*Result*: A gaussian process framework for solving forward and inverse problems involving nonlinear partial differential equations.

*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 been substantially advanced over the past few years, their performance is still very sensitive to the network's architecture, loss function, and optimization settings. 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 framework based on Gaussian processes (GPs) whose mean functions are parameterized via deep NNs. The resulting PIML model, abbreviated as NN-CoRes , can solve PDE systems without any labeled data inside the domain and is particularly attractive because it (1) naturally satisfies the boundary and initial conditions of a PDE system in arbitrary domains, and (2) can leverage any differentiable function approximator, e.g., deep NN architectures, in its mean function. To ensure computational efficiency and robustness, we devise a modular approach for NN-CoRes to separately estimate the parameters of the kernel and the deep NN. Our studies indicate that NN-CoRes consistently outperforms competing methods and considerably decreases 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. [ABSTRACT FROM AUTHOR]

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