Proceedings of the National Academy of Sciences· 2018Q1
Solving high-dimensional partial differential equations using deep learning
- 1,720citations
- Q1SCImago
- 2018year
Short summary
A novel deep learning algorithm effectively solves high-dimensional parabolic partial differential equations (PDEs) by reformulating them as backward stochastic differential equations and approximating solution gradients with neural networks.
AI-generated from the title and abstract; the full text is not read.
Key points
- Introduces a deep learning algorithm for high-dimensional parabolic PDEs.
- Reformulates PDEs as backward stochastic differential equations.
- Approximates the gradient of the unknown solution using neural networks.
- Demonstrates effectiveness on nonlinear Black-Scholes, Hamilton-Jacobi-Bellman, and Allen-Cahn equations.
AI-generated from the title and abstract; the full text is not read.
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 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 interrelationships.
The authors' abstract, as published at the source. Proceedings of the National Academy of Sciences, 2018 · DOI ↗
Continue with a free account
Ask the paper: 3 free questions a day about this paper; save it, get its citation, new summaries every day for your field. Takeaways are Premium.
Continue free on the webSign in with Google or Apple; no card needed. You come back to this paper.
On your phone:
Field: Statistical and Nonlinear Physics
Statistical and Nonlinear PhysicsPhysics and Astronomy