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Sequential Bayesian parameter-state estimation in dynamical systems with noisy and incomplete observations via a variational framework

This paper proposes a sequential variational inference framework that efficiently approximates the joint posterior of states and parameters in dynamical systems with noisy, incomplete observations by factorizing the distribution into a parameter posterior and a conditional state distribution, thereby enabling robust, uncertainty-quantified online estimation that outperforms joint ensemble Kalman filters in high-dimensional and chaotic settings.

Original authors: Liliang Wang, Alex Gorodetsky

Published 2026-06-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Liliang Wang, Alex Gorodetsky

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to drive a car through a thick fog at night. You have a map (your model of the car and the road), but the map might be slightly wrong, and your eyes (your sensors) are seeing blurry, incomplete shapes. You need to figure out two things simultaneously: where the car is right now (the state) and exactly how the car's engine and steering work (the parameters).

This is the problem of "joint parameter-state estimation." It's crucial for things like "Digital Twins"—virtual copies of real machines that need to stay updated to make predictions. But doing this in real-time, with noisy data, is incredibly hard.

Here is how the authors of this paper solved it, using a few creative analogies:

The Problem: The "Black Box" Puzzle

Usually, trying to guess both the car's location and its engine specs at the same time is like trying to solve a giant, tangled knot of math. If you try to untangle it all at once (like traditional "Particle Filters" do), you need millions of guesses to get it right, which is too slow for real-time use. If you try to simplify it too much (like "Kalman Filters" do), you assume everything is a perfect circle (Gaussian), but real life is often a weird, squiggly shape. When the math gets too complex, these simple assumptions break, and your guesses go wildly off track.

The Solution: The "Two-Stage Detective" (FBOVI)

The authors propose a new method called FBOVI (Factorization-Based Online Variational Inference). Instead of trying to solve the whole knot at once, they split the job into two manageable steps, like a detective team working together:

Stage 1: The Parameter Detective (The "Engine Expert")
- The Job: This detective looks at the blurry data and tries to figure out the "rules of the game" (the unknown parameters).
- The Trick: Instead of guessing blindly, they use a smart technique called "Variational Inference." Imagine they are trying to fit a flexible, stretchy sheet over a complex 3D object. They stretch the sheet until it hugs the object as closely as possible without tearing. This gives them the best possible guess for the engine specs based on the current evidence.
Stage 2: The State Detective (The "Location Tracker")
- The Job: Now that the Engine Expert has a good guess about the engine, the Location Tracker uses that specific guess to figure out exactly where the car is.
- The Trick: Because the engine specs are now "fixed" (based on Stage 1), the math for finding the location becomes much easier. They use standard, fast tracking tools (like Gaussian filters) to pinpoint the car's position.

The Magic Loop:
After the Location Tracker finishes, the team moves to the next second. The new data comes in, and the cycle repeats. The "Engine Expert" updates their guess using the new info, and the "Location Tracker" updates the position. They never look back at old data; they just use the latest "best guess" from the previous step. This makes it incredibly fast and efficient.

Why This is Better Than the Old Ways

Vs. The "Million Guess" Method (Particle Filters): The old way tries to throw a million darts at a board to find the answer. It's accurate but slow and gets confused easily if the board is too big (high dimensions). FBOVI is like using a laser pointer; it's precise and fast, even on a giant board.
Vs. The "Perfect Circle" Method (Joint Kalman Filters): The old way assumes the answer is always a perfect circle. If the real answer is a star or a blob, the old method fails. FBOVI is flexible; it can stretch its "sheet" to fit weird shapes, so it doesn't break when the real world gets messy.

What They Tested It On

The authors didn't just talk about theory; they tested this "Two-Stage Detective" on three very different challenges:

The Simple Swing (Linear Pendulum): A basic physics problem where the math is straightforward. Here, FBOVI was just as good as the heavy-hitting "Million Guess" method but much faster.
The Chaotic Weather (Lorenz '96 System): This is a system that is famous for being chaotic—tiny changes in the beginning lead to huge differences later (the "Butterfly Effect"). They tested it with:
- Wrong Models: When the map they were using was slightly wrong.
- Noisy Data: When the sensors were very blurry.
- Result: Even with bad maps and blurry sensors, FBOVI kept the car on the road, while the other methods crashed.
The Giant Wave (Convection-Diffusion): A complex fluid dynamics problem with 51 moving parts (high-dimensional). This is like trying to track a massive ocean wave. The "Million Guess" method would have been too slow to run, but FBOVI handled it smoothly, outperforming the standard "Joint Ensemble Kalman Filter."

