SDE Matching in Latent and Diffusion Models

• SDE Matching is a family of methods that align stochastic differential equations and their derived quantities with observed data or target dynamical laws.
• It employs simulation-free latent reparameterization, deterministic probability-flow ODEs, and drift–score matching to optimize model training without costly SDE integration.
• These approaches offer significant computational savings and improved scalability across applications such as temporal order recovery, motion capture, and image editing.

SDE Matching denotes a family of methods that align stochastic differential equation dynamics, or quantities derived from them, with observed data, prescribed probability paths, or auxiliary dynamical constraints. In one central formulation, it is a simulation-free method for training latent stochastic differential equations by parameterizing posterior marginals as $z_t = F_\phi(\epsilon, t, X)$, constructing a posterior SDE with the same diffusion as the prior, and optimizing a re-expressed ELBO without numerical SDE solves during training (Bartosh et al., 4 Feb 2025). In adjacent usage, the same term also covers score-based recovery of temporal order from unordered SDE trajectories (Tran et al., 28 Jan 2026), likelihood-based matching of simulated path distributions to sparse observations (Yildiz et al., 2018), trajectory-level flow matching for Neural SDEs (Zhang et al., 2024), and SDE-consistent stochastic samplers for flow-matching generators (Zou et al., 22 May 2026).

1. Scope of the term

Current usage is heterogeneous but structurally related: an SDE, or an SDE-induced object such as a drift, score, path distribution, or sampler, is matched to data or to a target dynamical law. The shared motif is local-in-time alignment of stochastic dynamics rather than purely static density fitting.

Setting	What is matched	Representative paper
Latent sequence modeling	Posterior marginals and prior/posterior local dynamics	(Bartosh et al., 4 Feb 2025)
Missing time information	Empirical drift and score-corrected drift	(Tran et al., 28 Jan 2026)
Irregular SDE learning	Simulated path and marginal state distributions to observations	(Yildiz et al., 2018)
Clinical time series	Neural SDE drift/diffusion to Brownian-bridge conditionals	(Zhang et al., 2024)
Flow-matching RL sampling	Reverse-time stochastic sampler to flow-matching marginals	(Zou et al., 22 May 2026)
Diffusion-based image editing	Edited and original marginals under reverse SDE	(Nie et al., 2023)

This suggests that “SDE Matching” is less a single algorithm than a methodological family. Some versions are explicitly variational and simulation-free, some are likelihood-based, some are score-based, and some operate at the level of sampler design rather than model fitting.

2. Simulation-free latent SDE training

In the formulation named “SDE Matching,” a latent SDE model assumes a prior process

$$d z_t = h_\theta(z_t, t)\,dt + g_\theta(z_t, t)\,d w_t,$$

with $z_0 \sim p_\theta(z_0)$, together with an observation model $x_{t_i} \sim p_\theta(x_{t_i}\mid z_{t_i})$ (Bartosh et al., 4 Feb 2025). Standard latent SDE training introduces a posterior SDE, estimates an ELBO, and requires simulation plus backpropagation through approximate SDE solutions. The central modification is to parameterize posterior marginals directly through

$$z_t = F_\phi(\epsilon, t, X), \qquad \epsilon \sim \mathcal{N}(0,I),$$

rather than by solving a posterior SDE during training (Bartosh et al., 4 Feb 2025).

From this reparameterization, one obtains a posterior ODE

$\frac{d z_t}{dt} = f_\phi(z_t, t, X),$

where $f_\phi$ is induced by the time derivative of $F_\phi$. Using the probability-flow relationship, this ODE is converted into a posterior SDE that shares diffusion $g_\theta$ with the prior and has the desired marginals $q_\phi(z_t\mid X)$. Its drift is

$$d z_t = h_\theta(z_t, t)\,dt + g_\theta(z_t, t)\,d w_t,$$0

This permits the standard latent SDE ELBO to be rewritten as expectations over independent samples from $$d z_t = h_\theta(z_t, t)\,dt + g_\theta(z_t, t)\,d w_t,$$1, rather than over simulated paths (Bartosh et al., 4 Feb 2025).

