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

Grigory Bartosh; Christian A. Naesseth

stat.MLOct 3, 2026

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

Authors: Grigory Bartosh, Christian A. Naesseth

Organizations: Informatics Institute University of Amsterdam

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.

Figures & tables

Method	Gauge	Cost	State-dependent $g$
Course and Nair (2023a) ; Course and Nair (2023b)	symmetric, fixed	$\mathcal{O}(d^{3})$	no
SDE Matching ( Bartosh et al., 2025 )	square root, fixed	$\mathcal{O}(d^{2})$	yes
Cholesky gauge (ablation)	$\Omega\equiv 0$ , fixed	$\mathcal{O}(d^{2})$	yes
Helmholtz- SDE ( Smith et al., 2026 )	order-one correction, computed	$\mathcal{O}(d^{3})$	no
Gauge Matching (ours)	learned	$\mathcal{O}(d^{2})$	yes

Table 1: Posterior processes with Gaussian marginals in simulation-free latent SDE learning. Cost is that of evaluating the drift coefficient at one

(t,z)

beyond the network evaluations, for a full

d\times d

covariance ( SDE Matching: diagonal or triangular

S

). The Cholesky gauge (

\Omega\equiv 0

) is Gauge Matching without the gauge.

Figure 1: Learning the gauge of the stationary Ornstein–Uhlenbeck spiral (

\alpha=0.2

\omega=2\pi

) without observations. A: one sample path of each fitted model, learned gauge (blue) and fixed gauge

O(t)\equiv I

(orange), over the true

2\sigma

disk (shaded) and each model’s own

2\sigma

circle (dashed): the fixed gauge cannot circulate and shrinks the marginals instead. B: lagged cross-covariance

[\mathrm{Cov}(Z_{t+\tau},Z_{t})]_{21}

of the true and the two fitted processes. C: rotation angle

\theta(t)

of the learned

O(t)

against

\omega t

Figure 2: Gauge Matching in the conjugate setting (

\sigma=0.3

; median and interquartile range over 16 trials). A: KL divergence to the true posterior process against the number of observations

N

\omega=2\pi

; the learned gauge is

4

–

34

times below the fixed gauges and within a nat of Helmholtz- SDE , whose gauge is exact here. B: learned gauge

\Omega_{12}(t)

for one trial with

N=10

, against the optimal gauge eq. 17 at the learned marginals and the gauge of the true posterior process, which jumps at the observation times (ticks). C: KL divergence against the rotation speed

\omega

N=10

Posterior process	Negative ELBO $\downarrow$	Latent LL $\uparrow$	Autocorr. error $\downarrow$	Gauge excess $\downarrow$
SDE Matching (square root)	$77.9\pm 0.6$	$-0.22\pm 0.01$	$0.134\pm 0.009$	$2.84\pm 0.08$
Cholesky gauge	$75.5\pm 0.8$	$-0.20\pm 0.01$	$0.111\pm 0.007$	$2.52\pm 0.09$
Symmetric gauge	$76.9\pm 0.8$	$-0.22\pm 0.01$	$0.111\pm 0.006$	$2.61\pm 0.11$
Helmholtz- SDE ( $\ell=1$ )	$53.0\pm 0.6$	$-0.06\pm 0.01$	$0.052\pm 0.006$	$0.001\pm 0.000$
Gauge Matching	$53.2\pm 0.7$	$-0.06\pm 0.01$	$0.056\pm 0.009$	$0.044\pm 0.003$
Gauge Matching, diagonal $L_{\phi}$	$55.3\pm 1.9$	$-0.10\pm 0.02$	$0.064\pm 0.006$	$0.022\pm 0.001$

Table 2: Noisy Lorenz attractor ( Smith et al., 2026 ) ,

d=4

; mean

\pm 2

standard errors over 5 seeds on held-out trials; arrows give the better direction. Negative ELBO : per trial, nats. Latent LL : log-density of the true state under the posterior predictive. Autocorr. error : distance between the lagged correlation matrices of the learned and the true prior process. Gauge excess :

\mathcal{J}(\Omega)-\mathcal{J}(\Omega^{\star})

of eq. 16 at the method’s own gauge and marginals, nats per unit time ( appendix D ).

