Physics informed kriging method improves partial differential equation solving
PDE-Residual-Informed Kriging
Machine Learning
Summary
Solving complex math problems called partial differential equations (PDEs) helps us understand many physical systems. This paper presents a new way to train a special kind of model, called a Gaussian process, that knows about these PDE rules everywhere in a flexible yet mathematically clean way. The authors show their method can find the best approximation to the solution by minimizing the total error over the whole problem area. Their approach also provides clear measures of how good the solution is and how certain the model is about it. They prove that the method works well and demonstrate it with examples.
What this means in practice
- •For engineering simulation teams: Improve numerical solution accuracy of physical models by incorporating global PDE residuals into probabilistic frameworks.
- •For environmental modeling groups: Enhance prediction of environmental phenomena by using PDE-informed kernels that quantify solution uncertainty and error.
Authors
Yu Tan, Shichang Du, Yongxiang Li
Abstract
Physics-informed Gaussian process (GP) methods provide a probabilistic framework for PDE solving. Existing approaches incorporate PDE information in different ways: pointwise conditioning uses the governing equation only at finitely many selected locations, globally constrained GPs restrict the admissible functions to satisfy the PDE exactly, and residual-penalized methods control the PDE mismatch over the whole domain through an additional penalty weight. What is still missing is a GP formulation that directly incorporates global PDE-residual information of the entire domain into its covariance structure, while retaining model flexibility and standard GP conditioning without a separate residual-weight parameter. To solve this issue, this study proposes PDE-residual-informed Kriging (PRIK) based on a PDE-residual kernel. The PDE-residual kernel is constructed by equipping a kernel-section space with an inner product defined by the $L^{2}$ inner product of PDE residuals. This construction makes PRIK yield the posterior mean with minimum integrated squared PDE residual. It also provides direct statistical interpretations: the variance-scale estimate equals the normalized PDE-residual energy (i.e., the integrated squared PDE residual of the posterior mean divided by the number of labeled observations), and the profile likelihood balances this residual energy against model complexity. Under the stated regularity and stability conditions, we establish $H^{2}$-quasi-optimality and convergence rates for the posterior mean, together with a nonasymptotic bound for its PDE-residual norm. We further show that the normalized posterior kernel converges to a variational Green-type kernel and derive a uniform nonasymptotic contraction bound for the posterior variance. Numerical studies provide empirical evidence consistent with the theory and demonstrate the PDE-solving capability of PRIK.