Finite-difference least square methods for solving Hamilton-Jacobi equations using neural networks
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909442804547584 |
|---|---|
| author | Esteve-Yagüe, Carlos Tsai, Richard Massucco, Alex |
| author_facet | Esteve-Yagüe, Carlos Tsai, Richard Massucco, Alex |
| contents | We present a simple algorithm to approximate the viscosity solution of Hamilton-Jacobi (HJ) equations by means of an artificial deep neural network. The algorithm uses a stochastic gradient descent-based method to minimize the least square principle defined by a monotone, consistent numerical scheme. We analyze the least square principle's critical points and derive conditions that guarantee that any critical point approximates the sought viscosity solution. The use of a deep artificial neural network on a finite difference scheme lifts the restriction of conventional finite difference methods that rely on computing functions on a fixed grid. This feature makes it possible to solve HJ equations posed in higher dimensions where conventional methods are infeasible. We demonstrate the efficacy of our algorithm through numerical studies on various canonical HJ equations across different dimensions, showcasing its potential and versatility. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_10758 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finite-difference least square methods for solving Hamilton-Jacobi equations using neural networks Esteve-Yagüe, Carlos Tsai, Richard Massucco, Alex Numerical Analysis Analysis of PDEs Optimization and Control 49L25, 35A15, 68T07, 49K20, 65N06 We present a simple algorithm to approximate the viscosity solution of Hamilton-Jacobi (HJ) equations by means of an artificial deep neural network. The algorithm uses a stochastic gradient descent-based method to minimize the least square principle defined by a monotone, consistent numerical scheme. We analyze the least square principle's critical points and derive conditions that guarantee that any critical point approximates the sought viscosity solution. The use of a deep artificial neural network on a finite difference scheme lifts the restriction of conventional finite difference methods that rely on computing functions on a fixed grid. This feature makes it possible to solve HJ equations posed in higher dimensions where conventional methods are infeasible. We demonstrate the efficacy of our algorithm through numerical studies on various canonical HJ equations across different dimensions, showcasing its potential and versatility. |
| title | Finite-difference least square methods for solving Hamilton-Jacobi equations using neural networks |
| topic | Numerical Analysis Analysis of PDEs Optimization and Control 49L25, 35A15, 68T07, 49K20, 65N06 |
| url | https://arxiv.org/abs/2406.10758 |