提出新评估框架，衡量神经PDE求解器何时比传统方法更划算。
Breakeven complexity: A new perspective on neural partial differential equation solvers
- 以‘盈亏复杂度’为指标，综合考虑训练与求解的总成本
- 在多个复杂场景下验证，神经求解器在高难度问题中更占优势
- 适合需频繁求解且计算成本高的工程仿真场景
神经代理求解器可显著提升偏微分方程（PDE）求解速度，尤其在需多次求解的场景中。但现有基于精度的评估未充分考虑两大核心问题：（1）神经求解器存在数据生成、训练与调参的高昂前期成本；（2）经典求解器在低精度要求下也能以极低成本完成模拟。为此，本文提出以‘盈亏复杂度’为核心的评估框架，该指标衡量在多少次前向求解后，学习型求解器相较误差相当的传统求解器更具成本优势。通过应用尺度定律确定数据生成的训练预算，并讨论跨场景误差匹配策略，我们在两个基准测试中评估了多种神经PDE求解器的表现：包括来自APEBench的三个二维周期域上的PDE问题，以及使用GPU原生PyFR代码生成的多障碍绕流新基准。结果表明，随着问题难度上升——如更高维度、更长滚动时间、更高雷诺数等物理情形——神经求解器的成本效益愈发显著。
原文摘要 · Abstract (English)
Neural surrogate solvers of partial differential equations (PDEs) promise dramatic speedups over numerical methods, especially in scenarios requiring many solves. However, current accuracy-based evaluations do not fully consider two central issues: (1) neural solvers incur substantial up-front costs for data generation, training, and tuning; and (2) classical solvers can also generate low-fidelity solutions at a sufficiently low simulation cost. To explicitly account for these realities and fully incorporate end-to-end costs, we propose an evaluation framework centered on breakeven complexity, a metric that counts the forward solves before a learned solver is cost-effective relative to an error-equivalent traditional solver. To evaluate this measure, we apply scaling laws to determine how much training budget to allocate to data generation and discuss how to achieve smooth error-matching in diverse settings. We evaluate the breakeven complexity of multiple neural PDE solvers on three PDEs on 2D periodic domains from APEBench and a novel benchmark of flows past multiple obstacles generated by the GPU-native PyFR code. Among other findings, our results suggest that neural PDE solvers become more effective as problems get harder in terms of cost, dimension, rollout, physics regime (e.g. higher Reynolds number), etc.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。