The resulting objective keeps the familiar prior, diffusion, and reconstruction decomposition. The diffusion term is evaluated from

$$d z_t = h_\theta(z_t, t)\,dt + g_\theta(z_t, t)\,d w_t,$$2

and penalizes $$d z_t = h_\theta(z_t, t)\,dt + g_\theta(z_t, t)\,d w_t,$$3. Training then requires only: sampling $$d z_t = h_\theta(z_t, t)\,dt + g_\theta(z_t, t)\,d w_t,$$4, sampling $$d z_t = h_\theta(z_t, t)\,dt + g_\theta(z_t, t)\,d w_t,$$5, computing $$d z_t = h_\theta(z_t, t)\,dt + g_\theta(z_t, t)\,d w_t,$$6, and evaluating $$d z_t = h_\theta(z_t, t)\,dt + g_\theta(z_t, t)\,d w_t,$$7, $$d z_t = h_\theta(z_t, t)\,dt + g_\theta(z_t, t)\,d w_t,$$8, $$d z_t = h_\theta(z_t, t)\,dt + g_\theta(z_t, t)\,d w_t,$$9, and the observation model at that point. No SDE or ODE is integrated during training (Bartosh et al., 4 Feb 2025).

The computational distinction is explicit in the paper’s asymptotic comparison:

Method	Memory	Time
Forward pathwise	$z_0 \sim p_\theta(z_0)$0	$z_0 \sim p_\theta(z_0)$1
Backprop through solver	$z_0 \sim p_\theta(z_0)$2	$z_0 \sim p_\theta(z_0)$3
Stochastic adjoint	$z_0 \sim p_\theta(z_0)$4	$z_0 \sim p_\theta(z_0)$5
Amortized reparameterization	$z_0 \sim p_\theta(z_0)$6	$z_0 \sim p_\theta(z_0)$7
SDE Matching	$z_0 \sim p_\theta(z_0)$8	$z_0 \sim p_\theta(z_0)$9

Empirically, on a 3D stochastic Lorenz system, the adjoint-based method required approximately $x_{t_i} \sim p_\theta(x_{t_i}\mid z_{t_i})$0 simulation steps per training iteration, whereas SDE Matching required only a single evaluation of the posterior drift per iteration; the reported per-iteration runtime was about $x_{t_i} \sim p_\theta(x_{t_i}\mid z_{t_i})$1 faster, the negative ELBO decreased much faster, and the combined effective speed-up was reported as over $x_{t_i} \sim p_\theta(x_{t_i}\mid z_{t_i})$2 (Bartosh et al., 4 Feb 2025). On motion capture, SDE Matching achieved $x_{t_i} \sim p_\theta(x_{t_i}\mid z_{t_i})$3 test MSE versus $x_{t_i} \sim p_\theta(x_{t_i}\mid z_{t_i})$4 for adjoint-trained Latent SDE and $x_{t_i} \sim p_\theta(x_{t_i}\mid z_{t_i})$5 for ARCTA, which the paper describes as comparable predictive accuracy with much lower computation (Bartosh et al., 4 Feb 2025).

3. Probability-flow, score, and flow-matching foundations

The conceptual foundation of many SDE-matching methods is the equivalence between stochastic dynamics and a deterministic probability-flow ODE with the same marginals. In the flow-matching literature, a stochastic interpolant

$x_{t_i} \sim p_\theta(x_{t_i}\mid z_{t_i})$6

induces an expected velocity field

$x_{t_i} \sim p_\theta(x_{t_i}\mid z_{t_i})$7

and the ODE

$x_{t_i} \sim p_\theta(x_{t_i}\mid z_{t_i})$8

specializes to the probability flow ODE of an underlying SDE when the interpolant matches diffusion marginals (Benton et al., 2023). This establishes a direct bridge between classical score-based SDE models, DDIM-style deterministic sampling, and flow matching.

Error analysis in this deterministic setting shows that if

$x_{t_i} \sim p_\theta(x_{t_i}\mid z_{t_i})$9

then the endpoint error obeys

$$z_t = F_\phi(\epsilon, t, X), \qquad \epsilon \sim \mathcal{N}(0,I),$$0

with $$z_t = F_\phi(\epsilon, t, X), \qquad \epsilon \sim \mathcal{N}(0,I),$$1 a spatial Lipschitz bound on the approximate drift (Benton et al., 2023). The same paper then controls the Lipschitz constant of the true velocity under a $$z_t = F_\phi(\epsilon, t, X), \qquad \epsilon \sim \mathcal{N}(0,I),$$2-regularity assumption and derives polynomial rates after Gaussian smoothing. In that sense, deterministic flow matching can be read as an ODE counterpart of SDE matching: the same marginal path is realized without stochastic sampling, but sensitivity to Lipschitz constants becomes explicit (Benton et al., 2023).

