Recovering dynamical systems from noisy observations is a recurring challenge across scientific domains, including neuroscience and physics. Latent stochastic differential equations (SDEs) address this by modeling the system as an unobserved state that evolves according to a learnable SDE and generates the observations. Variational inference (VI) provides a tractable objective for fitting latent SDEs. Traditional VI algorithms evaluate this objective by numerical simulation over a time discretization, trading fidelity for computational cost. A recent class of algorithms, simulation-free VI, sidesteps this tradeoff by parameterizing the posterior through its instantaneous marginals rather than its drift. In this work, we show that the efficiency of existing simulation-free VI algorithms comes at a price: their parameterizations restrict the approximate posterior to a subset of the SDEs available to simulation-based methods, degrading posterior inference and parameter learning. We propose Helmholtz-SDE, a simulation-free VI algorithm that closes this gap by optimizing over path laws compatible with a prescribed collection of marginals. Helmholtz-SDE recovers dynamics more faithfully than prior simulation-free methods, with the largest gains under high posterior uncertainty. It further matches the performance of simulation-based VI at a fraction of the runtime.

翻译：从含噪观测中恢复动力系统是包括神经科学和物理学在内的科学领域中的反复挑战。潜变量随机微分方程通过将系统建模为一个根据可学习随机微分方程演化并生成观测的未观测状态来解决这一问题。变分推断为拟合潜变量随机微分方程提供了可处理的目标。传统变分推断算法通过时间离散化上的数值模拟来评估该目标，从而在保真度与计算成本之间做出权衡。最近一类算法——无模拟变分推断——通过以其瞬时边际分布而非漂移项参数化后验分布，规避了这一权衡。在本工作中，我们证明现有无模拟变分推断算法的高效性是以牺牲精度为代价的：其参数化方式将近似后验限制在基于模拟方法可用的随机微分方程子集中，从而降低了后验推断和参数学习的效果。我们提出Helmholtz-SDE，一种通过优化与预设边际分布集兼容的路径律来弥合这一差距的无模拟变分推断算法。Helmholtz-SDE比先前的无模拟方法更忠实地恢复动力系统，在高后验不确定度下增益最为显著。它还能以极短的运行时间匹配基于模拟的变分推断性能。