Learning-based local RBF neural network with adaptively supported domain for solving partial differential equations.

The Local Radial Basis Function method (Local RBF) is attractive for its sparsity and efficiency. However, its accuracy and stability depend critically on the choice of the local support domain. Conventional methods rely on trial-and-error to select a uniform support domain size for all center points, lack theoretical guidance, and often exhibit significant sensitivity of numerical error to this choice. This paper introduces a learning-based Local RBF Neural Network with Adaptively Supported Domain (Local RBF-ASD), which mitigates this sensitivity by adaptively adjusting the effective support domain within a differentiable training loop. Both the model parameters and unknown field quantities are optimized through a composite loss function that combines interior PDE residuals with boundary constraints, while preserving the sparse structure of the discretization. Numerical benchmarks on (i) Helmholtz problems with complex geometries (including anisotropic variants), (ii) steady Stokes flows, and (iii) a three-dimensional near-singular Poisson problem demonstrate that Local RBF-ASD achieves accurate and stable solutions across a wide range of support domain sizes.