Neural network-enhanced $hr$-adaptive finite element algorithm for parabolic equations
Jiaxiong Hao, Yunqing Huang, Nianyu Yi, Peimeng Yin
TL;DR
The paper tackles the high cost of interpolation in $h$-adaptive finite element methods for parabolic equations, especially on non-nested meshes with evolving singularities. It replaces the mesh-dependent interpolation step by a mesh-free neural surrogate $u_{\theta}^{n-1}$ trained once per time step, and couples this with a mesh-size field to guide refined, non-nested meshes via Gmsh; the result is an efficient $hr$-AFEM that terminates within seven refinement iterations per time step while preserving accuracy. The authors develop a gradient-recovery–based error estimator that blends information from two consecutive time steps to drive refinement, and demonstrate notable performance gains in both 2D and 3D test problems, including rotating, diffusing, and splitting singularities. The approach significantly reduces computational cost and interpolation overhead, offering a scalable framework for time-dependent PDEs with localized features, with code publicly available on GitHub.
Abstract
In this paper, we propose a novel $hr$-adaptive finite element method, enhanced by neural networks, for parabolic equations. The main challenge of the conventional $h$-adaptive finite element method is interpolating the finite element solution from the previous step in the updated mesh. The interpolation dependent on the new mesh must be recomputed at each adaptive iteration, resulting in high computational costs. The new approach addresses this challenge by introducing a neural network to construct a mesh-free surrogate of the previous step finite element solution. Since the neural network is mesh-free, it only requires training once per time step, with its parameters initialized using the minimizer of the previous time step. This approach effectively overcomes the interpolation challenges associated with non-nested meshes in computation, making node insertion and movement more convenient and efficient. The new algorithm also emphasizes SIZE and GENERATE, allowing each refinement to roughly double the number of mesh nodes of the previous iteration and then redistribute them to form a new mesh that effectively captures the singularities. It significantly reduces the time required for repeated refinement of the conventional methods and achieves the desired accuracy in no more than seven space-adaptive iterations per time step. Numerical experiments confirm the efficiency of the proposed algorithm in capturing dynamic changes of singularities. The code is made publicly available on GitHub.