PHYSICS-INFORMED KERNEL FUNCTION-BASED NEURAL NETWORKS AND NEURAL OPERATORS

This paper addresses the issues of insufficient physical priors, limited training stability, and restricted generalization capability in traditional neural network models for solving partial differential equations (PDEs). To this end, we investigate the construction of physics-informed kernel functions and their integration with neural networks. First, we analyze the characteristics and limitations of traditional numerical methods, physics-informed neural networks, and neural operators in solving high-dimensional, complex-geometry, and nonlinear problems. Subsequently, we introduce approaches for constructing physics-informed kernel functions that incorporate either the governing PDEs or boundary condition information. Building upon these kernels, we develop three types of models: a governing-equation-based physics-informed kernel neural network, a boundary-condition-based physics-informed kernel neural network, and a physics-informed kernel neural operator. Their performance is validated through numerical examples. The results demonstrate that the proposed neural network and neural operator models not only maintain high solution accuracy but also improve training stability and computational efficiency, offering novel pathways toward efficient and interpretable intelligent solutions for complex mechanics and engineering problems.