Within latent SDE training, diffusion models appear as a special case. For a forward SDE with Gaussian marginals $$z_t = F_\phi(\epsilon, t, X), \qquad \epsilon \sim \mathcal{N}(0,I),$$3, the latent SDE ELBO reduces to reweighted denoising score matching,

$$z_t = F_\phi(\epsilon, t, X), \qquad \epsilon \sim \mathcal{N}(0,I),$$4

showing that standard diffusion training is itself an SDE-derived matching objective (Bartosh et al., 4 Feb 2025). This places score matching, flow matching, and latent SDE matching on a common axis: all learn local fields that are consistent with a target stochastic evolution.

4. Temporal and trajectory-level matching

One important extension treats temporal order itself as the latent object to be matched. For time-homogeneous linear SDEs with additive diffusion,

$$z_t = F_\phi(\epsilon, t, X), \qquad \epsilon \sim \mathcal{N}(0,I),$$5

the “ReTrace” framework assumes an equally spaced grid but only observes a permuted set of states. The key observation is that for irreversible SDEs the backward drift differs from the forward drift by a score term,

$$z_t = F_\phi(\epsilon, t, X), \qquad \epsilon \sim \mathcal{N}(0,I),$$6

and this asymmetry yields a drift–score discrepancy

$$z_t = F_\phi(\epsilon, t, X), \qquad \epsilon \sim \mathcal{N}(0,I),$$7

that distinguishes the correct temporal direction on average (Tran et al., 28 Jan 2026). Theorem 1 states that reversibility implies non-identifiability, Theorem 2 shows that the correct direction uniquely minimizes the score-based error under irreversibility, and Algorithm 1 aggregates pairwise decisions by a bubble-sort-like procedure interleaved with MLE of $$z_t = F_\phi(\epsilon, t, X), \qquad \epsilon \sim \mathcal{N}(0,I),$$8 (Tran et al., 28 Jan 2026).

This formulation is empirically strong on synthetic irreversible linear SDEs. With $$z_t = F_\phi(\epsilon, t, X), \qquad \epsilon \sim \mathcal{N}(0,I),$$9, Euler–Maruyama simulation at $\frac{d z_t}{dt} = f_\phi(z_t, t, X),$0, $\frac{d z_t}{dt} = f_\phi(z_t, t, X),$1 trajectories, and $\frac{d z_t}{dt} = f_\phi(z_t, t, X),$2 steps, ReTrace–MLE achieved $\frac{d z_t}{dt} = f_\phi(z_t, t, X),$3 sorting accuracy, while MST–MLE achieved $\frac{d z_t}{dt} = f_\phi(z_t, t, X),$4 and DPT–MLE $\frac{d z_t}{dt} = f_\phi(z_t, t, X),$5 (Tran et al., 28 Jan 2026). The same paper reports that accurate order reconstruction reduces Treatment Effect Bias and counterfactual RMSE in stochastic PKPD models, extending SDE parameter estimation to settings with missing temporal order (Tran et al., 28 Jan 2026).

Trajectory Flow Matching generalizes local matching to entire time series. It defines Gaussian Brownian bridges between consecutive observations,

$\frac{d z_t}{dt} = f_\phi(z_t, t, X),$6

with target drift

$\frac{d z_t}{dt} = f_\phi(z_t, t, X),$7

and trains a Neural SDE by regressing $\frac{d z_t}{dt} = f_\phi(z_t, t, X),$8 and optionally $\frac{d z_t}{dt} = f_\phi(z_t, t, X),$9 against these analytic bridge quantities, without backpropagation through SDE solvers (Zhang et al., 2024). In clinical time series, TFM and TFM-ODE reduced MSE by $f_\phi$0–$f_\phi$1 relative to baselines and TFM yielded the lowest RBF-MMD on increments in ICU GIB and ED GIB, indicating better matching of trajectory variability (Zhang et al., 2024).

A different line, predating these simulation-free formulations, learns non-parametric drift and diffusion functions by matching simulated path distributions to observations via likelihood rather than gradient matching. The Gaussian-process SDE model of Wenk et al. directly simulates SDE paths, approximates the likelihood by Monte Carlo, and optimizes inducing-point parameters so that the SDE-induced marginal distributions at observation times explain irregular and sparse data (Yildiz et al., 2018). In that terminology, SDE matching means likelihood-driven distribution matching between the SDE-generated path distribution and the empirical data, rather than score matching or OT/Wasserstein criteria (Yildiz et al., 2018).

