization-theoretic paradigm for physics-informed learning, bridging physical modeling fidelity with statistical learning guarantees under data dependence.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Security and privacy of machine learning and AI applications

Abstract

A major challenge in physics-informed machine learning is to understand how the incorporation of prior domain knowledge affects learning rates when data are dependent. Focusing on empirical risk minimization with physics-informed regularization, we derive complexity-dependent bounds on the excess risk in probability and in expectation. We prove that, when the physical prior information is aligned, the learning rate improves from the (slow) Sobolev minimax rate to the (fast) optimal i.i.d. one without any sample-size deflation due to data dependence.

Problem

• Physics-informed machine learning with dependent data
• Improving learning rates through physical knowledge
• Achieving optimal i.i.d. rates with aligned priors

Innovation

• Physics-informed regularization accelerates learning rates
• Aligned physical priors enable optimal i.i.d. convergence
• Overcomes slow Sobolev rates under dependent data conditions