"Coefficients-Learning Based Radial Basis Function Method for Partial Differential Equation Boundary Value Problems and Inverse"

"Partial differential equations (PDEs) have becomeindispensable tools for modeling complex systemsand processes across various fields, including physics,biology, and engineering. Traditional methods for solvingthese equations include mesh-based numerical techniques,such as the finite element method (FEM) and finite differencemethod, as well as mesh-free approaches like Kansa\u2019smethod, and deep learning approaches including Physicsinformedneural networks and deep operator networks. However,these methods are usually computationally expensive,which can limit their application in parameter exploration forPDEs. In this paper, we introduce the Coefficients-LearningbasedRadial Basis Function (CLRBF) method, a deeplearning-driven emulator designed for efficient resolutionof both forward and inverse PDE problems. The novelty ofthe proposed method lies in utilizing deep neural networksto learn the latent relationships between different coefficientsunder different parameters, where the coefficientsare derived by solving PDEs using radial basis functions.Our method requires fewer collocation points comparedto traditional mesh-based techniques such as FEM, and itsignificantly accelerates solution inference comparing withKansa\u2019s method. For example, in our real-world case study,CLRBF accurately solve the inverse problem with only 2,000collocation points, a 90% reduction compared to FEM, andaccelerates inference by approximately two orders of magnitudeobserved across all experiments. Extensive numericalexperiments, including real-world examples, demonstratethat the CLRBF method can accurately and efficiently predictsolutions with unknown parameters and effectively inferparameters in inverse problems."