学习离散化：基于扩散的具有谱引导的自适应网格
Learning to Discretize: Diffusion-Based Adaptive Mesh with Spectral Guidance
浏览论文内容
中文总结 AI 辅助
研究提出能否让替代模型在预测场演化前学习分辨率位置的问题，构建两阶段扩散框架解决自适应离散化，通过多种约束使网格生成器正则化，结果显示该方法在五个PDE regime有竞争力，表明离散化应依 regime学习，重构了自适应网格划分问题。
中文摘要 AI 辅助
大多数神经偏微分方程（PDE）替代模型在已选择网格后学习场如何演化。但在应用任何算子之前，网格就已决定了建模能力在空间、分辨率和谱带宽上的分配方式。我们认为这种隐藏的设计选择本身应是可学习的，从而引出一个不同于标准算子学习的问题：替代模型能否在预测场演化之前学习分辨率应存在的位置？我们将自适应离散化表述为关于有效网格位移的物理约束条件生成问题。扩散模型在PDE场预测中的成功表明其在类似结构约束下学习自适应离散化的潜力。这导致了一个两阶段扩散框架：第一阶段基于观测动力学学习r自适应位移网格，第二阶段从网格信息表示预测解的演化。网格生成器通过物理感知代理通道、几何有效性约束和局部谱浓度进行正则化，以使自适应在物理上可解释且在数值上合法。在五个PDE regime中，结果表明基于扩散的学习离散化与自适应网格和降阶基线具有竞争力，在固定或手工分配不足的 regime中收益尤其显著。主要结论不是存在通用的最优网格规则，而是离散化应以依赖 regime的方式学习：不同的空间和谱结构有利于不同的分配行为。这将神经PDE求解器的自适应网格划分从特定求解器的启发式方法重新构建为生成式表示学习问题。
英文摘要
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.
