In most neural partial differential equation (PDE) surrogates, the grid choice is predetermined, influencing the modeling capacity across space, resolution, and spectral bandwidth. This paper argues that the design choice of the grid should itself be learnable, leading to a novel question: can a surrogate learn where resolution should exist prior to predicting field evolution? We formalize adaptive discretization as a physics-constrained conditional generation problem involving valid mesh displacements. Given the success of diffusion models in PDE field predictions, we explore their potential for learning adaptive discretizations under similar structured constraints.
The approach consists of two stages: Stage 1 learns an r-adaptive displacement mesh conditioned on observed dynamics, while Stage 2 predicts solution evolution from the mesh-informed representation. The mesh generator is regularized by physics-aware proxy channels, geometric validity constraints, and local spectral concentration to ensure that adaptation remains physically interpretable and numerically legal. Results across five PDE regimes demonstrate that diffusion-based learned discretization is competitive with adaptive mesh and reduced-order baselines, particularly showing strong gains in regimes where fixed or handcrafted allocation is insufficient.
The main conclusion is that there is no universal optimal mesh rule, but that discretization should be learned in a regime-dependent manner: different spatial and spectral structures favor different allocation behaviors. This reframes adaptive meshing for neural PDE solvers from solver-specific heuristics into a generative representation-learning problem.
Blogger's Review: This paper profoundly reflects and innovates traditional adaptive mesh design by introducing diffusion models, illustrating how to combine physical constraints with machine learning to advance numerical simulation. This method not only holds theoretical significance but also provides new insights for practical applications, warranting further exploration.