Neural network-enhanced $hr$-adaptive finite element algorithm for parabolic equations

In this paper, we present a novel enhancement to the conventional $hr$-adaptive finite element methods for parabolic equations, integrating traditional $h$-adaptive and $r$-adaptive methods via neural networks. A major challenge in $hr$-adaptive finite element methods lies in projecting the previous step's finite element solution onto the updated mesh. This projection depends on the new mesh and must be recomputed for each adaptive iteration. To address this, we introduce 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 optimizer from 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 SIZING 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 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.