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Training nonlinear parametrizations such as deep neural networks tonumerically approximate solutions of partial differential equations is oftenbased on minimizing a loss that includes the residual, which is analyticallyavailable in limited settings only. At the same time, empirically estimatingthe training loss is challenging because residuals and related quantities canhave high variance, especially for transport-dominated and high-dimensionalproblems that exhibit local features such as waves and coherent structures.Thus, estimators based on data samples from un-informed, uniform distributionsare inefficient. This work introduces Neural Galerkin schemes that estimate thetraining loss with data from adaptive distributions, which are empiricallyrepresented via ensembles of particles. The ensembles are actively adapted byevolving the particles with dynamics coupled to the nonlinear parametrizationsof the solution fields so that the ensembles remain informative for estimatingthe training loss. Numerical experiments indicate that few dynamic particlesare sufficient for obtaining accurate empirical estimates of the training loss,even for problems with local features and with high-dimensional spatialdomains.

Coupling parameter and particle dynamics for adaptive sampling in Neural Galerkin schemes