Gaussian Flow Dynamics: Simulation-Free Neural SDE Learning Beyond One-Time Marginals

2026-10-03

AI Summary

This study addresses the limitations of simulation-free SDE training, where fixed Gaussian marginals neglect temporal structure, resulting in restricted posterior families and model bias. To overcome this, we propose a Gaussian flow dynamics framework. The core innovation lies in explicitly parameterizing the gauge freedom that preserves marginals, transcending conventional one-shot marginal constraints to support state-dependent diffusion coefficients. Methodologically, efficient learning is achieved by integrating an SDE matching objective, a gauge matching algorithm, and variational inference. Experimental results demonstrate that the proposed approach accurately recovers linear SDE posteriors while maintaining linear computational complexity. Furthermore, it achieves performance comparable to Helmholtz-SDE on nonlinear systems with broader applicability.

Abstract

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.

Problem

• Stochastic Differential Equations
• Simulation-free learning
• One-time marginals
• Latent SDE
• Posterior process

Innovation

• Gaussian Flow Dynamics
• Gauge Matching
• Simulation-Free SDE Learning
• Stochastic Differential Equations
• Gauge Degrees of Freedom