Solving High-Dimensional Partial Differential Equations Using Deep Learning

Solving High-Dimensional Partial Differential Equations Using Deep Learning

3 Jul 2018 | Jiequn Han, Arnulf Jentzen, Weinan E
This paper proposes a deep learning-based method for solving high-dimensional partial differential equations (PDEs). The method reformulates PDEs as backward stochastic differential equations (BSDEs) and uses deep neural networks to approximate the gradient of the solution. This approach is inspired by deep reinforcement learning, where the gradient acts as the policy function. The method is tested on examples including the nonlinear Black-Scholes equation, the Hamilton-Jacobi-Bellman equation, and the Allen-Cahn equation, showing effectiveness in high dimensions in terms of both accuracy and computational cost. The method allows for the simultaneous consideration of all agents, assets, resources, or particles, opening new possibilities in economics, finance, operational research, and physics. The deep BSDE method is shown to be effective in solving high-dimensional nonlinear PDEs, including the nonlinear Black-Scholes equation with default risk, the Hamilton-Jacobi-Bellman equation, and the Allen-Cahn equation. The method uses a deep neural network to approximate the solution and its gradient, with numerical results demonstrating high accuracy and efficiency. The method is also shown to be effective in handling high-dimensional problems where traditional methods are intractable due to the curse of dimensionality. The paper concludes that the deep BSDE method provides a promising approach for solving high-dimensional PDEs and has potential applications in various fields.This paper proposes a deep learning-based method for solving high-dimensional partial differential equations (PDEs). The method reformulates PDEs as backward stochastic differential equations (BSDEs) and uses deep neural networks to approximate the gradient of the solution. This approach is inspired by deep reinforcement learning, where the gradient acts as the policy function. The method is tested on examples including the nonlinear Black-Scholes equation, the Hamilton-Jacobi-Bellman equation, and the Allen-Cahn equation, showing effectiveness in high dimensions in terms of both accuracy and computational cost. The method allows for the simultaneous consideration of all agents, assets, resources, or particles, opening new possibilities in economics, finance, operational research, and physics. The deep BSDE method is shown to be effective in solving high-dimensional nonlinear PDEs, including the nonlinear Black-Scholes equation with default risk, the Hamilton-Jacobi-Bellman equation, and the Allen-Cahn equation. The method uses a deep neural network to approximate the solution and its gradient, with numerical results demonstrating high accuracy and efficiency. The method is also shown to be effective in handling high-dimensional problems where traditional methods are intractable due to the curse of dimensionality. The paper concludes that the deep BSDE method provides a promising approach for solving high-dimensional PDEs and has potential applications in various fields.
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[slides and audio] Solving high-dimensional partial differential equations using deep learning