Solving partial differential equations (PDEs) is a fundamental problem in engineering and science. While neural PDE solvers can be more efficient than established numerical solvers, they often require large amounts of training data that is costly to obtain. Active Learning (AL) could help surrogate models reach the same accuracy with smaller training sets by querying classical solvers with more informative initial conditions and PDE parameters. While AL is more common in other domains, it has yet to be studied extensively for neural PDE solvers. To bridge this gap, we introduce AL4PDE, a modular and extensible active learning benchmark. It provides multiple parametric PDEs and state-of-the-art surrogate models for the solver-in-the-loop setting, enabling the evaluation of existing and the development of new AL methods for PDE solving. We use the benchmark to evaluate batch active learning algorithms such as uncertainty- and feature-based methods. We show that AL reduces the average error by up to 71% compared to random sampling and significantly reduces worst-case errors. Moreover, AL generates similar datasets across repeated runs, with consistent distributions over the PDE parameters and initial conditions. The acquired datasets are reusable, providing benefits for surrogate models not involved in the data generation.

本研究应对传统神经偏微分方程(PDE)求解器在大量训练数据需求上的不足，提出了AL4PDE，一个模块化和可扩展的主动学习基准。通过使用主动学习方法，研究表明与随机采样相比，主动学习可以将平均误差降低最多71%，并且生成的数据集具有重复利用性和一致性，对后续的模型提高有显著作用。

神经偏微分方程求解器的主动学习