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

arXiv: https://arxiv.org/abs/2509.24801 · HTML: https://mlsanity.com/paper/2509.24801/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Sep 2025.

## Built on by (most cited first, 3 of 3)

- [Complexity Dependent Error Rates for Physics-informed Statistical Learning via the Small-ball Method](https://mlsanity.com/paper/2510.23149/) (2025)
- [A PAC-Bayesian View of Generalisation for Physics-Informed Machine Learning](https://mlsanity.com/paper/2605.26341/) (2026)
- [Learning switched non-linear dynamical systems from a single trajectory](https://mlsanity.com/paper/2607.23502/) (2026)

## Abstract (arXiv)

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.