英文标题：
《DGM: A deep learning algorithm for solving partial differential
equations》
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作者：
Justin Sirignano and Konstantinos Spiliopoulos
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最新提交年份：
2018
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英文摘要：
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.
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中文摘要：
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层神经网络逼近解来求解高维偏微分方程，该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，因为网格在高维中变得不可行。神经网络不是形成网格，而是在一批随机采样的时间和空间点上进行训练。该算法在一类高维自由边界偏微分方程上进行了测试，我们能够在高达200美元的维度上精确求解。该算法也在高维Hamilton-Jacobi-Bellman偏微分方程和Burgers方程上进行了测试。深度学习算法近似于不同边界条件和物理条件（可视为高维空间）的连续统的Burgers方程的一般解。我们将该算法称为“深伽辽金方法（DGM）”，因为它与伽辽金方法在精神上相似，其解由神经网络近似，而不是基函数的线性组合。此外，我们还证明了一类拟线性抛物型偏微分方程的神经网络逼近能力定理。
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分类信息：

一级分类：Quantitative Finance 数量金融学
二级分类：Mathematical Finance 数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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一级分类：Mathematics 数学
二级分类：Numerical Analysis 数值分析
分类描述：Numerical algorithms for problems in analysis and algebra, scientific computation
分析和代数问题的数值算法，科学计算
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一级分类：Quantitative Finance 数量金融学
二级分类：Computational Finance 计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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一级分类：Statistics 统计学
二级分类：Machine Learning 机器学习
分类描述：Covers machine learning papers (supervised, unsupervised, semi-supervised learning, graphical models, reinforcement learning, bandits, high dimensional inference, etc.) with a statistical or theoretical grounding
覆盖机器学习论文（监督，无监督，半监督学习，图形模型，强化学习，强盗，高维推理等）与统计或理论基础
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