Solving high-dimensional partial differential equations using deep learning (1707.02568v3)

Abstract: 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 new 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 inter-relationships.

• The paper presents a novel deep BSDE method that reformulates high-dimensional PDEs as backward stochastic processes to mitigate the curse of dimensionality.
• It employs neural network approximations with Euler discretization and the Adam optimizer, yielding low relative errors in benchmark tests.
• The method demonstrates robust performance on nonlinear Black-Scholes, HJB, and Allen-Cahn equations, highlighting its scalability and practical impact.

Solving High-Dimensional Partial Differential Equations Using Deep Learning

Overview

The paper "Solving High-Dimensional Partial Differential Equations Using Deep Learning" by Jiequn Han, Arnulf Jentzen, and Weinan E introduces a novel approach to address the challenge of solving high-dimensional parabolic partial differential equations (PDEs). The methodology leverages deep learning techniques to overcome the computational bottlenecks traditionally associated with the curse of dimensionality. Specifically, the paper reformulates PDEs using backward stochastic differential equations (BSDEs) and approximates the gradient of the unknown solution through neural networks.

Key Methodology

The approach taken in this paper involves several steps:

1. BSDE Reformulation: The PDEs are first recast into the framework of BSDEs. This allows the problem of solving a PDE to be transformed into solving a stochastic process.
2. Neural Network Approximation: The gradient of the solution to the BSDE is approximated using deep neural networks. This is akin to the policy function in deep reinforcement learning.
3. Temporal Discretization: An Euler scheme is applied to discretize the temporal component, facilitating numerical approximation.
4. Optimization: A stochastic gradient descent-type (SGD) algorithm, specifically the Adam optimizer, is employed to train the neural networks and optimize the parameters.

Numerical Results

The proposed deep BSDE method was tested on several high-dimensional examples, demonstrating its efficacy in terms of both computational cost and accuracy.

1. Nonlinear Black-Scholes Equation with Default Risk:
   • Dimension: 100
   • Relative Error: 0.46%
   • Computational Time: 1607 seconds
2. Hamilton-Jacobi-BeLLMan (HJB) Equation:
   • Dimension: 100
   • Relative Error: 0.17%
   • Computational Time: 330 seconds
3. Allen-Cahn Equation:
   • Dimension: 100
   • Relative Error: 0.30%
   • Computational Time: 647 seconds

These results underscore the method's robustness in handling various types of high-dimensional PDEs.

Implications and Future Directions

The deep BSDE method expands the toolkit available for solving high-dimensional PDEs, with significant implications in fields such as economics, finance, and operational research. By enabling the consideration of multiple interacting components without simplifying assumptions, this approach enhances the precision and applicability of models used in these domains.

Theoretically, this methodology highlights the potential of neural networks to serve as powerful function approximators in the context of stochastic processes. Practically, the algorithm's scalability to higher dimensions without encountering exponential growth in computational demand is particularly noteworthy.

Conclusion

While the paper presents a robust and generalizable methodology for solving high-dimensional PDEs, there remain challenges, particularly in extending the approach to quantum many-body problems due to the Pauli exclusion principle. Future work could explore refining the neural network architecture, improving training algorithms, and extending applicability to a broader class of PDEs, thereby further cementing the role of deep learning in computational mathematics.