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3: KL divergence to the true posterior process in the conjugate setting for every combination of observation noise $\sigma$ and number of observations $N$ , for the five posterior processes of section 5 , at $\omega=2\pi$ (top) and $\omega=0$ (bottom). Median over 16 trials. At $\omega=0$ the prior drift is a gradient field, so the symmetric gauge is optimal and Helmholtz- SDE coincides with it; the learned gauge lies within a factor of $1.5$ of both, and the square root gauge retains a gap because the posterior covariance is not isotropic under a generic $C$ . The residual error grows with $N$ and shrinks with $\sigma$ : it is the error of a continuously differentiable posterior at the observation times. At $\sigma=0.05$ every method is instead limited by optimization ( fig. 4 and the text).

Figure 4: What the variational family admits versus what training reaches. Same grid as fig. 3 , but each entry is the minimum of the closed-form KL divergence over the variational family of that gauge, found by L-BFGS directly on the time grid (functions smoothed by a Gaussian kernel of width $0.02$ ), rather than the divergence of a trained network. Median over 16 trials. The gap of the fixed gauges is a property of the family: their optima are $12$ – $28$ nats at $\omega=2\pi$ , and the trained models of fig. 3 lie within $0.04$ – $2.2$ nats of them at $\sigma\geq 0.3$ . The gauge-corrected families admit $0.02$ – $2.2$ nats there, and at $\sigma=0.05$ they admit $1.8$ – $15$ nats against the $5$ – $56$ nats that training reaches. Helmholtz- SDE at order one and the learned gauge have the same optimum, since the order-one correction is exact for a linear prior drift; at $\omega=0$ the symmetric gauge coincides with both.

Figure 5: Scaling in the latent dimension on the rotating Ornstein–Uhlenbeck system. A: gap to the true posterior process, generative model known (median and interquartile range over 16 trials and 3 seeds). B: negative ELBO per trial, generative model learned (mean

\pm 2

standard errors over 3 seeds). C: GPU time per training iteration, models as in B up to

d=128

(batch of 256 trials; median and interquartile range over 50 iterations).

Figure 6: Dimension sweep at an identifiable setting (learned prior drift, full observation,

N=21

\omega=\pi

; see the text). A: negative ELBO minus that of Helmholtz- SDE , paired per seed as the median over held-out trials of the per-trial difference (mean and

2

standard errors over 3 seeds), with the observation model fixed. B: gauge excess (Helmholtz- SDE is zero by construction and drawn at the floor of the axis). C: as A, with the observation model learned: the full-covariance posteriors drop latent directions (numbers: mean over seeds for the diagonal

\mid

full-covariance posteriors) and lose by hundreds of nats to the diagonal one.

Figure 7: Noisy Lorenz attractor under nine observation protocols: spacing

\Delta t\in\{0.125,0.25,0.5\}

(

N=41,21,11

observations per trial) and noise

\sigma\in\{0.15,0.3,0.6\}

(light to dark),

d=4

, the model and training of table 2 , three seeds per protocol. A: negative ELBO relative to Helmholtz- SDE , paired over seeds. B: gauge excess (log scale). C: autocorrelation error of the learned prior process. Mean

\pm 2

standard errors over seeds.

			Negative ELBO $\downarrow$				Gauge excess $\downarrow$
$\Delta t$	$N$	$\sigma$	Sq. root	Cholesky	Helmholtz	Gauge M.	Sq. root	Gauge M.
$0.125$	$41$	$0.15$	$45.8\pm 2.1$	$46.0\pm 1.1$	$25.7\pm 0.7$	$27.5\pm 2.1$	$2.97\pm 0.02$	$0.059\pm 0.014$
$0.125$	$41$	$0.3$	$98.2\pm 0.4$	$96.4\pm 0.8$	$75.0\pm 0.3$	$74.3\pm 0.7$	$3.02\pm 0.09$	$0.040\pm 0.005$
$0.125$	$41$	$0.6$	$164.6\pm 0.1$	$162.4\pm 0.2$	$144.1\pm 0.8$	$145.0\pm 2.1$	$1.98\pm 0.06$	$0.032\pm 0.009$
$0.25$	$21$	$0.15$	$60.8\pm 1.1$	$59.7\pm 0.5$	$36.8\pm 0.5$	$37.2\pm 1.4$	$3.12\pm 0.04$	$0.070\pm 0.007$
$0.25$	$21$	$0.3$	$77.8\pm 0.5$	$75.7\pm 0.3$	$52.6\pm 0.4$	$53.5\pm 0.8$	$2.86\pm 0.10$	$0.046\pm 0.001$

