Deep neural networks have been shown to provide accurate function approximations in high dimensions. However, fitting network parameters requires training data that may not be available beforehand, which is particularly challenging in science and engineering applications where often it is even unclear how to collect new informative training data in the first place. This work proposes Neural Galerkin schemes based on deep learning that generate training data samples with active learning for numerically solving high-dimensional partial differential equations. Neural Galerkin schemes train networks by minimizing the residual sequentially over time, which enables adaptively collecting new training data in a self-informed manner that is guided by the dynamics described by the partial differential equations, which is in stark contrast to many other machine learning methods that aim to fit network parameters globally in time without taking into account training data acquisition. Our finding is that the active form of gathering training data of the proposed Neural Galerkin schemes is key for numerically realizing the expressive power of networks in high dimensions. Numerical experiments demonstrate that Neural Galerkin schemes have the potential to enable simulating phenomena and processes with many variables for which traditional and other deep-learning-based solvers fail, especially when features of the solutions evolve locally such as in high-dimensional wave propagation problems and interacting particle systems described by Fokker-Planck and kinetic equations.
翻译:深心神经网络被证明能够提供高维的准确功能近似值;然而,适当的网络参数需要培训数据,而这种培训数据可能事先无法提供,在科学和工程应用方面尤其具有挑战性,因为通常甚至无法首先收集新的信息培训数据。这项工作提出了基于深层学习的神经Galerkin计划,这种深层学习能够产生培训数据样本,并积极学习数字解析高维部分差异方程式。神经加热金计划通过在时间上尽可能减少残缺来培训网络,从而能够以自我知情的方式适应性地收集新的培训数据,这种数据以部分差异方程式描述的动态为指导,这在科学和工程应用方面尤其与许多其他机器学习方法形成鲜明的对比,这些方法的目的是在不考虑培训数据获取的情况下及时使网络参数适合全球。我们的发现是,为拟议的神经加热金计划收集培训数据的积极形式对于从数字上认识高维度网络的表达力十分关键。 数值实验表明,神经加热金计划有可能以自适应性地收集新的培训数据和过程,许多变量,而基于部分差异方程式的溶解解解解的系统则无法进行。