The Bottom Line

The paper claims that this new framework is a "Goldilocks" solution: it's not too slow like the million-guess methods, and not too rigid like the simple circle methods. It provides a way to update our knowledge of a system's location and its internal rules in real-time, even when the data is messy, incomplete, or the model is slightly wrong. It comes with a mathematical "safety guarantee" (a theorem) that proves the error won't get out of hand, making it a reliable tool for keeping Digital Twins accurate and trustworthy.

Technical Summary: Sequential Bayesian Parameter–State Estimation via a Variational Framework

Problem Statement

The paper addresses the challenge of online joint estimation of unknown model parameters ( $\theta$ ) and system states ( $X_k$ ) for dynamical systems, specifically under conditions of noisy and incomplete observations. This task is critical for applications such as Digital Twins (DTs), which require continuous updates of model knowledge to support prediction and decision-making.

The core difficulty lies in computing the joint posterior distribution $p(X_k, \theta \mid Y_k)$ , where $Y_k$ represents all observations up to time $k$ . This distribution provides accurate uncertainty quantification by capturing dependencies between states and parameters. However, the joint posterior is generally intractable. Existing online methods face significant limitations:

Joint Gaussian Filters (e.g., Joint UKF, Joint EnKF): Assume the joint posterior is Gaussian. This assumption is overly restrictive for nonlinear systems, leading to large approximation errors.
Joint Particle Filters (Joint PFs): Avoid Gaussian assumptions but suffer from severe particle weight degeneracy when static parameters are augmented into the state vector, requiring an intractable number of particles for high-dimensional problems.
SMC2: While effective, its computational cost grows with time, rendering it unsuitable for strictly online tasks.
Existing Variational Inference (VI) methods: Most are designed for offline settings or approximate the joint posterior as a product of independent marginals (mean-field), neglecting the intrinsic dependence between parameters and states.

Furthermore, few existing methods provide theoretical guarantees on approximation accuracy, which is essential for establishing trust in safety-critical applications.

Methodology: Factorization-Based Online Variational Inference (FBOVI)

The authors propose FBOVI, a flexible online variational inference framework that approximates the joint posterior $p(X_k, \theta \mid Y_k)$ with a distribution $q_k(X_k, \theta)$ .

Core Framework

Instead of approximating the true joint posterior directly, FBOVI introduces "online targets" and factorizes the approximation as:
$q_k(X_k, \theta) = \rho_k(X_k \mid \theta) \nu_k(\theta)$
where:

$\nu_k(\theta)$ approximates the online parameter posterior (the marginal distribution of parameters).
$\rho_k(X_k \mid \theta)$ approximates the online conditional filtering distribution (the state distribution conditioned on parameters).

The framework operates in a two-stage recursive procedure at each time step $k$ :

Stage 1: Parameter Posterior Approximation ( $\nu_k$ )

The goal is to minimize the Kullback-Leibler (KL) divergence $KL(\nu_k \parallel \nu^*_k)$ , where $\nu^*_k$ is the online target parameter posterior.
This is achieved by maximizing an Evidence Lower Bound (ELBO).
The integral required for the ELBO is computed differently based on the system type:
- Linear Systems: Closed-form expressions are derived using the Kalman filter logic.
- Nonlinear Systems: The predictive distribution is approximated by a Gaussian (via Unscented Transform or Gaussian Quadrature), allowing the integral to be evaluated. Alternatively, Monte Carlo sampling can be used.