Appendix

Table 3: Noisy Lorenz attractor under nine observation protocols (spacing

\Delta t

N

observations per trial, noise

\sigma

); the row

(0.25,21,0.3)

is the protocol of table 2 , here over its first three seeds. Negative ELBO per test trial and gauge excess in nats per unit time, mean

\pm 2

standard errors over three seeds; the gauge excess of the Cholesky gauge is

10\%

below the square root’s where the deficit is large, and that of Helmholtz- SDE is

0.000

–

0.005

Figure 8: Three samples from the true prior process of the noisy Lorenz attractor and from the prior processes learned with each posterior process. The fixed gauges miss the outward spiral within each lobe of the attractor; the learned gauge and Helmholtz- SDE recover it.

Posterior process	Negative ELBO $\downarrow$	Latent LL $\uparrow$	Autocorr. error $\downarrow$	Gauge excess $\downarrow$
SDE Matching (square root)	$64.5\pm 0.1$	$-1.89\pm 0.01$	$0.054\pm 0.001$	$0.685\pm 0.020$
Cholesky gauge	$64.3\pm 0.1$	$-1.89\pm 0.02$	$0.053\pm 0.002$	$0.658\pm 0.015$
Symmetric gauge	$64.5\pm 0.2$	$-1.89\pm 0.01$	$0.054\pm 0.001$	$0.686\pm 0.017$
Helmholtz- SDE ( $\ell=1$ )	$54.1\pm 0.1$	$-1.67\pm 0.01$	$0.043\pm 0.002$	$0.135\pm 0.006$
Gauge Matching	$52.5\pm 0.3$	$-1.69\pm 0.01$	$0.044\pm 0.004$	$0.014\pm 0.003$
Gauge Matching, diagonal $L_{\phi}$	$52.9\pm 0.2$	$-1.71\pm 0.01$	$0.045\pm 0.003$	$0.012\pm 0.003$

Appendix

Table 4: Stuart–Landau oscillator with shear

\kappa=3

d=2

, generative model known, posterior process learned. Columns as in table 2 without the time, with the autocorrelation error over lags in

[0,4]

; the negative ELBO excludes held-out trials whose path-space KL term exceeds

10^{3}

nats (one of

5\times 2048

for Helmholtz- SDE , none otherwise). Mean

\pm 2

standard errors over 5 seeds; the seed-paired ELBO difference between Gauge Matching and Helmholtz- SDE is

1.6\pm 0.3

nats.

Figure 9: Stuart–Landau oscillator with shear

\kappa=3

, known generative model. A: the true drift field with the trajectory and observations (dots) of one test trial, and three sample paths of the true prior process started from the initial marginal

q_{\phi}(0,z\mid X)

fitted by SDE Matching, Helmholtz- SDE and Gauge Matching; since the generative model is fixed at the truth, the panels differ only through the fitted initial marginal. B: the gauge

\Omega_{12}(t)

that each of the two gauge-corrected methods uses for this trial, against the optimal gauge eq. 17 at its own marginals (thick, light). C: gauge excess

\mathcal{J}(\Omega)-\mathcal{J}(\Omega^{\star})

over time, averaged over 64 test trials and all seeds, for the five posterior processes (log scale): the order-one correction is off by the closed-form amount

8\kappa^{2}\operatorname{tr}P\det P/s^{2}\approx 0.13

nats per unit time, the learned gauge by a tenth of it.

