Learning to Discretize: Diffusion-Based Adaptive Mesh with Spectral Guidance

2026-07-12

AI Summary

This work addresses a key limitation of conventional neural PDE surrogates, which model field evolution on fixed grids and thereby overlook the critical role of mesh design in allocating spatial resolution and spectral bandwidth. The study introduces, for the first time, adaptive discretization as a physics-constrained conditional generation task, proposing a two-stage diffusion framework: it first generates an r-adaptive displacement grid conditioned on observed dynamics and then predicts solution evolution on this adaptive mesh. By incorporating physics-aware regularization, geometric validity constraints, and local spectral concentration, the method achieves learnable, interpretable, and numerically stable mesh adaptation. Extensive experiments across five classes of PDE problems demonstrate substantial improvements over traditional adaptive and reduced-order methods, with particularly notable gains in complex domains.

Abstract

Most neural partial differential equation (PDE) surrogates learn how fields evolve after a grid has already been chosen. However, before any operator is applied, the grid has already determined how modeling capacity is allocated across space, resolution, and spectral bandwidth. We argue that this hidden design choice should itself be learnable, leading to a question different from standard operator learning: can a surrogate learn where resolution should exist before predicting field evolution? We formulate adaptive discretization as a physics-constrained conditional generation problem over valid mesh displacements. The success of diffusion models in PDE field prediction suggests their potential for learning adaptive discretizations under similar structured constraints. This leads to a two-stage diffusion framework: Stage 1 learns an r-adaptive displacement mesh conditioned on the observed dynamics, while Stage 2 predicts the solution evolution from the mesh-informed representation. The mesh generator is regularized by physics-aware proxy channels, geometric validity constraints, and local spectral concentration so that adaptation remains physically interpretable and numerically legal. Across five PDE regimes, the results show that diffusion-based learned discretization is competitive with adaptive-mesh and reduced-order baselines, with particularly strong gains in regimes where fixed or handcrafted allocation is insufficient. The main conclusion is not that there exists a 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 a solver-specific heuristic into a generative representation-learning problem.

Problem

• adaptive mesh
• neural PDE surrogates
• discretization
• resolution allocation
• physics-constrained generation

Innovation

• diffusion models
• adaptive mesh generation
• neural PDE solvers
• spectral guidance
• learnable discretization