DGM: A deep learning algorithm for solving partial differential equations

High-dimensional PDEs have been a longstanding computational challenge. We propose to solve high-dimensional PDEs by approximating the solution with a deep neural network which is trained to satisfy the differential operator, initial condition, and boundary conditions. Our algorithm is meshfree, whi...

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Bibliographic Details
Published in:Journal of computational physics Vol. 375; pp. 1339 - 1364
Main Authors: Sirignano, Justin, Spiliopoulos, Konstantinos
Format: Journal Article
Language:English
Published: Cambridge Elsevier Inc 15.12.2018
Elsevier Science Ltd
Subjects:
Algorithms
Approximation
Artificial intelligence
Artificial neural networks
Basis functions
Boundary conditions
Burgers equation
Computational physics
Deep learning
Free boundaries
Galerkin method
High-dimensional partial differential equations
Machine learning
Meshless methods
Neural networks
Operators (mathematics)
Parabolic differential equations
Partial differential equations
Deep learning
High-dimensional partial differential equations
Partial differential equations
Machine learning
ISSN:0021-9991, 1090-2716
Online Access:Get full text
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Summary:High-dimensional PDEs have been a longstanding computational challenge. We propose to solve high-dimensional PDEs by approximating the solution with a deep neural network which is trained to satisfy the differential operator, initial condition, and boundary conditions. Our algorithm is meshfree, which is key since meshes become infeasible in higher dimensions. Instead of forming a mesh, the neural network is trained on batches of randomly sampled time and space points. The algorithm is tested on a class of high-dimensional free boundary PDEs, which we are able to accurately solve in up to 200 dimensions. The algorithm is also tested on a high-dimensional Hamilton–Jacobi–Bellman PDE and Burgers' equation. The deep learning algorithm approximates the general solution to the Burgers' equation for a continuum of different boundary conditions and physical conditions (which can be viewed as a high-dimensional space). We call the algorithm a “Deep Galerkin Method (DGM)” since it is similar in spirit to Galerkin methods, with the solution approximated by a neural network instead of a linear combination of basis functions. In addition, we prove a theorem regarding the approximation power of neural networks for a class of quasilinear parabolic PDEs. •We develop a deep learning algorithm for solving high-dimensional PDEs.•The algorithm is meshfree, which is key since meshes become infeasible in higher dimensions.•We accurately solve a class of high-dimensional free boundary PDEs in up to 200 dimensions.•We prove a theorem regarding the approximation power of neural networks for quasilinear PDEs.
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ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2018.08.029