论文标题

部分分化方程的大规模神经求解器

Large-scale Neural Solvers for Partial Differential Equations

论文作者

Stiller, Patrick, Bethke, Friedrich, Böhme, Maximilian, Pausch, Richard, Torge, Sunna, Debus, Alexander, Vorberger, Jan, Bussmann, Michael, Hoffmann, Nico

论文摘要

求解部分微分方程(PDE)是许多科学分支的必不可少的部分,因为许多过程都可以根据PDE进行建模。但是,最近的数值求解器需要手动离散基础方程以及用于分布式计算的复杂的,量身定制的代码。扫描基础模型的参数会显着增加运行时,因为必须冷启动每个参数配置的模拟。基于机器学习的替代模型表示在输入,参数和解决方案之间学习复杂关系的有希望的方法。但是,最近的生成神经网络需要大量的培训数据,即完整的模拟运行,使其昂贵。相比之下,我们研究了连续无网神经求解器在部分微分方程中的适用性,物理信息的神经网络(PINN),这些神经网络(PINN)仅需要初始/边界值和验证点的训练点,但没有模拟数据。通过学习一个域分解,该域分解可引导每单位体积的数量并显着改善运行时,可以接近降低的诅咒。大规模集群系统上的分布式培训也有望大大利用大量GPU,我们通过一项全面的评估研究进行评估。最后,我们讨论了GatedPinn相对于分析溶液的准确性 - 以及最新的数值求解器,例如光谱求解器。

Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. However, recent numerical solvers require manual discretization of the underlying equation as well as sophisticated, tailored code for distributed computing. Scanning the parameters of the underlying model significantly increases the runtime as the simulations have to be cold-started for each parameter configuration. Machine Learning based surrogate models denote promising ways for learning complex relationship among input, parameter and solution. However, recent generative neural networks require lots of training data, i.e. full simulation runs making them costly. In contrast, we examine the applicability of continuous, mesh-free neural solvers for partial differential equations, physics-informed neural networks (PINNs) solely requiring initial/boundary values and validation points for training but no simulation data. The induced curse of dimensionality is approached by learning a domain decomposition that steers the number of neurons per unit volume and significantly improves runtime. Distributed training on large-scale cluster systems also promises great utilization of large quantities of GPUs which we assess by a comprehensive evaluation study. Finally, we discuss the accuracy of GatedPINN with respect to analytical solutions -- as well as state-of-the-art numerical solvers, such as spectral solvers.

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