Deep ReLU neural networks overcome the curse of dimensionality for partial integrodifferential equations

Author:

Gonon Lukas1,Schwab Christoph2

Affiliation:

1. Department of Mathematics, University of Munich, Theresienstrasse 39, 80333 Munich, Germany

2. Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland

Abstract

Deep neural networks (DNNs) with ReLU activation function are proved to be able to express viscosity solutions of linear partial integrodifferential equations (PIDEs) on state spaces of possibly high dimension d. Admissible PIDEs comprise Kolmogorov equations for high-dimensional diffusion, advection, and for pure jump Lévy processes. We prove for such PIDEs arising from a class of jump-diffusions on [Formula: see text], that for any suitable measure [Formula: see text] on [Formula: see text], there exist constants [Formula: see text] such that for every [Formula: see text] and for every [Formula: see text] the DNN [Formula: see text]-expression error of viscosity solutions of the PIDE is of size [Formula: see text] with DNN size bounded by [Formula: see text]. In particular, the constant [Formula: see text] is independent of [Formula: see text] and of [Formula: see text] and depends only on the coefficients in the PIDE and the measure used to quantify the error. This establishes that ReLU DNNs can break the curse of dimensionality (CoD for short) for viscosity solutions of linear, possibly degenerate PIDEs corresponding to suitable Markovian jump-diffusion processes. As a consequence of the employed techniques, we also obtain that expectations of a large class of path-dependent functionals of the underlying jump-diffusion processes can be expressed without the CoD.

Funder

Deutsche Forschungsgemeinschaft

Publisher

World Scientific Pub Co Pte Ltd

Subject

Applied Mathematics,Analysis

Cited by 13 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3