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.
Related
- AKAPINN: Adaptive Kolmogorov-Arnold Physics-Informed Neural Networks for approximating solutions to quasilinear partial differential equations
- Rigidity results via a Poincaré type formula for stable solutions of quasilinear anisotropic elliptic equations
- Liouville Theorems for Stable Solutions to Anisotropic Quasilinear Equations
- Existence of positive solutions to anisotropic elliptic equations in the entire Euclidean space
- On the stability of solutions to the 3-D anisotropic Navier-Stokes equations in the critical space
BibTeX
@misc{qu2026adaptive,
title = {{An Adaptive Transferable Neural Network for Partial Differential Equations with Anisotropic or Sharp Solutions}},
author = {Jiacong Qu and Liyong Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40201},
eprint = {2610.40201},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40201},
}
RIS
TY - UNPB
TI - An Adaptive Transferable Neural Network for Partial Differential Equations with Anisotropic or Sharp Solutions
AU - Jiacong Qu
AU - Liyong Zhu
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40201
UR - https://arxiv.org/abs/2610.40201
AB - 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.
ER -