Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations
This work addresses the challenge of solving high-dimensional PDEs and BSDEs, which is critical for applications in physics and finance, by introducing a novel deep learning-based method that shows competitive performance on specific benchmarks.
The authors tackled the problem of solving high-dimensional parabolic PDEs and BSDEs by proposing a new algorithm that analogizes BSDEs to reinforcement learning, using neural networks to approximate the policy function. Numerical results demonstrated efficiency and accuracy for 100-dimensional nonlinear PDEs from physics and finance, such as the Allen-Cahn and Hamilton-Jacobi-Bellman equations.
We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function, and the loss function given by the error between the prescribed terminal condition and the solution of the BSDE. The policy function is then approximated by a neural network, as is done in deep reinforcement learning. Numerical results using TensorFlow illustrate the efficiency and accuracy of the proposed algorithms for several 100-dimensional nonlinear PDEs from physics and finance such as the Allen-Cahn equation, the Hamilton-Jacobi-Bellman equation, and a nonlinear pricing model for financial derivatives.