A Gaussian Process Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations
This work addresses a key bottleneck in physics-informed machine learning for solving PDEs, offering a more robust and efficient approach that could renew interest in kernel methods for this domain.
The paper tackles the sensitivity of physics-informed machine learning models to neural network architecture and loss function by introducing kernel-weighted Corrective Residuals (CoRes), which integrates kernel methods with deep neural networks to solve nonlinear PDEs, resulting in a framework that consistently outperforms competing methods on benchmark problems with simplified training and negligible inference cost increases.
Physics-informed machine learning (PIML) has emerged as a promising alternative to conventional numerical methods for solving partial differential equations (PDEs). PIML models are increasingly built via deep neural networks (NNs) whose architecture and training process are designed such that the network satisfies the PDE system. While such PIML models have substantially advanced over the past few years, their performance is still very sensitive to the NN's architecture and loss function. Motivated by this limitation, we introduce kernel-weighted Corrective Residuals (CoRes) to integrate the strengths of kernel methods and deep NNs for solving nonlinear PDE systems. To achieve this integration, we design a modular and robust framework which consistently outperforms competing methods in solving a broad range of benchmark problems. This performance improvement has a theoretical justification and is particularly attractive since we simplify the training process while negligibly increasing the inference costs. Additionally, our studies on solving multiple PDEs indicate that kernel-weighted CoRes considerably decrease the sensitivity of NNs to factors such as random initialization, architecture type, and choice of optimizer. We believe our findings have the potential to spark a renewed interest in leveraging kernel methods for solving PDEs.