DGM: A deep learning algorithm for solving partial differential equations
DGM: A deep learning algorithm for solving partial differential equations
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DOI：
10.1016/j.jcp.2018.08.029
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发表时间：
2018-12-15
影响因子：
4.1
通讯作者：
Spiliopoulos, Konstantinos
中科院分区：
文献类型：
--
作者：
Sirignano, Justin;Spiliopoulos, Konstantinos
高维偏微分方程一直是一个长期的计算挑战。我们建议通过使用深度神经网络来近似解决高维偏微分方程，该深度神经网络经过训练以满足微分算子，初始条件和边界条件。我们的算法是无网格的，这是关键，因为网格在更高的维度变得不可行。神经网络不是形成一个网格，而是在一批随机采样的时间和空间点上进行训练。该算法是测试一类高维自由边界偏微分方程，我们能够准确地解决在多达200个维度。该算法还测试了高维Hamilton-Jacobi-Bellman偏微分方程和Burgers方程。深度学习算法近似Burgers方程的一般解，用于不同边界条件和物理条件的连续体（可以被视为高维空间）。我们称该算法为“深层伽辽金方法（DGM）”，因为它在精神上类似于伽辽金方法，用神经网络而不是基函数的线性组合来近似解。此外，我们证明了一个关于神经网络对一类拟线性抛物型偏微分方程的逼近能力的定理。(C)2018爱思唯尔公司All rights reserved.
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. (C) 2018 Elsevier Inc. All rights reserved.