Physics-informed machine learning as a kernel method
This work addresses the challenge of integrating physical interpretability into machine learning for regression tasks, offering a theoretical framework that could benefit fields like computational physics and engineering, though it is incremental as it builds on existing kernel methods.
The paper tackles the problem of combining data-driven machine learning with physical models by regularizing empirical risk with a partial differential equation, proving that for linear differential priors the problem reduces to kernel regression and achieving convergence rates at least at the Sobolev minimax rate, with potential for faster rates depending on physical error.
Physics-informed machine learning combines the expressiveness of data-based approaches with the interpretability of physical models. In this context, we consider a general regression problem where the empirical risk is regularized by a partial differential equation that quantifies the physical inconsistency. We prove that for linear differential priors, the problem can be formulated as a kernel regression task. Taking advantage of kernel theory, we derive convergence rates for the minimizer of the regularized risk and show that it converges at least at the Sobolev minimax rate. However, faster rates can be achieved, depending on the physical error. This principle is illustrated with a one-dimensional example, supporting the claim that regularizing the empirical risk with physical information can be beneficial to the statistical performance of estimators.