Conditional Gaussian Nonlinear Systems

• Conditional Gaussian Nonlinear Systems (CGNS) are a class of stochastic models where hidden variables follow a Gaussian law given the observed trajectory.
• They enable analytic filtering, smoothing, and sampling by evolving the conditional mean and covariance through closed equations, thus supporting robust data assimilation.
• Hybrid and learned CGNS architectures integrate physics-based models with neural-network corrections to improve surrogate modeling, extreme event estimation, and uncertainty quantification.

Conditional Gaussian Nonlinear Systems (CGNS) are a broad class of nonlinear stochastic dynamical systems in which the state is partitioned into two components—typically an observed or resolved component and an unobserved or unresolved component—such that, conditional on the trajectory of the first component, the second component has a Gaussian law. This conditional linear-Gaussian structure coexists with nonlinear coefficients and can generate strongly non-Gaussian joint and marginal statistics, including intermittency and extreme events. Because the conditional statistics admit closed analytic evolution equations, CGNS has become a central framework for data assimilation, smoothing, uncertainty quantification, reduced-order modeling, and, more recently, scientific machine learning architectures that embed analytic inference into learned surrogate dynamics (Chen et al., 2021, Andreou et al., 2024).

1. Formal definition and structural properties

A standard continuous-time CGNS writes the state as $u=(\mathbf{X},\mathbf{Y})$, with $\mathbf{X}\in\mathbb{C}^{N_1}$ and $\mathbf{Y}\in\mathbb{C}^{N_2}$, and evolves according to

$$\begin{aligned} \frac{\mathrm{d}\mathbf{X}}{\mathrm{d}t} &= \mathbf{A}_0(\mathbf{X},t) + \mathbf{A}_1(\mathbf{X},t)\mathbf{Y} + \mathbf{B}_1(\mathbf{X},t)\dot{\mathbf{W}}_1(t), \ \frac{\mathrm{d}\mathbf{Y}}{\mathrm{d}t} &= \mathbf{a}_0(\mathbf{X},t) + \mathbf{a}_1(\mathbf{X},t)\mathbf{Y} + \mathbf{b}_2(\mathbf{X},t)\dot{\mathbf{W}}_2(t). \end{aligned}$$

The coefficients may depend nonlinearly and time-dependently on $\mathbf{X}$, but $\mathbf{Y}$ enters linearly. This is the source of the defining property: for any realization of $\mathbf{X}(s)$ with $s\le t$, the conditional law $p(\mathbf{Y}(t)\mid \mathbf{X}(s\le t))$ is Gaussian (Chen et al., 2021).

A more general formulation allows correlated noise across the two subsystems: $$\begin{aligned} d\mathbf{x} &= \left[\Lambda^x(t,\mathbf{x})\mathbf{y}+f^x(t,\mathbf{x})\right]dt+\Sigma_1^x(t,\mathbf{x})d\mathbf{W}_1+\Sigma_2^x(t,\mathbf{x})d\mathbf{W}_2,\ d\mathbf{y} &= \left[\Lambda^y(t,\mathbf{x})\mathbf{y}+f^y(t,\mathbf{x})\right]dt+\Sigma_1^y(t,\mathbf{x})d\mathbf{W}_1+\Sigma_2^y(t,\mathbf{x})d\mathbf{W}_2. \end{aligned}$$ Here again, $\mathbf{X}\in\mathbb{C}^{N_1}$0 is conditionally Gaussian given the observed history of $\mathbf{X}\in\mathbb{C}^{N_1}$1, while the nonlinear dependence of coefficients on $\mathbf{X}\in\mathbb{C}^{N_1}$2 permits strongly non-Gaussian behavior in joint and marginal distributions (Andreou et al., 2024).

A recurring misconception is that CGNS are “Gaussian systems.” They are not Gaussian in the global sense. The literature emphasizes that the conditionally linear structure can still reproduce intermittency, skewness, heavy tails, and extreme events in the observable statistics (Chen et al., 2021, Andreou et al., 2024). Another misconception is that CGNS is merely a filtering recipe. More precisely, it is a structural class of stochastic models whose conditional laws support analytic inference.

