Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations
Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations
This paper introduces a novel deep learning-based numerical method, termed the deep BSDE solver, for solving high-dimensional parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs). The key idea is to leverage the nonlinear Feynman-Kac formula to reformulate the PDE as a BSDE, which is then interpreted as a stochastic control problem. In this analogy, the gradient of the PDE solution serves as the policy function, and the loss function is the squared error between the BSDE's terminal value and the prescribed terminal condition. The policy function is approximated by a deep neural network, and the parameters are optimized using stochastic gradient descent to minimize the loss. This approach effectively transforms the PDE solving problem into a deep reinforcement learning task.
The algorithm is tested on several 100-dimensional nonlinear PDEs from physics and finance, including the Allen-Cahn equation, the Hamilton-Jacobi-Bellman equation, and a nonlinear pricing model for financial derivatives. Numerical results demonstrate that the deep BSDE solver achieves high accuracy and efficiency, successfully overcoming the curse of dimensionality that plagues traditional numerical methods. The method's ability to handle high dimensions opens new possibilities for solving complex PDEs arising in various scientific and engineering fields. The paper also provides a detailed description of the algorithm, including time discretization and neural network architecture, and discusses potential extensions to more general PDEs and BSDEs.
Highlights
- 1Proposes a deep learning-based algorithm (deep BSDE solver) for solving high-dimensional parabolic PDEs and BSDEs, overcoming the curse of dimensionality.
- 2Establishes a connection between BSDEs and reinforcement learning, where the gradient of the PDE solution acts as the policy function.
- 3Approximates the gradient of the solution using deep neural networks and optimizes via stochastic gradient descent to minimize the terminal condition error.
- 4Demonstrates efficiency and accuracy on 100-dimensional nonlinear PDEs from physics and finance, including Allen-Cahn, Hamilton-Jacobi-Bellman, and nonlinear pricing models.
Methods
- MNonlinear Feynman-Kac formula to reformulate PDEs as BSDEs.
- MForward Euler discretization of the BSDE to obtain a time-discrete approximation.
- MDeep neural network approximation of the spatial gradient of the solution (policy function).
- MStochastic gradient descent optimization of the expected squared terminal error loss function.
Results
- RThe deep BSDE solver successfully solves 100-dimensional semilinear parabolic PDEs with high accuracy.
- RNumerical experiments on the Allen-Cahn equation show accurate approximation of the solution at initial time.
- RFor the Hamilton-Jacobi-Bellman equation, the algorithm achieves low relative errors compared to analytical solutions.
- RIn a nonlinear pricing model for financial derivatives, the method provides reliable solutions in high dimensions.
- RThe algorithm scales well with dimension, demonstrating potential to break the curse of dimensionality for a class of PDEs.
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