Stage 2: Conditional State Distribution Approximation ( $\rho_k$ )

Using the computed $\nu_k(\theta)$ , the framework minimizes the expected KL divergence $E_{\theta \sim \nu_k}[KL(\rho_k(\cdot \mid \theta) \parallel \rho^*_k(\cdot \mid \theta))]$ .
$\rho_k(X_k \mid \theta)$ is modeled as a conditional Gaussian: $N(X_k; m_k(\theta), C_k(\theta))$ .
Neural Network Representation: To ensure computational efficiency and prevent cost growth over time, the mean function $m_k(\theta)$ and covariance function $C_k(\theta)$ are represented by neural networks with fixed architectures. These networks are trained to approximate the target mean and covariance derived from a Gaussian filter (KF, UKF, or EnKF) applied to the online target.
EnKF Integration: For high-dimensional systems, the framework incorporates the Ensemble Kalman Filter (EnKF). Since state particles are not propagated forward (only the functional representations $m_k, C_k$ are), a novel every-step resampling procedure is used to generate training data for the neural networks.

Theoretical Guarantees

The paper establishes Theorem 1, which provides an upper bound on the error (measured by Total Variation or Hellinger distance) between the true joint posterior and the FBOVI approximation. The bound depends on the KL divergences between the approximations and their respective online targets. For linear systems, Theorem 2 provides a more specific, estimable error bound.

Key Contributions

Flexible Variational Framework: A novel online VI framework that factorizes the joint posterior to preserve dependencies between parameters and states, avoiding the restrictive Gaussian assumptions of joint filters and the degeneracy issues of joint particle filters.
Theoretical Error Bounds: Derivation of upper bounds on the approximation error (Theorem 1 and Theorem 2), providing a theoretical foundation for the method's effectiveness and guiding algorithmic design.
Scalability and Modularity: The framework supports various Gaussian filtering methods (KF, UKF, EnKF) within Stage 2. By using neural networks to represent conditional moments, it achieves constant per-step computational costs and scales to high-dimensional systems.
Comprehensive Validation: Extensive numerical experiments demonstrating performance across diverse scenarios, including partially unknown models, model misspecification, chaotic dynamics, and high-dimensional PDE discretizations.

Experimental Results

The method was evaluated against Joint PF, Joint UKF, and Joint EnKF in three distinct experiments:

Linear Single Pendulum (Partially Unknown Models):
- FBOVI achieved estimation and prediction performance comparable to or better than Joint PF and Joint UKF.
- It successfully tracked true posteriors even when the initial prior was inaccurate, whereas Joint UKF diverged.
- It handled partially unknown dynamical and observation models effectively.
Chaotic Lorenz '96 System (10-dimensional):
- Correct Model: FBOVI significantly outperformed Joint UKF in estimating parameters and unobserved states, capturing non-Gaussian features in the joint posterior that Joint UKF missed.
- Model Misspecification: Under structural model errors (using Euler vs. RK4), FBOVI remained robust, providing reasonable state estimates and predictions, though accuracy degraded slightly near trajectory peaks/troughs.
- High Noise: The method remained robust even with highly noisy measurements ( $\gamma=10$ ), though estimation accuracy near extrema decreased.
Nonlinear Convection–Diffusion Equation (51-dimensional):
- FBOVI, integrated with EnKF, successfully tracked the reference solution and provided accurate predictions.
- It outperformed the Joint EnKF, particularly in the early stages where the true posterior is highly non-Gaussian. The Joint EnKF's Gaussian assumption limited its predictive capability in this high-dimensional, nonlinear setting.

Significance and Claims

The paper claims that FBOVI offers a reliable, scalable, and adaptable solution for online parameter-state estimation. Its significance lies in:

Bridging the gap between the accuracy of particle methods and the efficiency of Gaussian filters without suffering from weight degeneracy or restrictive Gaussian assumptions.
Providing theoretical guarantees for approximation accuracy, which is crucial for establishing trust in Digital Twins and risk-aware decision-making.
Enabling high-dimensional inference by leveraging neural networks and EnKF, making it suitable for complex physical systems described by PDEs.

The authors conclude that the framework is a powerful tool for emerging data-driven techniques, particularly in scenarios requiring continuous updates under uncertainty, noise, and model discrepancies.