An Adaptive Transferable Neural Network for Partial Differential Equations with Anisotropic or Sharp Solutions

Abstract

Transferable neural network (TransNet) has demonstrated great potential for solving partial differential equations (PDEs) with high accuracy and efficiency. However, it struggles significantly when dealing with PDEs featuring anisotropy or sharp solutions. In this work, we propose an adaptive transferable neural network (A-TransNet), a novel machine learning framework that adaptively expands neurons in regions with large solution gradients to effectively capture complex solution behaviors. Instead of relying on training or random sampling, the location parameters of newly added neurons are determined by integrating the active subspaces method and local gradient analysis based on posterior gradient information. This adaptive strategy enables neurons to be allocated where they are most needed, thereby improving the resolution of anisotropic and sharp features without additional computational costs. Moreover, the shape parameters of new neurons are automatically determined from the neuron density induced by their locations, further enhancing the network representation capability. An iterative neuron expansion strategy is developed, together with rigorous analysis of stability, a posteriori error estimation, and convergence of the proposed A-TransNet method. Extensive numerical experiments on challenging PDE problems with anisotropic and sharp solutions demonstrate that A-TransNet achieves superior accuracy, efficiency, and robustness compared with existing methods.