Near-optimal learning of Banach-valued, high-dimensional functions via deep neural networks

The past decade has seen increasing interest in applying Deep Learning (DL) to Computational Science and Engineering (CSE). Driven by impressive results in applications such as computer vision, Uncertainty Quantification (UQ), genetics, simulations and image processing, DL is increasingly supplanting classical algorithms, and seems poised to revolutionize scientific computing. However, DL is not yet well-understood from the standpoint of numerical analysis. Little is known about the efficiency and reliability of DL from the perspectives of stability, robustness, accuracy, and sample complexity. In particular, approximating solutions to parametric PDEs is an objective of UQ for CSE. Training data for such problems is often scarce and corrupted by errors. Moreover, the target function is a possibly infinite-dimensional smooth function taking values in the PDE solution space, generally an infinite-dimensional Banach space. This paper provides arguments for Deep Neural Network (DNN) approximation of such functions, with both known and unknown parametric dependence, that overcome the curse of dimensionality. We establish practical existence theorems that describe classes of DNNs with dimension-independent architecture size and training procedures based on minimizing the (regularized) $\ell^2$-loss which achieve near-optimal algebraic rates of convergence. These results involve key extensions of compressed sensing for Banach-valued recovery and polynomial emulation with DNNs. When approximating solutions of parametric PDEs, our results account for all sources of error, i.e., sampling, optimization, approximation and physical discretization, and allow for training high-fidelity DNN approximations from coarse-grained sample data. Our theoretical results fall into the category of non-intrusive methods, providing a theoretical alternative to classical methods for high-dimensional approximation.Comment: 49 page

Error analysis for deep neural network approximations of parametric hyperbolic conservation laws

We derive rigorous bounds on the error resulting from the approximation of the solution of parametric hyperbolic scalar conservation laws with ReLU neural networks. We show that the approximation error can be made as small as desired with ReLU neural networks that overcome the curse of dimensionality. In addition, we provide an explicit upper bound on the generalization error in terms of the training error, number of training samples and the neural network size. The theoretical results are illustrated by numerical experiments

Deep Operator Learning Lessens the Curse of Dimensionality for PDEs

Deep neural networks (DNNs) have achieved remarkable success in numerous domains, and their application to PDE-related problems has been rapidly advancing. This paper provides an estimate for the generalization error of learning Lipschitz operators over Banach spaces using DNNs with applications to various PDE solution operators. The goal is to specify DNN width, depth, and the number of training samples needed to guarantee a certain testing error. Under mild assumptions on data distributions or operator structures, our analysis shows that deep operator learning can have a relaxed dependence on the discretization resolution of PDEs and, hence, lessen the curse of dimensionality in many PDE-related problems including elliptic equations, parabolic equations, and Burgers equations. Our results are also applied to give insights about discretization-invariant in operator learning