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

Gonon, Lukas und Schwab, Christoph (2022): Deep ReLU neural networks overcome the curse of dimensionality for partial integrodifferential equations. In: Analysis and Applications, Bd. 21, Nr. 1: S. 1-47

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 Levy processes. We prove for such PIDEs arising from a class of jump-diffusions on R-d, that for any suitable measure mu(d) on R-d, there exist constants C, p, q > 0 such that for every epsilon is an element of(0, 1] and for every d is an element of N the DNN L-2(mu(d))-expression error of viscosity solutions of the PIDE is of size epsilon with DNN size bounded by Cd-p epsilon(-q). In particular, the constant C > 0 is independent of d is an element of N and of epsilon is an element of(0, 1] 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.

• Dokumententyp: Zeitschriftenartikel
• Fakultät: Mathematik, Informatik und Statistik > Informatik
• Themengebiete: 000 Informatik, Informationswissenschaft, allgemeine Werke > 004 Informatik
• ISSN: 0219-5305
• Sprache: Englisch
• DFG: Gefördert durch die Deutsche Forschungsgemeinschaft (DFG) - 464123384