Computational Mathematics and Scientific Computing Seminar

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Solving nonlinear partial differential equations (PDEs) remains a central challenge in scientific computing. Traditional numerical methods are backed by theoretical guarantees but often require problem-specific designs and struggle with the curse of dimensionality, while neural-network-based approaches provide flexibility yet face difficulties such as nonconvex optimization, over-parameterization, and limited interpretability. In this talk, I will present a sparse radial basis function (RBF) network framework that, on one hand, functions as an adaptive collocation PDE solver, and on the other hand, as a shallow neural network with efficient training procedures and greater interpretability. The key idea lies in extending classical RBF collocation by optimizing nonlinear features, with sparsity-promoting regularization that prevents over-parameterization and removes redundant features. 

On the theoretical side, the method is built on Reproducing Kernel Banach Spaces (RKBS) induced by one-hidden-layer neural networks of possibly infinite width. We prove a representer theorem showing that the sparse optimization problem in the RKBS admits a finite solution and establishes error bounds that offer a foundation for generalizing classical numerical analysis. On the computational side, the method is implemented via a three-phase algorithm that combines adaptive feature selection, second-order optimization, and pruning of inactive neurons. I will illustrate the framework with numerical experiments, showing both its effectiveness and scenarios where it provides clear advantages over Gaussian Process and RKHS-based methods. This framework exemplifies a new pathway toward adaptive PDE solvers that combine rigorous analysis with efficient, learning-inspired algorithms.