Coord.	Posterior process	Period err. $\downarrow$	Lag err. $\downarrow$	Autocorr. err. $\downarrow$	Gauge excess $\downarrow$
log	SDE Matching (square root)	$0.12\pm 0.04$	$0.09\pm 0.05$	$0.119\pm 0.019$	$0.215\pm 0.006$
	Helmholtz- SDE ( $\ell=1$ )	$0.09\pm 0.03$	$0.07\pm 0.03$	$0.100\pm 0.015$	$0.000\pm 0.000$
	Gauge Matching	$0.07\pm 0.02$	$0.05\pm 0.03$	$0.093\pm 0.018$	$0.004\pm 0.001$
original	SDE Matching (square root)	$0.22\pm 0.05$	$0.42\pm 0.03$	$0.191\pm 0.022$	$0.081\pm 0.002$
	Cholesky gauge	$0.22\pm 0.05$	$0.44\pm 0.04$	$0.192\pm 0.024$	$0.082\pm 0.004$
	Gauge Matching, diagonal $L_{\phi}$	$0.13\pm 0.05$	$0.35\pm 0.03$	$0.139\pm 0.022$	$0.002\pm 0.001$

Appendix

Table 5: Stochastic Lotka–Volterra system with sparse noisy observations,

d=2

, in log coordinates, where the diffusion coefficient is state-independent and fixed at its true value, and in the original coordinates, where it is multiplicative and learned, and Helmholtz- SDE does not apply. Columns: relative error of the period of the cycle; relative error of the lag of the predator behind the prey; autocorrelation error as in table 2 , lags in

[0,10]

, all for the learned prior process against the true system; gauge excess in nats per unit time as in table 2 , in the original coordinates from the state-dependent linear system of section C.2 estimated by Monte Carlo (

10^{3}

samples at

100

times on

64

test trials). Mean

\pm 2

standard errors over 10 seeds.

Figure 10: Learned prior dynamics of the stochastic Lotka–Volterra system. Drift field and three sample paths of the true system and of the models learned with each posterior process, in log coordinates with the diffusion coefficient fixed at its true value (top) and in the original coordinates with a learned state-dependent diffusion coefficient (bottom). In log coordinates all three posterior processes learn a drift field that sustains the cycle. In the original coordinates the learned multiplicative noise is larger than the true one, which widens the sample paths, and the cycle of the fixed gauge is smaller and less regular than that of the learned gauge.

Model	Posterior process	Neg. ELBO $\downarrow$	Latent LL $\uparrow$	Autocorr. err. $\downarrow$	Gauge excess $\downarrow$
known	SDE Matching (square root)	$154.8\pm 0.8$	$-1.03\pm 0.04$	—	$1.40\pm 0.02$
	Cholesky gauge	$155.1\pm 0.8$	$-1.04\pm 0.04$	—	$1.29\pm 0.02$
	Gauge Matching	$148.2\pm 0.8$	$-1.09\pm 0.03$	—	$0.049\pm 0.002$
	Gauge Matching, diagonal $L_{\phi}$	$147.3\pm 0.5$	$-1.08\pm 0.03$	—	$0.041\pm 0.001$
learned	SDE Matching (square root)	$150.5\pm 0.5$	$-1.05\pm 0.02$	$0.072\pm 0.003$	$0.88\pm 0.02$
	Cholesky gauge	$150.8\pm 0.2$	$-1.06\pm 0.01$	$0.071\pm 0.003$	$0.85\pm 0.02$

Appendix

Table 6: Rotating Ornstein–Uhlenbeck process in

d=8

with multiplicative noise, where Helmholtz- SDE and the symmetric gauge do not apply. Top: generative model fixed at the truth, posterior process learned. Bottom: prior drift, diffusion coefficient and initial law learned jointly with the posterior process. Columns as in table 2 without the time, with the gauge excess evaluated with the state-dependent optimal gauge of section C.2 and, for the learned model, the autocorrelation error over lags in

[0,2]

; the negative ELBO excludes held-out trials above

10^{3}

nats (

0

–

3

2048

per seed). Mean

\pm 2

standard errors over 5 seeds.

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Closing the Approximation Gap in Simulation-free Latent SDEs

Henry D. Smith, Brian L. Trippe, Scott W. Linderman

Stanford University

Parameter Estimation Stochastic Differential Equations Latent Dynamics Modeling System Identification Variational Inference

Open paper AskPDF arXiv

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. Code is available at https://github.com/lindermanlab/helmholtz-sde .

Abstract

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