从偏微分方程的数值模拟器到神经模拟器再回到数值模拟器

中文摘要 AI 辅助

模拟是现代工程与科学的核心，但当需要快速或多查询评估时，偏微分方程（PDE）数值求解器的成本仍是瓶颈。基于求解器生成数据训练的神经模拟器有望大幅提速，不过它们通常被视为产生其训练信号的方法的不透明替代方案。本论文认为这两种范式的相似之处多于差异：神经架构与经典离散化方法相呼应，其误差可接受相同的谱分析，且双向的见解流动均能获益。我们通过梳理求解器在模拟器学习流程中扮演的多重角色来探究二者关系，而模式级傅里叶分析则提供了一种通用语言，可同时解读求解器误差、架构归纳偏置与训练目标。综上，本研究完成三项贡献：（1）APEBench，一套用于PDE自回归神经模拟器的综合基准套件，使用JAX中的快速可微伪谱求解器；（2）渐进式精细化可微物理，研究未收敛求解器对替代模型训练的影响；（3）神经模拟器优越性，分析数值误差与架构归纳偏置的影响。

英文摘要

Simulation is central to modern engineering and science, but the cost of numerical solvers for partial differential equations (PDEs) remains a bottleneck whenever fast or many-query evaluations are required. Neural emulators trained on solver-generated data promise significant speedups, yet they are usually framed as opaque alternatives to the very methods that produce their training signal. This thesis argues the two paradigms are more alike than different: neural architectures mirror classical discretizations, their errors are amenable to the same spectral analysis, and insight flows profitably in both directions. We approach the relationship by disentangling the multiple roles a solver plays in the emulator learning pipeline. Mode-wise Fourier analysis then provides a common language in which solver errors, architectural inductive biases, and training objectives can all be read off simultaneously. Taken together, this allows synthesizing three contributions. (1) APEBench, a comprehensive benchmarking suite for autoregressive neural emulators of PDEs that uses fast differentiable pseudo-spectral solvers in JAX. (2) Progressively Refined Differentiable Physics, an investigation of the effect of unconverged solvers on surrogate training. (3) Neural Emulator Superiority, an analysis of the influence of numerical errors and architectural inductive biases.

发表机构

• TU Braunschweig（布伦瑞克工业大学）
• TU Munich（慕尼黑工业大学）

机构由 AI 辅助整理，请以论文原文为准。