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

Technical Summary: Neural Network-Enhanced $hr$-Adaptive Finite Element Algorithm for Parabolic Equations

Problem Statement
The paper addresses the computational challenges inherent in applying adaptive finite element methods (AFEM) to time-dependent parabolic equations, particularly those exhibiting moving singularities or multiscale features. While $hr$-adaptive methods (combining mesh refinement $h$ and node movement $r$) offer theoretical advantages, their practical implementation for parabolic problems suffers from three primary bottlenecks:

1. Inefficient Refinement: Conventional $h$-adaptive strategies often require excessive iterations (30–40) per time step to achieve convergence due to layer-by-layer bisection.
2. Interpolation Challenges: Transferring the solution from a previous time step's mesh to a newly refined, non-nested mesh requires complex and computationally expensive point-location algorithms and interpolation operators ($\Pi_h^n$).
3. Redundant Initialization: Standard approaches inherit the final, highly refined mesh from the previous time step as the starting point for the next, leading to an inflated number of degrees of freedom (DOF) and necessitating a separate, intricate coarsening step to reduce computational cost.

Methodology
The authors propose a novel Neural Network-Enhanced $hr$-Adaptive Finite Element Algorithm designed to overcome these limitations through three integrated mechanisms:

1. Mesh-Free Surrogate via Neural Networks (Interpolation Replacement):
   Instead of performing geometric interpolation between non-nested meshes, the method employs a neural network (NN) to construct a continuous, mesh-free surrogate of the finite element solution from the previous time step ($u_{h}^{n-1}$).

   • The NN is trained on the vertices and solution values of the previous mesh.
   • To strictly enforce Dirichlet boundary conditions, the surrogate is constructed as $u_{\theta}^{n-1}(x) = d(x)u_{\theta}(x) + \tilde{g}(x, t_{n-1})$, where $d(x)$ vanishes on the boundary and $\tilde{g}$ extends boundary data.
   • Warm-Start Strategy: The NN parameters are initialized using the converged minimizer from the previous time step, requiring only brief fine-tuning (few epochs) rather than full retraining at each step.
   • This surrogate replaces the interpolation operator in the fully discrete scheme, allowing direct function evaluation at any spatial point on the new mesh.

2. Controlled Mesh Generation ($h$-Adaptivity):
   The algorithm utilizes the Gmsh library to generate meshes based on a user-defined mesh size field (SIZE).

   • A mesh size field is computed from local error estimators, targeting high density in regions of high error.
   • Unlike standard bisection, this approach allows for the generation of unstructured, non-nested meshes that can drastically increase resolution in a single step.

3. Predictive Iteration Strategy ($r$-Adaptivity and Loop Control):
   The algorithm limits the adaptive loop to a maximum of seven iterations per time step through a predictive workflow:

   • Iterations 1–2: "Start-up" phase to stabilize the solution.
   • Iterations 3–5: Data collection to fit a power-law model ($\eta_h \approx cN^{-p}$) relating the error estimator to the number of vertices (NOV).
   • Iteration 6: The fitted model predicts the required NOV to meet the tolerance. A new mesh is generated to match this target.
   • Iteration 7: A final verification step ensures the prescribed tolerance is met.
   • Combined Error Estimator: The error indicator incorporates information from both the current solution ($u_h^n$) and the NN surrogate of the previous solution ($u_{\theta}^{n-1}$) to guide mesh generation effectively.

Key Contributions

• Elimination of Interpolation Costs: By replacing the mesh-dependent interpolation operator with a mesh-free neural network surrogate, the method bypasses the algorithmic complexity and computational cost of transferring data between non-nested meshes.
• Drastic Reduction in Iterations: The predictive mesh generation strategy reduces the number of adaptive iterations per time step from the typical 30–40 down to a maximum of 7.
• Efficient Mesh Reset: The method re-initializes the mesh to a coarse, uniform state at the beginning of each time step (rather than inheriting the refined mesh), avoiding the need for a separate coarsening step and preventing the accumulation of unnecessary DOFs.
• Dimensional Agnosticism: The framework is designed to be dimension-agnostic, demonstrated by successful application in both 2D and 3D without structural changes to the algorithm.

Numerical Results
The paper validates the algorithm through extensive numerical experiments on parabolic equations with moving singularities (rotation, diffusion, and splitting) in 2D and 3D, as well as nonlinear Allen–Cahn equations and problems on L-shaped domains.

• Efficiency: The proposed method achieves the desired accuracy in no more than seven space-adaptive iterations per time step.
• Training Overhead: While the initial time step requires significant training (e.g., ~2000 epochs), subsequent steps require minimal fine-tuning (<100 epochs), leveraging the warm-start strategy.
• Performance Gains: Comparisons with conventional $h$-adaptive methods show a substantial reduction in total CPU runtime (e.g., reducing 818s to 132s in a 2D rotation example) and a significant reduction in the total number of degrees of freedom throughout the simulation.
• Accuracy: The method successfully captures dynamic changes in singularities and achieves quasi-optimal convergence rates for gradient errors.

Significance and Claims
The paper claims that the proposed algorithm offers a comprehensive efficiency gain by addressing the fundamental bottlenecks of conventional AFEM for parabolic problems. By integrating neural networks not as standalone solvers but as specialized, data-driven modules for data transfer and function representation, the method achieves a robust and efficient workflow. The authors emphasize that this approach makes node insertion and movement more convenient, significantly reduces the time required for repeated refinement, and achieves high accuracy with a controlled number of degrees of freedom. The code is made publicly available to facilitate further research and application.