Stochastic Latent Differential Inference (SLDI)

• Stochastic Latent Differential Inference (SLDI) is a unified probabilistic framework that models complex continuous-time dynamics using latent stochastic differential equations.
• It employs variational inference, neural network parameterization, and Euler–Maruyama discretization to capture nonlinear, irregular, and nonstationary behavior in high-dimensional data.
• SLDI enables rigorous uncertainty quantification, scalable training, and interpretable gradient computation, with applications in neuroscience, clinical time series, and structured dynamical systems.

Stochastic Latent Differential Inference (SLDI) is a unified probabilistic framework for modeling and inferring continuous-time stochastic dynamics in high-dimensional observed data using latent stochastic differential equations (SDEs). The SLDI approach leverages variational inference, deep generative modeling, stochastic calculus, and control-theoretic principles, enabling rigorous uncertainty quantification, flexible modeling of temporal structure, and efficient, scalable learning in complex dynamical scenarios (Rice, 8 Jan 2026).

1. Latent SDE Generative Modeling

SLDI posits that observed structured or temporal data $\{x_t\}_{t=1}^T$ are generated by first drawing an initial latent state $z_0 \sim p(z_0)$ (typically standard Gaussian) and then evolving $\{z_t\}$ under a continuous-time Itô SDE: $$dz_t = \mu_\theta(z_t, t) dt + \Sigma_\theta(z_t, t)dW_t,$$ where $\mu_\theta$ (drift) and $\Sigma_\theta$ (diffusion) are parameterized by neural networks, and $W_t$ denotes standard $m$-dimensional Brownian motion. Observations are decoded from the latent using $p_\psi(x_t|z_t)$, which can be Gaussian, categorical, or count-based (Rice, 8 Jan 2026, ElGazzar et al., 2024, Aslanimoghanloo et al., 20 Nov 2025).

SLDI generalizes standard state-space, RNN, and ODE latent variable models to handle:

• Irregular sampling schemes
• Nonlinear and nonstationary dynamics
• Continuous-time latent evolution
• Complex, high-dimensional temporal or structured data

2. Variational Inference and the Evidence Lower Bound

Exact marginal likelihood computation in deep latent SDE models is intractable. SLDI introduces a variational family for the posterior, typically $q_\phi(z_0|x_{1:T})$ for the initial state (via neural encoder) and an implicit path distribution $q_\phi(z_{1:T}|z_0)$, induced by the SDE with learned initial condition (Rice, 8 Jan 2026, Aslanimoghanloo et al., 20 Nov 2025).

The ELBO for SLDI takes the form: $$\mathcal{L}(\theta,\psi,\phi) = \mathbb{E}_{q_\phi(z_{0:T}|x_{1:T})}\left[\sum_{t=1}^T \log p_\psi(x_t|z_t)\right] - \mathrm{KL}[q_\phi(z_0|x_{1:T}) || p(z_0)] - \mathbb{E}_{q_\phi(z_{0:T})}\left[\sum_{t=1}^T \mathrm{KL}\left(q(z_t|z_{t-1}) || p(z_t|z_{t-1})\right)\right].$$ When the variational and generative diffusions match, Girsanov’s theorem reduces the pathwise KL to a time-integral of squared drift differences under the diffusion metric. This pathwise formulation grants SLDI superior handling of uncertainty and process noise (Rice, 8 Jan 2026, Aslanimoghanloo et al., 20 Nov 2025, ElGazzar et al., 2024).

3. Numerical Integration and Gradient Computation

Inference in SLDI requires backpropagating through SDE solvers. The canonical scheme is Euler–Maruyama discretization: $$z_{t+\Delta t} = z_t + \mu_\theta(z_t, t)\Delta t + \Sigma_\theta(z_t, t)\sqrt{\Delta t}\,\varepsilon_t,\quad \varepsilon_t\sim\mathcal{N}(0,I).$$ All steps are built to support reparameterization, enabling gradients to flow through sampled Wiener paths and initial state encodings (Rice, 8 Jan 2026, Aslanimoghanloo et al., 20 Nov 2025, Ryder et al., 2018).

SLDI advances gradient computation by introducing a co-parameterized adjoint SDE/ODE system: $$\frac{d a_t}{dt} = -a_t^\top \left(\frac{\partial \mu}{\partial z} - \sum_{i=1}^m \frac{\partial \Sigma_i}{\partial z} \frac{\partial \Sigma_i}{\partial z}^\top \right),$$ with a learned adjoint field $\mathcal{A}_\theta(z_t,t)$. Training enforces consistency between the learned and analytic adjoint by pathwise-regularized adjoint loss: $$\tilde{\ell} = \ell + \beta \int_0^T \|\mathcal{A}_\theta(z_t,t) - \hat{H}_t \|^2 dt,$$ where $\hat{H}_t$ is the true analytic adjoint time-derivative (Rice, 8 Jan 2026).

This approach yields memory- and compute-efficient training, variance reduction, and accurate recovery of gradient flow for deep latent SDEs. Alternative scalable inference is achieved through amortized reparametrization strategies, where linear SDE variational families further reduce computational complexity, making SLDI practical for long and stiff time series (Course et al., 2023).

