Deep learning for non-local partial differential equations

• Applicants: Professor Dr. Arnulf Jentzen; Professorin Dr. Gitta Kutyniok, since 10/2022
• Subject Area: Mathematics
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 464123384

Partial differential equations (PDEs) are ubiquitously used in order to formally describe and solve problems in natural sciences, technology and man-made systems. For instance, PDEs arise in the areas of valuation and hedging of financial derivatives or in models of biodiversity aiming at an improved understanding of changes in the ecosystem induced by climate change. The PDEs that emerge in the applications that we just mentioned are often non-local non-linear high-dimensional PDEs, where for instance in the case of financial derivatives the dimension of the PDE corresponds precisely to the number of considered financial assets (e.g. stocks, commodities, bonds, ...) and in the case of models for biodiversity the dimension corresponds, roughly speaking, to the number of biological traits of the individuals in the considered ecosystem. In these applications, realistic PDE models are often non-local, because of the potential jump behaviour of the asset prices in the case of financial derivatives and because each individual is subject to competition with individuals with all possible values of traits in the case of biodiversity models. In nearly all cases it is impossible to solve non-local non-linear PDEs explicitly and to date there is no numerical method which could be used to approximately solve these high-dimensional PDEs efficiently. It is the key contribution of this project to propose and analyze deep learning based methods for approximately solving such high-dimensional non-local non-linear PDEs.This project will cover the full spectrum from numerical simulations in the concrete above mentioned applications to rigorous mathematical convergence results. In particular, we will rigorously prove that the deep neural networks in the proposed deep learning algorithms have indeed the fundamental power to overcome the curse of dimensionality in the numerical approximation of such non-local non-linear PDEs in the sense that the number of parameters in the neural networks grow at most polynomially in both the reciprocal of the prescribed approximation accuracy and the PDE dimension. We also intend to develop a full error analysis for the proposed deep learning algorithm ensuring convergence for the approximation error, the generalizaton error and the optimization error and thus proving in particular convergence of the proposed approximation algorithm. Finally, in certain situations we also intend to prove completely dimension-independent convergence rates for all error components and thereby show that deep learning algorithms are able to overcome the curse of dimensionality in the numerical solution of these PDEs. We expect that the error analysis techniques discovered within this project will have a high impact on the mathematical error analysis of more general deep learning based algorithms for other problems much beyond the specific PDE applications considered in this project.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2298: Theoretical Foundations of Deep Learning
• Ehemaliger Antragsteller: Professor Dr. Lukas Gonon, until 9/2022