Neural Network-Based Mesh Optimization for Arbitrary Node Valence

The quality of computational meshes plays an important role in the accuracy and efficiency of simulations, particularly in finite element analysis frameworks. Traditional smoothing techniques for unstructured meshes, such as Laplacian and optimization-based methods, typically employ Gauss–Seidel-style iterative schemes, where interior nodes are updated sequentially over multiple passes. While effective at capturing local, high-frequency changes, these methods often suffer from high computational costs and slow convergence. Recent advances in machine learning, particularly neural networks, offer a promising alternative. Existing neural-network-based mesh smoothing approaches commonly rely on separate models for each valence configuration, which limits their generalizability across diverse mesh topologies.

This work proposes a valence-aware, feedforward neural network architecture that learns to surrogate traditional quality metric optimization processes. By explicitly encoding the local valence degree of mesh umbrellas, the network can operate across varying topological configurations within a single model. The method specifically targets improvement of the weighted inverse-Jacobian quality metric for triangular elements. A fully connected network is trained on synthetic unstructured meshes paired with optimal node placements, which are derived from conventional optimization routines. Numerical experiments demonstrate that the proposed approach provides a faster alternative to traditional smoothing and optimization methods while achieving high-quality meshes. The improvement in the quality/time ratio underscores the potential of neural networks to address complex, non-linear relationships in mesh optimization tasks.