用神经网络模拟曲面上的扩散过程，无需网格和坐标系。
Neural Pushforward Samplers for the Fokker-Planck Equation on Embedded Riemannian Manifolds
- 通过神经映射将概率分布投影到曲面，自动保持分布守恒。
- 在球面上求解双阱问题，成功捕捉多峰稳态分布。
- 完全无网格、无局部坐标，适合复杂曲面建模。
本文将弱对抗神经前推方法拓展至紧致嵌入黎曼流形上的福克-普朗克方程。该方法通过神经前推映射表示解为概率分布，利用收缩层约束映射结果始终位于流形上，从而保证流形隶属关系与概率守恒。训练基于弱对抗目标，采用环境平面波测试函数，其内在微分算子由嵌入几何直接导出，实现完全无网格、无坐标图的算法。发展了稳态与瞬态两种形式，数值结果在二维球面上的双阱问题中验证了该方法在曲面上捕捉多峰不变分布的能力。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。