神经推前采样器用于嵌入黎曼流形上的福克-计划克方程
Neural Pushforward Samplers for the Fokker-Planck Equation on Embedded Riemannian Manifolds
AI总结：
本文扩展了弱对抗神经推前方法，用于嵌入黎曼流形上的福克-计划克方程，通过神经推前映射和重traction层约束流形，实现概率守恒，采用弱对抗目标训练，无需网格或坐标图，验证了方法在曲面多模态不变分布中的能力。
摘要：
在本文中，我们扩展了弱对抗神经推前方法，用于在紧致嵌入黎曼流形上的福克-计划克方程。该方法通过神经推前映射将解表示为概率分布，该映射通过重traction层约束于流形，从而强制流形成员资格和概率守恒。训练由弱对抗目标引导，使用环境平面波测试函数，其内在微分算子从嵌入几何中闭合形式导出，从而得到完全无网格和无坐标图的算法。开发了稳态和时间依赖形式，并在双井问题上的两球体上进行了数值实验，验证了该方法在曲面上捕捉多模态不变分布的能力。
英文摘要：
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.
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