Solving high-dimensional partial differential equations using deep learning

By Jiequn Han, Arnulf Jentzen, W. E. 2017

2,069 citations recorded by the source corpus.

Significance Partial differential equations (PDEs) are among the most ubiquitous tools used in modeling problems in nature. However, solving high-dimensional PDEs has been notoriously difficult due to the “curse of dimensionality.” This paper introduces a practical algorithm for solving nonlinear PDEs in very high (hundreds and potentially thousands of) dimensions. Numerical results suggest that the proposed algorithm is quite effective for a wide variety of problems, in terms of both accuracy and speed. We believe that this opens up a host of possibilities in economics, finance, operational research, and physics, by considering all participating agents, assets, resources, or particles together at the same time, instead of making ad hoc assumptions on their interrelationships. Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the “curse of dimensionality.” This paper introduces a deep learning-based approach that can handle general high-dimensional parabolic PDEs. To this end, the PDEs are reformulated using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy function. Numerical results on examples including the nonlinear Black–Scholes equation, the Hamilton–Jacobi–Bellman equation, and the Allen–Cahn equation suggest that the proposed algorithm is quite effective in high dimensions, in terms of both accuracy and cost. This opens up possibilities in economics, finance, operational research, and physics, by considering all participating agents, assets, resources, or particles together at the same time, instead of making ad hoc assumptions on their interrelationships.

Subfields: Artificial Intelligence, Computational Mechanics, Statistical and Nonlinear Physics

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Citations in the corpus

Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data (2019, background): “, for solving a handful of computer vision problems [48], deterministic PDEs [33, 49, 50], high-dimensional stochastic partial differential equations (SPDE), and backward stochastic differential equations (BSDE) [51–55].”
Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks (2020, background): “For instance, various studies deal with problems of data-driven modeling, especially ones using deep learning to solve differential equations [11].”
Deep Autoencoder based Energy Method for the Bending, Vibration, and Buckling Analysis of Kirchhoff Plates (2020, methodology): “E et al. developed a deep learning-based numerical method for high-dimensional parabolic PDEs and back-forward stochastic diﬀerential equations [38, 39].”
A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applications (2020, methodology): “- Physics-informed ML models trained using governing PDEs: Instead of solving PDEs using FE tools to generate training datasets, in several studies governing PDEs have been used directly to train ML models (Han et al., 2018; Raissi, 2018; Raissi et al., 2019, 2017a, 2017b; Sirignano and Spiliopoulos”
Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism (2020, methodology): “ces in GPU capabilities and training algorithms for deep neural networks [5, 6], as well as the invention of automatic differentiation methods [7, 8], which can be used to solve ordinary [9], partial [10], or stochastic [11] differential equations. A great advantage of PINNs over traditional time-st”
Solving Irregular and Data-enriched Differential Equations using Deep Neural Networks (2019, background): “[7], Berg and Nyström [8], Sirignano and Spiliopoulos [9], Raissi et al.”
Hamiltonian Neural Networks for solving differential equations (2020, methodology): “In addition to the data-driven studies, equation-driven unsupervised NNs have been used to solve ordinary and partial DEs relevant to a variaty of physical systems [16]–[19].”
PhyGeoNet: Physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain (2020, methodology): “The effectiveness of the data-free FC-NN based PDE solution algorithm has been demonstrated on a number of canonical PDEs [41, 42] and stochastic PDEs [1, 43, 44].”
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