Physics-informed machine learning improves function prediction with noisy data
Fast Learning Rates for Physics-Informed Kernel Methods
Machine Learning
Summary
Predicting how physical systems behave can be tricky when data is noisy or limited. The authors study a method that uses both value measurements and information from physical laws expressed as differential equations. They show how the amount of physical information available can speed up learning rates, improving predictions. They also explain under what conditions the extra physical data gives the most benefit and provide mathematical guarantees and examples.
What this means in practice
- •For computational physicists: Improve accuracy and speed of simulations by integrating noisy measurement data with physical constraints for predictive models.
- •For engineering modelers: Enhance system models by combining sensor data and known physics to better predict states governed by differential equations.
Authors
Luc Brogat-Motte, Joachim Bona-Pellissier, Giacomo Meanti, Lorenzo Rosasco
Abstract
In physics-informed machine learning, a target function $u^*$ is learned from noisy value observations $y_i=u^*(x_i)+ \varepsilon_i$, together with differential information, given either by noisy observations $d_j=(Du^*)(z_j)+ξ_j$ or by a known physical constraint $Du^*=v$. We consider the setting where $D$ is a linear differential operator and analyze a physics-informed kernel estimator $\hat u$ combining $n$ value observations and $m$ differential observations. In this context, we ask how much can differential information improve predictions, and how does this improvement depend quantitatively on $n$, $m$, and $D$. We prove finite-sample bounds, supported by numerical simulations, revealing a two-regime structure for the prediction error. When $m$ is limited, the rate depends jointly on $n$ and $m$; when $m$ exceeds a problem-dependent threshold, the rate saturates and matches the oracle rate obtained when the perfect constraint $D \hat u = Du^*$ is imposed. Examples are discussed for Sobolev spaces which are reproducing kernel Hilbert spaces and include partial Laplacian constraints on the torus and gradient observations on bounded domains. These examples illustrate the range of possible learning rate improvements --- from the standard nonparametric $n^{-1/4}$ to the parametric rate $n^{-1/2}$. Finally, we derive physically consistent rates in a stronger norm that jointly controls the errors in $\hat u$ and $D\hat u$.