Convergence of the Deep Galerkin Method for Mean Field Control Problems
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866910456942166016 |
|---|---|
| author | Hofgard, William Sun, Jingruo Cohen, Asaf |
| author_facet | Hofgard, William Sun, Jingruo Cohen, Asaf |
| contents | We establish the convergence of the deep Galerkin method (DGM), a deep learning-based scheme for solving high-dimensional nonlinear PDEs, for Hamilton-Jacobi-Bellman (HJB) equations that arise from the study of mean field control problems (MFCPs). Based on a recent characterization of the value function of the MFCP as the unique viscosity solution of an HJB equation on the simplex, we establish both an existence and convergence result for the DGM. First, we show that the loss functional of the DGM can be made arbitrarily small given that the value function of the MFCP possesses sufficient regularity. Then, we show that if the loss functional of the DGM converges to zero, the corresponding neural network approximators must converge uniformly to the true value function on the simplex. We also provide numerical experiments demonstrating the DGM's ability to generalize to high-dimensional HJB equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_13346 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence of the Deep Galerkin Method for Mean Field Control Problems Hofgard, William Sun, Jingruo Cohen, Asaf Optimization and Control Machine Learning 91A07, 35Q89, 68T07, 49L12, 49N10, 35A35, 60J27 We establish the convergence of the deep Galerkin method (DGM), a deep learning-based scheme for solving high-dimensional nonlinear PDEs, for Hamilton-Jacobi-Bellman (HJB) equations that arise from the study of mean field control problems (MFCPs). Based on a recent characterization of the value function of the MFCP as the unique viscosity solution of an HJB equation on the simplex, we establish both an existence and convergence result for the DGM. First, we show that the loss functional of the DGM can be made arbitrarily small given that the value function of the MFCP possesses sufficient regularity. Then, we show that if the loss functional of the DGM converges to zero, the corresponding neural network approximators must converge uniformly to the true value function on the simplex. We also provide numerical experiments demonstrating the DGM's ability to generalize to high-dimensional HJB equations. |
| title | Convergence of the Deep Galerkin Method for Mean Field Control Problems |
| topic | Optimization and Control Machine Learning 91A07, 35Q89, 68T07, 49L12, 49N10, 35A35, 60J27 |
| url | https://arxiv.org/abs/2405.13346 |