AI Learns Optimal Mesh Discretization for PDEs

In the realm of neural partial differential equation (PDE) surrogates, the choice of computational grid or mesh is typically a pre-determined design decision that dictates how modeling capacity is distributed across space, resolution, and spectral bandwidth. This research argues that this critical choice should itself be learnable, posing the question of whether a surrogate can learn to optimally allocate resolution *before* predicting field evolution. The study frames adaptive discretization as a physics-constrained conditional generation problem over valid mesh displacements. Leveraging the success of diffusion models in PDE field prediction, a two-stage diffusion framework is proposed. Stage 1 learns an r-adaptive displacement mesh, conditioned on observed dynamics, while Stage 2 then predicts the solution evolution from this mesh-informed representation. The mesh generator is rigorously regularized using physics-aware proxy channels, geometric validity constraints, and local spectral concentration to ensure that the adaptation remains physically interpretable and numerically sound. Across five different PDE regimes, the diffusion-based learned discretization proved competitive with, and often superior to, traditional adaptive-mesh and reduced-order baselines. The most significant gains were observed in regimes where fixed or handcrafted mesh allocation proved insufficient. The core conclusion is that optimal discretization is regime-dependent and should be learned, transforming adaptive meshing from a heuristic into a generative representation-learning challenge for neural PDE solvers.