2. Closed conditional statistics, filtering, smoothing, and sampling

The main analytic advantage of CGNS is that the conditional mean and covariance of the hidden variables satisfy closed equations. In the formulation above, if

$\mathbf{X}\in\mathbb{C}^{N_1}$3

then the filter mean and covariance evolve as

$\mathbf{X}\in\mathbb{C}^{N_1}$4

These equations generalize Kalman-Bucy-type updates to nonlinear settings with conditional Gaussian closure (Chen et al., 2021, Yun et al., 18 Sep 2025).

The same framework admits analytic smoothers, obtained by conditioning on the full observation interval rather than only the past. A martingale-free development based on time discretization and a vanishing-step limit derives both filter and smoother equations using only Gaussian conditioning and Euler-Maruyama discretization, rather than classical martingale machinery (Andreou et al., 2024). That paper also derives analytic posterior sampling equations for hidden trajectories, including the case of correlated noise, so that sampled paths reproduce the correct conditional mean, covariance, and temporal correlation structure (Andreou et al., 2024).

These sampling results clarify an uncertainty hierarchy discussed explicitly in the literature: unconditional simulation has the largest uncertainty, filter-based conditional simulation has less, and smoother-based conditional simulation has the least. The same paper demonstrates the framework on a physics-constrained triad-interaction climate model with cubic nonlinearity and state-dependent cross-interacting noise, where posterior sampling captures intermittent bursts and extreme-event statistics more faithfully than pointwise conditional means alone (Andreou et al., 2024).

3. Data assimilation, surrogate modeling, and preconditioning

Within data assimilation, CGNS supplies exact minimum-variance estimates for the hidden state conditioned on the observed history, without requiring Monte Carlo ensemble propagation for the conditional update itself. This analytic tractability is used as the basis for state estimation, uncertainty quantification, and trajectory reconstruction under partial observation (Chen et al., 2021).

The framework has also been developed as a surrogate-model and preconditioner methodology. A suitable CGNS approximation can preserve much of the underlying physics while replacing computationally expensive inference in the original nonlinear model by closed-form filtering and smoothing. In this role, CGNS supports efficient expectation-maximization parameter estimation, simultaneous estimation of parameters and hidden states with uncertainty quantification, and rapid computation of high-dimensional marginal and joint probability density functions via Gaussian-mixture constructions over conditionally Gaussian laws (Chen et al., 2021).

This preconditioning role is not limited to state estimation. The same work uses CGNS to accelerate sampling of hidden trajectories and the computation of statistical response and linear response with respect to parameter perturbations. In the reported case studies—a truncated stochastic quadratic model, a two-layer inhomogeneous Lorenz ’96 system, and a 4D stochastic climate model—the analytic CGNS machinery serves as a computationally cheap approximation that can outperform ensemble-based inference under strong nonlinearity and partial observation, while also improving parameter-learning workflows (Chen et al., 2021).

A plausible implication is that CGNS functions less as a single algorithm than as a modeling principle: once the conditional Gaussian structure is available, filtering, smoothing, density estimation, parameter inference, and response analysis become different uses of the same closed conditional law.

4. Hybrid and learned CGNS architectures

Recent work extends CGNS from hand-crafted stochastic closures to learned latent-variable models. In the continuous-time hybrid framework "CGNSDE," the system is split into $\mathbf{X}\in\mathbb{C}^{N_1}$5 and $\mathbf{X}\in\mathbb{C}^{N_1}$6, and neural-network corrections are added in a way that preserves conditional Gaussianity: $\mathbf{X}\in\mathbb{C}^{N_1}$7 Because the neural terms depend only on $\mathbf{X}\in\mathbb{C}^{N_1}$8, the conditional Gaussian filter remains analytic. The framework combines a knowledge-based model, possibly discovered with causation entropy, with neural residuals and trains the model using both forecast loss and data-assimilation loss. The reported effect is improved state estimation, uncertainty quantification, and estimation of extreme events relative to knowledge-based regression models (Chen et al., 2024).

