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Deep neural networks have been shown to provide accurate function approximations in high dimensions. However, fitting network parameters requires informative training data that are often challenging to collect in science and engineering applications. This work proposes Neural Galerkin schemes based on deep learning that generate training data with active learning for numerically solving high-dimensional partial differential equations. Neural Galerkin schemes build on the Dirac-Frenkel variational principle to 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. This is in contrast to 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. • Coupling training of deep networks with adaptive data acquisition to solve high-dimensional evolution equations. • Enables predicting local features in high dimensions for which training with classical, uniform data acquisition fails. • Sequentially collects data over time and so avoids requirement of having rich initial data set over the time-space domain. • Suited for transport problems, for which collocation-based deep-network methods with uniform data sampling are inefficient. [ABSTRACT FROM AUTHOR]

In this paper, we study a physics-informed algorithm for Wasserstein Generative Adversarial Networks (WGANs) for uncertainty quantification in solutions of partial differential equations. By using groupsort activation functions in adversarial network discriminators, network generators are utilized to learn the uncertainty in solutions of partial differential equations observed from the initial/boundary data. Under mild assumptions, we show that the generalization error of the computed generator converges to the approximation error of the network with high probability, when the number of samples are sufficiently taken. According to our established error bound, we also find that our physics-informed WGANs have higher requirement for the capacity of discriminators than that of generators. Numerical results on synthetic examples of partial differential equations are reported to validate our theoretical results and demonstrate how uncertainty quantification can be obtained for solutions of partial differential equations and the distributions of initial/boundary data. However, the quality or the accuracy of the uncertainty quantification theory in all the points in the interior is still the theoretical vacancy, and required for further research. • Wasserstein generative adversarial networks need little priors for uncertainty quantifications. • Physics-informed regularization term enforces the satisfaction of governing PDEs. • The proposed probabilistic model (WGAN-PINNs) is able to deal with non-Gaussian and spatially dependent noise. • WGAN-PINNs have good convergence and generalization properties. • Theoretically we have higher requirements on discriminators than generators. [ABSTRACT FROM AUTHOR]

• An approach for training neural networks combined with partial differential equations. • Combines a finite element framework with a machine learning library. • Learns missing physics in the partial differential equation. We present a methodology combining neural networks with physical principle constraints in the form of partial differential equations (PDEs). The approach allows to train neural networks while respecting the PDEs as a strong constraint in the optimisation as apposed to making them part of the loss function. The resulting models are discretised in space by the finite element method (FEM). The method applies to both stationary and transient as well as linear/nonlinear PDEs. We describe implementation of the approach as an extension of the existing FEM framework FEniCS and its algorithmic differentiation tool dolfin-adjoint. Through series of examples we demonstrate capabilities of the approach to recover coefficients and missing PDE operators from observations. Further, the proposed method is compared with alternative methodologies, namely, physics informed neural networks and standard PDE-constrained optimisation. Finally, we demonstrate the method on a complex cardiac cell model problem using deep neural networks. [ABSTRACT FROM AUTHOR]