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Closing the Approximation Gap in Simulation-free Latent SDEs

NeurIPS 2026 · Probabilistic and scientific modeling › Dynamical systems · approach: State-space inference for stochastic systems · 0 citations (OpenAlex)

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Proposes Helmholtz-SDE, a simulation-free variational inference method for latent SDEs that optimizes over all path laws sharing given Gaussian one-time marginals by adding a Helmholtz (divergence-free) correction to a reference drift.

Problem

Simulation-free VI for latent SDEs restricts the variational posterior to a strictly smaller family than simulation-based VI, because one-time marginals do not determine the path law, degrading posterior inference and parameter learning.

Approach

Starting from a reference drift that realizes prescribed Gaussian marginals, add a q-divergence-free correction (from the q-weighted Helmholtz decomposition of the residual prior-posterior drift, truncated to Hermite polynomials of degree ℓ) that preserves marginals while lowering the path-space KL, closing the gap to simulation-based VI.

Key results

• On the Lorenz attractor, Helmholtz-SDE achieves a ~20 nat smaller negative ELBO than the other simulation-free methods (SDE Matching, SVISE).
• On the Lorenz attractor, Helmholtz-SDE reaches per-trial nELBO below 50 nat in 123.7 ± 3.4 s on an H100, versus 3695.6 ± 503.8 s for SING (>25x slower).
• On predator-prey data, Helmholtz-SDE recovers a median period of 6.92 days and lag of 1.66 days, versus 13.33 days and 3.23 days for SDE Matching.

Surprising findings

• Better variational approximations do not translate monotonically into better learned dynamics, consistent with prior work.
• Under the OU spiral setting, SDE Matching's square root gauge incurs a KL gap to the posterior that grows linearly in the rotation speed omega, even though the posterior is in the conjugate setting.

Limitations

• Closed-form correction relies on state-independent diffusion and Gaussian one-time marginals.
• Higher-order polynomial corrections scale unfavorably with latent dimension, O(K^max(l+1,3)) complexity.
• Gains beyond linear and quadratic corrections were not observed in experiments.

Experimental setup

Datasets: Ornstein-Uhlenbeck spiral (synthetic) noisy Lorenz attractor (synthetic) rotifer-algae predator-prey data 2D cylinder flow at Re 2000 (POD reduced order model)

Benchmarks: OU spiral noisy Lorenz attractor predator-prey forecasting fluid dynamics ROM forecasting

Metrics: negative ELBO symmetric KL to posterior symmetric KL to true prior lagged autocorrelation RMSE forecast MSE runtime

Baselines: SDE Matching SVISE SING Latent SDE

Builds on: SDE Matching SVISE Latent SDE Girsanov theorem

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