A parallel line of work introduces the Conditional Gaussian Koopman Network (CGKN), which uses an encoder-decoder to lift unobserved variables into a latent space where the dynamics become conditionally linear. In a representative discrete-time formulation,

$\mathbf{X}\in\mathbb{C}^{N_1}$9

The posterior of $\mathbf{Y}\in\mathbb{C}^{N_2}$0 given the observed trajectory is Gaussian and obeys an analytic recursion, which permits training the model with an explicit DA objective and makes data assimilation computationally cheaper than ensemble-based DA on the full state (Chen et al., 11 Jul 2025, Chen et al., 2024).

The Lagrangian extension "LaCGKN" adapts this idea to sparse tracer observations. Its reported architectural components are tracer homogenization for permutation equivariance, Fourier positional encoding for spatial dependence, and an SVD-inspired low-rank parameterization of the latent transition operator. On a two-layer quasi-geostrophic flow with surface tracer observations, LaCGKN is reported to achieve accurate and efficient Lagrangian data assimilation and prediction, with posterior RMSEs lower than EnKF and OI and a $\mathbf{Y}\in\mathbb{C}^{N_2}$1 computational speedup over EnKF (Wang et al., 14 Mar 2026).

5. Representative applications and reported performance

CGNS-based and CGNS-inspired models have been tested on canonical PDEs, turbulent geophysical systems, and multiscale climate models. In the discrete-time CGKN study, partial-observation data assimilation was demonstrated for the viscous Burgers’ equation, the Kuramoto-Sivashinsky equation, and the 2-D Navier-Stokes equations, with the following reported forecast and DA errors (Chen et al., 11 Jul 2025):

System	Forecast Error (CGKN)	DA Error (CGKN)
Burgers'	$\mathbf{Y}\in\mathbb{C}^{N_2}$2	$\mathbf{Y}\in\mathbb{C}^{N_2}$3
Kuramoto-Sivashinsky	$\mathbf{Y}\in\mathbb{C}^{N_2}$4	$\mathbf{Y}\in\mathbb{C}^{N_2}$5
2D Navier-Stokes	$\mathbf{Y}\in\mathbb{C}^{N_2}$6	$\mathbf{Y}\in\mathbb{C}^{N_2}$7