5. Sampler consistency, editing, and inverse problems

In diffusion-based image editing, SDE matching takes the form of contraction of an edited distribution toward the model distribution. If $f_\phi$2 is the marginal of an edited reverse SDE and $f_\phi$3 is the marginal of the original reverse SDE, then under the paper’s assumptions

$f_\phi$4

whereas for the probability-flow ODE the KL divergence is invariant in time (Nie et al., 2023). This provides a precise sense in which stochastic reverse dynamics “match” edited latents back toward the model’s native marginals. The paper then shows consistent empirical improvements for SDE counterparts of inpainting, DiffEdit, DDIB, and SDE-Drag (Nie et al., 2023).

For RL post-training of flow-matching generators, the issue is not model fitting but sampler design. “Precise” begins from a reverse-time SDE

$f_\phi$5

chosen so that, in ideal continuous time, it preserves the flow-matching marginals for any $f_\phi$6 (Zou et al., 22 May 2026). The paper analyzes exploration versus denoising stability through logSNR and proposes

$f_\phi$7

then derives a finite-step transition by freezing the clean-latent posterior mean within each step. The resulting sampler is described as SDE-consistent because it approximates a specific reverse-time SDE rather than introducing heuristic excess noise or covariance contraction. In RL experiments it achieved state-of-the-art alignment scores while requiring $f_\phi$8–$f_\phi$9 less wall-clock training time to match the best in-domain performance of prior samplers (Zou et al., 22 May 2026).

Plug-and-play image restoration supplies another continuous-time reinterpretation. Starting from the discrete PnP-Flow iteration, the continuous limit is an SDE surrogate

$F_\phi$0

which combines data-consistency and flow-matching prior transport (Jia et al., 3 Dec 2025). The SDE analysis motivates a geometric schedule $F_\phi$1, Lipschitz regularization of the learned vector field through a Hutchinson estimator of the Jacobian Frobenius norm, and an extrapolated rescaled SDE for acceleration (Jia et al., 3 Dec 2025). Numerical results on denoising, deblurring, super-resolution, and inpainting significantly outperform baseline PnP-Flow and other state-of-the-art methods (Jia et al., 3 Dec 2025).

6. Assumptions, limitations, and open directions

Across formulations, SDE matching is only as identifiable as the stochastic structure it exploits. For temporal-order recovery, reversibility or zero probability current makes time direction non-identifiable, and inaccurate score estimation, degenerate covariance, extreme noise, or very high diffusion can make the drift–score criterion unreliable (Tran et al., 28 Jan 2026). For trajectory flow matching, exact coupling preservation is proved only under assumptions such as a Monge map from initial state, unique recent history, or unique trajectory-level covariates, and the method does not provide explicit causal modeling (Zhang et al., 2024).

Simulation-free latent SDE training imposes structural constraints on the posterior parameterization and diffusion. The map $F_\phi$2 must be smooth, invertible in $F_\phi$3, and amenable to efficient score computation, while a fully general $F_\phi$4 makes the divergence term $F_\phi$5 expensive; practical implementations therefore prefer affine posterior maps and structured diffusions (Bartosh et al., 4 Feb 2025). More broadly, deterministic flow matching is sensitive to local Lipschitz constants of the learned drift, and Gaussian smoothing is used to keep the resulting bounds polynomial rather than exponentially unstable (Benton et al., 2023).

Sampler-oriented formulations introduce a different class of limitations. In Precise, the frozen clean-latent posterior mean is a local approximation rather than an exact identity, and the derivation is tailored to the linear flow-matching path $F_\phi$6 (Zou et al., 22 May 2026). In the PnP-Flow SDE surrogate, convergence conditions depend on the interplay between $F_\phi$7, $F_\phi$8, and the curvature of $F_\phi$9, so schedule design and Lipschitz regularization become part of the method rather than secondary implementation detail (Jia et al., 3 Dec 2025).

A plausible implication is that SDE matching should be viewed as a design principle rather than a single algorithmic template. The principle is to replace expensive or weakly identified global objectives with local stochastic consistency conditions: posterior-versus-prior drift matching, drift-versus-score matching, Brownian-bridge drift regression, or sampler-versus-marginal matching. The technical differences are substantial, but the family resemblance is the use of analytically or structurally specified stochastic laws to avoid, constrain, or reinterpret simulation.