Solving high-dimensional partial differential equations using deep learning.

Area of Science:

• Computational Mathematics
• Machine Learning
• Numerical Analysis

Background:

• High-dimensional partial differential equations (PDEs) pose significant computational challenges due to the "curse of dimensionality."
• Existing numerical methods often struggle with scalability and accuracy in high-dimensional spaces.

Purpose of the Study:

• To develop a novel deep learning-based algorithm for solving general high-dimensional parabolic PDEs.
• To address the limitations of traditional methods in handling complex, high-dimensional problems.

Main Methods:

• Reformulation of PDEs using backward stochastic differential equations.
• Approximation of the solution's gradient using neural networks, inspired by deep reinforcement learning.
• Implementation of a deep learning framework to approximate the solution.

Main Results:

• The proposed algorithm demonstrates high effectiveness and computational efficiency in high dimensions.
• Successful application to benchmark problems including the nonlinear Black-Scholes, Hamilton-Jacobi-Bellman, and Allen-Cahn equations.
• Validation of the method's accuracy and cost-effectiveness compared to existing approaches.

Conclusions:

• The deep learning approach offers a viable solution for high-dimensional parabolic PDEs.
• This method has broad implications for fields such as economics, finance, operational research, and physics.
• Enables simultaneous consideration of all interacting components without ad hoc assumptions.