Search

Recently proposed numerical algorithms for solving high-dimensional nonlinear partial differential equations (PDEs) based on neural networks have shown their remarkable performance. We review some of them and study their convergence properties. The methods rely on probabilistic representation of PDEs by backward stochastic differential equations and their iterated time discretization. Our proposed algorithm, called the deep backward multistep (MDBDP) scheme, is a machine learning version of the least square multistep-forward dynamic programming scheme of Gobet and Turkedjiev [Math. Comp., 85, 2016, pp. 1359--1391]. It estimates simultaneously by backward induction the solution and its gradient by neural networks through sequential minimizations of suitable quadratic loss functions that are performed by stochastic gradient descent. Our main theoretical contribution is to provide an approximation error analysis of the MDBDP scheme as well as the deep splitting scheme for semilinear PDEs designed by Beck, Becker, Cheridito, Jentzen, and Neufeld [SIAM J. Sci. Comput., 43 (2021), pp. A3135--A3154]. We also supplement the error analysis of the deep learning backward dynamic programming (DBDP) scheme of Hur\'e, Pham, and Warin [Math. Comp., 89 (2020), pp. 1547--1580]. This yields notably convergence rate in terms of the number of neurons for a class of deep Lipschitz continuous GroupSort neural networks when the PDE is linear in the gradient of the solution for the MDBDP scheme, and in the semilinear case for the DBDP scheme. We illustrate our results with some numerical tests that are compared with some other machine learning algorithms in the literature. [ABSTRACT FROM AUTHOR]

Currently, the dominating constraint in many high performance computing applications is data capacity and bandwidth, in both internode communications and even moreso in intranode data motion. A new approach to address this limitation is to make use of data compression in the form of a compressed data array. Storing data in a compressed data array and converting to standard IEEE-754 types as needed during a computation can reduce the pressure on bandwidth and storage. However, repeated conversions (lossy compression and decompression) introduce additional approximation errors, which need to be shown to not significantly affect the simulation results. We extend recent work [J. Diffenderfer et al., SIAM J. Sci. Comput., 41 (2019), pp. A1867-A1898] that analyzed the error of a single use of compression and decompression of the ZFP compressed data array representation [P. Lindstrom, IEEE Trans. Vis. Comput. Graph., 20 (2014), pp. 2674-2683; P. Lindstrom, ZFP version 0.5.5, May 2019] to the case of time-stepping and iterative schemes, where an advancement operator is repeatedly applied in addition to the conversions. We show that the accumulated error for iterative methods involving fixed-point and time evolving iterations is bounded under standard constraints. An upper bound is established on the number of additional iterations required for the convergence of stationary fixed-point iterations. An additional analysis of traditional forward and backward error of stationary iterative methods using ZFP compressed arrays is also presented. The results of several 1D, 2D, and 3D test problems are provided to demonstrate the correctness of the theoretical bounds. [ABSTRACT FROM AUTHOR]