发表机构
Informatics Institute; University of Amsterdam（信息学研究所; 阿姆斯特丹大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出高斯流动力学，显式参数化保持边际的规范自由度，结合SDE匹配目标形成规范匹配方法，以二次成本学习潜在SDE的时间结构，适用于状态相关噪声。
AI 中文摘要
潜在随机微分方程（SDE）的无模拟训练依赖于一个变分后验过程，其单时边际是可解的，通常是高斯的。然而，这样的边际并不决定潜在的动力学：许多过程共享相同的边际，但在时间结构上有所不同，现有的参数化隐式地固定了这种结构，这限制了后验族并使学习到的模型产生偏差。我们引入了高斯流动力学，它直接从平滑演化的高斯边际构造随机过程，同时使保持边际的（或规范）自由度显式化且可参数化。该构造允许状态相关的扩散系数，并恢复所有具有加性噪声和非退化高斯初始分布的线性SDE。在此基础上，我们提出了规范匹配（Gauge Matching），一种用于潜在SDE学习的无模拟方法，它将高斯流动力学与SDE匹配目标相结合。规范匹配每步在潜在维度上的成本最多为二次，如同SDE匹配，但它学习了后验超越其单时边际的时间结构。在线性基准上，它达到了与Helmholtz-SDE相差一个纳特（nat）的精度，而Helmholtz-SDE从先验雅可比以三次成本计算规范，在已知精确后验的情况下，规范匹配在非线性系统上与之相当，并适用于Helmholtz-SDE不适用的情况，即状态相关噪声。
英文摘要
Simulation-free training of latent Stochastic Differential Equations (SDEs) relies on a variational posterior process whose one-time marginals are tractable, typically Gaussian. Such marginals, however, do not determine the underlying dynamics: many processes share the same marginals while differing in their temporal structure, and existing parameterizations fix this structure implicitly, which restricts the posterior family and biases the learned model. We introduce Gaussian flow dynamics, which construct stochastic processes directly from smoothly evolving Gaussian marginals while making the marginal-preserving, or gauge, degrees of freedom explicit and parameterizable. The construction admits state-dependent diffusion coefficients and recovers every linear SDE with additive noise and a non-degenerate Gaussian initial distribution. Building on it, we propose Gauge Matching, a simulation-free method for latent SDE learning that combines Gaussian flow dynamics with the SDE Matching objective. Gauge Matching costs at most quadratically in the latent dimension per step, like SDE Matching, but learns the temporal structure of the posterior beyond its one-time marginals. It comes within a nat of Helmholtz-SDE, which computes the gauge from the prior Jacobian at cubic cost, on the linear benchmark where the exact posterior is known, matches it on nonlinear systems, and applies where Helmholtz-SDE does not, to state-dependent noise.
