发表机构
Purdue University; University of California, Los Angeles（普渡大学; 加利福尼亚大学洛杉矶分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出首个无网格自监督神经算子NFIST，通过结合概率流ODE与可逆归一化流Transformer解决随机平均场控制问题，可实现未见任务的一次前向求解，在多任务上展现零样本泛化并降低计算成本。
AI 中文摘要
随机平均场控制（MFC）为在不确定性下协调大量相互作用智能体提供了基础框架，具有广泛应用。现有数值和深度学习方法每次求解一个MFC问题实例，任务变化时必须重新优化。本研究将随机MFC表述为算子学习问题，开发了据我们所知首个无网格、自监督的随机MFC神经算子。主要挑战在于受控Fokker-Planck方程中的扩散项排除了确定性传输映射表示。我们通过将概率流常微分方程与基于可逆归一化流的Transformer相结合解决该挑战，将动力学重新表述为确定性连续性方程，并通过归一化流的精确逆和解析对数行列式实现闭式得分评估，固定规模网络中每个粒子的计算复杂度为O(d)。通过基于Transformer的上下文学习，以紧凑分布参数或原始粒子云表示的任务提示对传输映射进行条件设置，使单个预训练算子能在一次前向传播中求解未见任务。所得可逆归一化解Transformer（NFIST）通过直接最小化随机控制目标进行端到端训练，无需预计算数值解用于训练。我们进一步证明所提算子学习表述与逐任务优化的一致性。在随机最优控制、薛定谔桥、系统风险控制及避障路径规划上的数值实验表明，其具备有效的零样本泛化能力，同时大幅降低求解大规模随机MFC问题族的计算成本。
英文摘要
Stochastic mean-field control (MFC) provides a fundamental framework for coordinating large populations of interacting agents under uncertainty, with a wide range of applications. Existing numerical and deep-learning methods solve one MFC problem instance at a time and must be re-optimized whenever the task changes. In this work, we formulate stochastic MFC as an operator-learning problem and develop, to the best of our knowledge, the first mesh-free, self-supervised neural operator for stochastic MFC. The main challenge is that the diffusion term in the controlled Fokker--Planck equation precludes deterministic transport-map representations. We address this challenge by combining the probability-flow ODE with an invertible normalizing-flow-based transformer, which recasts the dynamics as a deterministic continuity equation and enables closed-form score evaluation through the exact inverse and analytical log-determinant of the normalizing flow, with $\mathcal{O}(d)$ cost per particle for networks of fixed size. Through transformer-based in-context learning, task prompts, represented by compact distribution parameters or raw particle clouds, condition the transport map, enabling a single pretrained operator to solve unseen tasks in one forward pass. The resulting \emph{Normalizing Flow Invertible Solution Transformer} (NFIST) is trained end-to-end by minimizing the stochastic control objective directly, requiring no precomputed numerical solutions for training. We further prove the consistency of the proposed operator-learning formulation with task-by-task optimization. Numerical experiments on stochastic optimal control, Schrödinger bridge, systemic-risk control, and obstacle-avoiding path planning demonstrate effective zero-shot generalization while substantially reducing the computational cost of solving large families of stochastic MFC problems.
