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This is a Quantitative Study study.

October 20, 2025Open Access

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

ASAnna ScampicchioUniversity of Padua LTLeonardo F. TosoColumbia University RRRahel RickenbachETH Zurich

Key Points

Learning rates improve significantly when integrating prior domain knowledge in machine learning.
Incorporation of physics-informed regularization changes the excess risk dynamics within the learning process.
Results demonstrate a transition from Sobolev minimax rates to optimal i.i.d. rates without sample-size issues.
The findings highlight the benefits of utilizing physical knowledge in combating data dependence challenges.

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.

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