4. Model Flexibility: Mechanistic Structure and Constraints

SLDI subsumes a hierarchy of architectures:

• General neural SDEs: Both drift and diffusion parameterized by neural networks (Aslanimoghanloo et al., 20 Nov 2025, ElGazzar et al., 2024).
• Hybrid mechanistic-neural SDEs: Partially hand-structured drift (e.g., coupled Hopf oscillators for neural population dynamics), with neural modulation (ElGazzar et al., 2024).
• SDEs on manifolds: Restriction to homogeneous spaces (e.g., spheres), enabling closed-form KL terms, geometric Euler–Maruyama solvers, and numerically stable manifold-constrained inference (Zeng et al., 2023).
• Hierarchical SDE compositions: Multiple layers (e.g., Brownian bridge latent manifold plus observation-driven SDE), optimized by EM-type or SMC-based procedures for scalability and interpretability (Rajaei et al., 29 Jul 2025).

Each variant adapts SLDI principles to the statistical, geometric, and computational demands of the application context.

5. Applications and Empirical Evaluations

SLDI has been systematically evaluated in diverse domains:

• Neuroscience: Modeling neural population dynamics from spiking or calcium imaging with continuous-time latent SDEs, outperforming Neural ODEs, LFADS, and RNN-based models in both predictive performance and parameter efficiency (ElGazzar et al., 2024, Rajaei et al., 29 Jul 2025).
• Clinical time series: Patient-specific latent SDEs for individual treatment effect estimation, probabilistic forecasting, and uncertainty quantification, handling irregular sampling and multi-modal emission distributions (Aslanimoghanloo et al., 20 Nov 2025).
• Structured dynamical systems: Continuous-time interpretable models of nonlinear latent dynamics, robust identification of fixed points and Jacobian structure, state-of-the-art performance on time series interpolation, classification, and recovery of latent manifold structure (Zeng et al., 2023, Duncker et al., 2019, Rajaei et al., 29 Jul 2025).
• General dynamical inference: Black-box SDE parameter recovery, uncertainty-robust video modeling, and population-level drift/diffusion identification from cross-sectional or Langevin-type data (Hasan et al., 2020, Tsourtis et al., 2020, Genkin et al., 2020, Ryder et al., 2018).

SLDI also provides theoretical guarantees of identifiability (up to isometry with generic latent mappings and continuous nondeterministic noise), consistency of variational inference (pathwise variational equivalence theorems), and interpretable control-theoretic and action-functional perspectives on latent trajectory learning (Rice, 8 Jan 2026, Hasan et al., 2020).

6. Training Stability, Regularization, and Scalability

Multiple algorithmic innovations underpin SLDI:

• Pathwise-regularized adjoint loss: Reduces stochastic gradient variance for stable, large-scale SDE training (Rice, 8 Jan 2026).
• Antithetic sampling and moving-average gradient clipping: Provide sub-Gaussian concentration of gradient noise under ergodic latent processes for robust optimization (Rice, 8 Jan 2026).
• Amortized blockwise inference: Scales SLDI variational inference to long and stiff time series, with compute cost independent of trajectory length or discretization granularity (Course et al., 2023).
• SMC and particle-based inference: Powers latent manifold SDE learning with linear scaling in observational data and supports high-dimensional, non-Gaussian latent trajectories (Rajaei et al., 29 Jul 2025, Deng et al., 2022).

These techniques enable practical deployment of SLDI in real-world scenarios involving high-dimensional and long-horizon dynamical systems.

7. Theoretical Insights and Interpretability

SLDI provides an information-geometric and control-theoretic interpretation of learning in deep latent SDEs:

• The energy regularization term parallels the classical action, leading to Pontryagin-type optimality equations in the small-step limit (Rice, 8 Jan 2026).
• The stochastic adjoint field learns a "natural gradient" on SDE path space, aligning parameter updates with intrinsic latent geometry.
• For SDEs on homogeneous spaces, SLDI achieves uninformative priors, closed-form KL divergences, and geometric integration, providing interpretable latent trajectories with competitive or state-of-the-art performance (Zeng et al., 2023).

SLDI supports analytic gradient computation for non-stationary Langevin models, weak-projection identification for population snapshot data, and interpretable phase-plane and fixed-point analysis for nonlinear dynamics (Genkin et al., 2020, Tsourtis et al., 2020, Duncker et al., 2019).

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Table: Core Components of SLDI Frameworks

Component	Typical Formulation	References
Latent SDE (generative)	$$dz_t = \mu_\theta(z_t, t)dt + \Sigma_\theta(z_t, t)dW_t$$	(Rice, 8 Jan 2026, Aslanimoghanloo et al., 20 Nov 2025)
Variational Posterior	$q_\phi(z_0|x_{1:T})$, SDE-flow for $q_\phi(z_{1:T}|z_0)$	(Rice, 8 Jan 2026, ElGazzar et al., 2024)
ELBO/Objective	Reconstruction + prior/posterior KL + path-KL (Girsanov)	(Rice, 8 Jan 2026, Course et al., 2023)
Adjoint System	$da_t/dt$, co-parameterized $\mathcal{A}_\theta(z_t, t)$	(Rice, 8 Jan 2026)
Variance Reduction	Pathwise-regularization, antithetic sampling, gradient clipping	(Rice, 8 Jan 2026)
EM/SMC/Particle Methods	For hierarchical and population-level models	(Rajaei et al., 29 Jul 2025, Deng et al., 2022)

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SLDI constitutes a comprehensive statistical and computational framework for latent dynamical system identification and uncertainty-aware generative modeling in continuous time, distinguishing itself by principled integration of stochastic calculus, neural parameterization, scalable variational inference, and control-inspired optimization (Rice, 8 Jan 2026, ElGazzar et al., 2024, Aslanimoghanloo et al., 20 Nov 2025, Rajaei et al., 29 Jul 2025, Course et al., 2023).