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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.16239 |
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| _version_ | 1866917350940344320 |
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| author | He, Andrew Qing Cai, Wei |
| author_facet | He, Andrew Qing Cai, Wei |
| contents | In this paper, we extend the Weak Adversarial Neural Pushforward Method to the Fokker--Planck equation on compact embedded Riemannian manifolds. The method represents the solution as a probability distribution via a neural pushforward map that is constrained to the manifold by a retraction layer, enforcing manifold membership and probability conservation by construction. Training is guided by a weak adversarial objective using ambient plane-wave test functions, whose intrinsic differential operators are derived in closed form from the geometry of the embedding, yielding a fully mesh-free and chart-free algorithm. Both steady-state and time-dependent formulations are developed, and numerical results on a double-well problem on the two-sphere demonstrate the capability of the method in capturing multimodal invariant distributions on curved spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_16239 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Neural Pushforward Samplers for the Fokker-Planck Equation on Embedded Riemannian Manifolds He, Andrew Qing Cai, Wei Numerical Analysis Machine Learning 65N75, 68T07, 58J65, 35Q84 In this paper, we extend the Weak Adversarial Neural Pushforward Method to the Fokker--Planck equation on compact embedded Riemannian manifolds. The method represents the solution as a probability distribution via a neural pushforward map that is constrained to the manifold by a retraction layer, enforcing manifold membership and probability conservation by construction. Training is guided by a weak adversarial objective using ambient plane-wave test functions, whose intrinsic differential operators are derived in closed form from the geometry of the embedding, yielding a fully mesh-free and chart-free algorithm. Both steady-state and time-dependent formulations are developed, and numerical results on a double-well problem on the two-sphere demonstrate the capability of the method in capturing multimodal invariant distributions on curved spaces. |
| title | Neural Pushforward Samplers for the Fokker-Planck Equation on Embedded Riemannian Manifolds |
| topic | Numerical Analysis Machine Learning 65N75, 68T07, 58J65, 35Q84 |
| url | https://arxiv.org/abs/2603.16239 |