A supervised learning scheme for computing Hamilton-Jacobi equation via density coupling
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909886236852224 |
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| author | Cui, Jianbo Liu, Shu Zhou, Haomin |
| author_facet | Cui, Jianbo Liu, Shu Zhou, Haomin |
| contents | We propose a supervised learning scheme for the first order Hamilton--Jacobi PDEs in high dimensions. The scheme is designed by using the geometric structure of Wasserstein Hamiltonian flows via a density coupling strategy. It is equivalently posed as a regression problem using the Bregman divergence, which provides the loss function in learning while the data is generated through the particle formulation of Wasserstein Hamiltonian flow. We prove a posterior estimate on $L^1$ residual of the proposed scheme based on the coupling density. Furthermore, the proposed scheme can be used to describe the behaviors of Hamilton--Jacobi PDEs beyond the singularity formations on the support of coupling density. Several numerical examples with different Hamiltonians are provided to support our findings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_15954 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A supervised learning scheme for computing Hamilton-Jacobi equation via density coupling Cui, Jianbo Liu, Shu Zhou, Haomin Numerical Analysis 65M75, 65P10, 49Q22, 68T07 We propose a supervised learning scheme for the first order Hamilton--Jacobi PDEs in high dimensions. The scheme is designed by using the geometric structure of Wasserstein Hamiltonian flows via a density coupling strategy. It is equivalently posed as a regression problem using the Bregman divergence, which provides the loss function in learning while the data is generated through the particle formulation of Wasserstein Hamiltonian flow. We prove a posterior estimate on $L^1$ residual of the proposed scheme based on the coupling density. Furthermore, the proposed scheme can be used to describe the behaviors of Hamilton--Jacobi PDEs beyond the singularity formations on the support of coupling density. Several numerical examples with different Hamiltonians are provided to support our findings. |
| title | A supervised learning scheme for computing Hamilton-Jacobi equation via density coupling |
| topic | Numerical Analysis 65M75, 65P10, 49Q22, 68T07 |
| url | https://arxiv.org/abs/2401.15954 |