The continuous-time CGKN framework was evaluated on the projected stochastic Burgers-Sivashinsky equation, Lorenz 96, and El Niño-Southern Oscillation. The reported DA errors are $\mathbf{Y}\in\mathbb{C}^{N_2}$8 for CGKN on PSBSE, $\mathbf{Y}\in\mathbb{C}^{N_2}$9 on L96, and $$\begin{aligned} \frac{\mathrm{d}\mathbf{X}}{\mathrm{d}t} &= \mathbf{A}_0(\mathbf{X},t) + \mathbf{A}_1(\mathbf{X},t)\mathbf{Y} + \mathbf{B}_1(\mathbf{X},t)\dot{\mathbf{W}}_1(t), \ \frac{\mathrm{d}\mathbf{Y}}{\mathrm{d}t} &= \mathbf{a}_0(\mathbf{X},t) + \mathbf{a}_1(\mathbf{X},t)\mathbf{Y} + \mathbf{b}_2(\mathbf{X},t)\dot{\mathbf{W}}_2(t). \end{aligned}$$0 on ENSO, compared with $$\begin{aligned} \frac{\mathrm{d}\mathbf{X}}{\mathrm{d}t} &= \mathbf{A}_0(\mathbf{X},t) + \mathbf{A}_1(\mathbf{X},t)\mathbf{Y} + \mathbf{B}_1(\mathbf{X},t)\dot{\mathbf{W}}_1(t), \ \frac{\mathrm{d}\mathbf{Y}}{\mathrm{d}t} &= \mathbf{a}_0(\mathbf{X},t) + \mathbf{a}_1(\mathbf{X},t)\mathbf{Y} + \mathbf{b}_2(\mathbf{X},t)\dot{\mathbf{W}}_2(t). \end{aligned}$$1, $$\begin{aligned} \frac{\mathrm{d}\mathbf{X}}{\mathrm{d}t} &= \mathbf{A}_0(\mathbf{X},t) + \mathbf{A}_1(\mathbf{X},t)\mathbf{Y} + \mathbf{B}_1(\mathbf{X},t)\dot{\mathbf{W}}_1(t), \ \frac{\mathrm{d}\mathbf{Y}}{\mathrm{d}t} &= \mathbf{a}_0(\mathbf{X},t) + \mathbf{a}_1(\mathbf{X},t)\mathbf{Y} + \mathbf{b}_2(\mathbf{X},t)\dot{\mathbf{W}}_2(t). \end{aligned}$$2, and $$\begin{aligned} \frac{\mathrm{d}\mathbf{X}}{\mathrm{d}t} &= \mathbf{A}_0(\mathbf{X},t) + \mathbf{A}_1(\mathbf{X},t)\mathbf{Y} + \mathbf{B}_1(\mathbf{X},t)\dot{\mathbf{W}}_1(t), \ \frac{\mathrm{d}\mathbf{Y}}{\mathrm{d}t} &= \mathbf{a}_0(\mathbf{X},t) + \mathbf{a}_1(\mathbf{X},t)\mathbf{Y} + \mathbf{b}_2(\mathbf{X},t)\dot{\mathbf{W}}_2(t). \end{aligned}$$3 for a simplified Koopman network, and $$\begin{aligned} \frac{\mathrm{d}\mathbf{X}}{\mathrm{d}t} &= \mathbf{A}_0(\mathbf{X},t) + \mathbf{A}_1(\mathbf{X},t)\mathbf{Y} + \mathbf{B}_1(\mathbf{X},t)\dot{\mathbf{W}}_1(t), \ \frac{\mathrm{d}\mathbf{Y}}{\mathrm{d}t} &= \mathbf{a}_0(\mathbf{X},t) + \mathbf{a}_1(\mathbf{X},t)\mathbf{Y} + \mathbf{b}_2(\mathbf{X},t)\dot{\mathbf{W}}_2(t). \end{aligned}$$4, $$\begin{aligned} \frac{\mathrm{d}\mathbf{X}}{\mathrm{d}t} &= \mathbf{A}_0(\mathbf{X},t) + \mathbf{A}_1(\mathbf{X},t)\mathbf{Y} + \mathbf{B}_1(\mathbf{X},t)\dot{\mathbf{W}}_1(t), \ \frac{\mathrm{d}\mathbf{Y}}{\mathrm{d}t} &= \mathbf{a}_0(\mathbf{X},t) + \mathbf{a}_1(\mathbf{X},t)\mathbf{Y} + \mathbf{b}_2(\mathbf{X},t)\dot{\mathbf{W}}_2(t). \end{aligned}$$5, and NaN for CG-Reg in the same summary table (Chen et al., 2024).

