DeepMartNet -- A Martingale Based Deep Neural Network Learning Method for Dirichlet BVPs and Eigenvalue Problems of Elliptic PDEs in R^d
Abstract: In this paper, we propose DeepMartNet - a Martingale based deep neural network learning method for solving Dirichlet boundary value problems (BVPs) and eigenvalue problems for elliptic partial differential equations (PDEs) in high dimensions or domains with complex geometries. The method is based on Varadhan's Martingale problem formulation for the BVPs/eigenvalue problems where a loss function enforcing the Martingale property for the PDE solution is used for an efficient optimization by sampling the stochastic processes associated with corresponding elliptic operators. High dimensional numerical results for BVPs of the linear and nonlinear Poisson-Boltzmann equation and eigenvalue problems of the Laplace equation and a Fokker-Planck equation demonstrate the capability of the proposed DeepMartNet learning method in solving high dimensional PDE problems.
- Cai W. DeepMartNet – A Martingale based Deep Neural Network Learning Algorithm for Eigenvalue/BVP Problems and Optimal Stochastic Controls. arXiv preprint arXiv:2307.11942. 2023 Jul 21.
- Cohen SN, Elliott RJ. Stochastic calculus and applications. New York: Birkhäuser; 2015 Nov 18.
- Davis MH. Martingale methods in stochastic control. InStochastic Control Theory and Stochastic Differential Systems: Proceedings of a Workshop of the Sonderforschungsbereich 72 der Deutschen Forschungsgemeinschaft an der Universität Bonn “which took place in January 1979 at Bad Honnef 2005 Oct 6 (pp. 85-117). Berlin, Heidelberg: Springer Berlin Heidelberg.
- Hsu P. Probabilistic approach to the Neumann problem. Communications on pure and applied mathematics. 1985 Jul;38(4):445-72.
- Karatzas I, Shreve S. Brownian motion and stochastic calculus. Springer Science & Business Media; 2012 Dec 6.
- Kingma DP, Ba J. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980. 2014 Dec 22.
- Klebaner FC. Introduction to stochastic calculus with applications. World Scientific Publishing Company; 2012 Mar 21.
- Papanicolaou VG. The probabilistic solution of the third boundary value problem for second order elliptic equations, Probab. Theory Relat. Fields 87 (1990) 27-77.
- Raissi M. Forward-backward stochastic neural networks: Deep learning of high-dimensional partial differential equations. arXiv preprint arXiv:1804.07010. 2018 Apr 19.
- Schuss Z. Brownian dynamics at boundaries and interfaces. Springer-Verlag New York; 2015.
- Zhang W, Cai W. FBSDE based neural network algorithms for high-dimensional quasilinear parabolic PDEs. Journal of Computational Physics. 2022 Dec 1;470:111557.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.