用傅里叶分析打通物理模拟与神经代理的壁垒，实现双向赋能。
From Numerical Simulators of PDEs to Neural Emulators and Back
- 通过模态傅里叶分析统一解析求解器与神经网络的误差机制。
- 提出 APEBench 基准测试套件，支持快速可微伪谱求解器。
- 揭示数值误差与模型先验对神经代理性能的关键影响，适合算法设计者。
仿真在现代工程与科学中至关重要，但偏微分方程（PDE）数值求解器的计算成本仍是快速或多次查询场景下的瓶颈。基于求解器生成数据训练的神经代理虽能显著提速，却常被视为与原方法对立的黑箱。本论文主张二者本质更相似：神经架构映射经典离散化，误差可进行同源谱分析，且洞见可双向流动。通过解耦求解器在代理学习流程中的多重角色，模态傅里叶分析成为共通语言，同步读取求解器误差、模型归纳偏置与训练目标。据此，提出三项贡献：(1) APEBench，一个使用 JAX 中快速可微伪谱求解器的自回归神经代理 PDE 基准测试套件；(2) 渐进式精炼可微物理，研究未收敛求解器对代理训练的影响；(3) 神经代理优越性，分析数值误差与架构归纳偏置的影响。
原文摘要 · Abstract (English)
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.
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