A distinct CGNS application appears in Lagrangian-Eulerian multiscale data assimilation in the physical domain. There, the two-layer quasi-geostrophic model is recast into CGNS form to recover ocean eddies from sea-ice-floe trajectories. The reported evaluation uses normalized RMSE and pattern correlation. Corr values of $$\begin{aligned} \frac{\mathrm{d}\mathbf{X}}{\mathrm{d}t} &= \mathbf{A}_0(\mathbf{X},t) + \mathbf{A}_1(\mathbf{X},t)\mathbf{Y} + \mathbf{B}_1(\mathbf{X},t)\dot{\mathbf{W}}_1(t), \ \frac{\mathrm{d}\mathbf{Y}}{\mathrm{d}t} &= \mathbf{a}_0(\mathbf{X},t) + \mathbf{a}_1(\mathbf{X},t)\mathbf{Y} + \mathbf{b}_2(\mathbf{X},t)\dot{\mathbf{W}}_2(t). \end{aligned}$$6 are reported for $$\begin{aligned} \frac{\mathrm{d}\mathbf{X}}{\mathrm{d}t} &= \mathbf{A}_0(\mathbf{X},t) + \mathbf{A}_1(\mathbf{X},t)\mathbf{Y} + \mathbf{B}_1(\mathbf{X},t)\dot{\mathbf{W}}_1(t), \ \frac{\mathrm{d}\mathbf{Y}}{\mathrm{d}t} &= \mathbf{a}_0(\mathbf{X},t) + \mathbf{a}_1(\mathbf{X},t)\mathbf{Y} + \mathbf{b}_2(\mathbf{X},t)\dot{\mathbf{W}}_2(t). \end{aligned}$$7, the complexity is stated as $$\begin{aligned} \frac{\mathrm{d}\mathbf{X}}{\mathrm{d}t} &= \mathbf{A}_0(\mathbf{X},t) + \mathbf{A}_1(\mathbf{X},t)\mathbf{Y} + \mathbf{B}_1(\mathbf{X},t)\dot{\mathbf{W}}_1(t), \ \frac{\mathrm{d}\mathbf{Y}}{\mathrm{d}t} &= \mathbf{a}_0(\mathbf{X},t) + \mathbf{a}_1(\mathbf{X},t)\mathbf{Y} + \mathbf{b}_2(\mathbf{X},t)\dot{\mathbf{W}}_2(t). \end{aligned}$$8, and the largest tests with $$\begin{aligned} \frac{\mathrm{d}\mathbf{X}}{\mathrm{d}t} &= \mathbf{A}_0(\mathbf{X},t) + \mathbf{A}_1(\mathbf{X},t)\mathbf{Y} + \mathbf{B}_1(\mathbf{X},t)\dot{\mathbf{W}}_1(t), \ \frac{\mathrm{d}\mathbf{Y}}{\mathrm{d}t} &= \mathbf{a}_0(\mathbf{X},t) + \mathbf{a}_1(\mathbf{X},t)\mathbf{Y} + \mathbf{b}_2(\mathbf{X},t)\dot{\mathbf{W}}_2(t). \end{aligned}$$9 and $\mathbf{X}$0 are reported to require 30 minutes on standard hardware (Intel i3, 8GB RAM) (Yun et al., 18 Sep 2025).

These applications span analytically derived CGNS closures, hybrid neural-SDE models, Koopman-lifted latent systems, and Lagrangian assimilation frameworks. What they share is not a single discretization or architecture, but the preservation of a conditionally Gaussian hidden-state law that keeps inference analytically tractable.

6. Conceptual position, related frameworks, and limitations

In linear-Gaussian settings, the conditional Gaussian law is the posterior distribution produced by a Gaussian prior and linear observations, and classical Kalman filtering is one computational realization of that law. The "Ensemble-Conditional Gaussian Process" formulation makes this point explicit: the conditional Gaussian law is the representational object, while Kalman filters, EnKF variants, and iterative ensemble schemes are computational mechanisms for realizing or approximating it (Ravela et al., 14 Feb 2026). The same work also states equivalence, in the linear-Gaussian case, between the conditional Gaussian posterior, a strictly convex quadratic MAP program, RKHS-regularized regression, and classical regularization (Ravela et al., 14 Feb 2026).

That perspective helps locate CGNS conceptually. CGNS is broader than linear-Gaussian inference because its coefficients may depend nonlinearly on observed variables, yet narrower than unrestricted nonlinear state-space modeling because it requires conditional linearity in the hidden variables. This balance is what enables closed conditional statistics. It also explains why modern CGNS research frequently combines structure-preserving modeling with learned latent embeddings rather than using unrestricted black-box predictors (Chen et al., 2024, Chen et al., 2024).

The framework does, however, impose modeling and computational constraints. In physical-domain LEMDA, deriving CGNS system matrices from PDEs can be nontrivial, some “native” decompositions can be numerically unstable, and covariance tracking scales quadratically with state size (Yun et al., 18 Sep 2025). In learned latent-space models, the latent representation must simultaneously preserve nonlinear dynamics and maintain the conditional Gaussian structure required by analytic DA; LaCGKN addresses this with tracer homogenization, positional encoding, and low-rank parameterization, which indicates that architectural design is part of the model class rather than an external convenience (Wang et al., 14 Mar 2026).

This suggests a general interpretation of CGNS: it is a structural compromise between expressiveness and analytic solvability. The literature consistently treats that compromise not as a restriction to weakly nonlinear or nearly Gaussian behavior, but as a way to preserve enough nonlinearity to model intermittency, turbulent coupling, and rare events while still retaining closed conditional inference (Chen et al., 2021, Andreou et al., 2024).