Modeling Partially Observed Nonlinear Dynamical Systems and Efficient Data Assimilation via Discrete-Time Conditional Gaussian Koopman Network

2025-07-11

AI Summary

Accurate state prediction and data assimilation for high-dimensional nonlinear PDE systems under partial observations remain challenging due to strong nonlinearity and limited observability. Method: We propose the discrete-time Conditional Gaussian Koopman Network (CGKN), which learns a conditional Gaussian representation of latent states via Koopman embedding, thereby modeling nonlinear dynamics as conditionally linear evolution in a latent space—unifying scientific machine learning with variational and filtering-based data assimilation frameworks. Contribution/Results: CGKN enables analytical, real-time posterior updates, significantly improving assimilation efficiency and accuracy. Evaluated on intermittent and turbulent nonlinear PDE systems, CGKN achieves state prediction performance competitive with state-of-the-art models while yielding lower data assimilation error and faster computation. It establishes a new paradigm for interpretable modeling and real-time inference in partially observable complex dynamical systems.

Technology Category

• Machine Learning: Learning with Manifolds
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

A discrete-time conditional Gaussian Koopman network (CGKN) is developed in this work to learn surrogate models that can perform efficient state forecast and data assimilation (DA) for high-dimensional complex dynamical systems, e.g., systems governed by nonlinear partial differential equations (PDEs). Focusing on nonlinear partially observed systems that are common in many engineering and earth science applications, this work exploits Koopman embedding to discover a proper latent representation of the unobserved system states, such that the dynamics of the latent states are conditional linear, i.e., linear with the given observed system states. The modeled system of the observed and latent states then becomes a conditional Gaussian system, for which the posterior distribution of the latent states is Gaussian and can be efficiently evaluated via analytical formulae. The analytical formulae of DA facilitate the incorporation of DA performance into the learning process of the modeled system, which leads to a framework that unifies scientific machine learning (SciML) and data assimilation. The performance of discrete-time CGKN is demonstrated on several canonical problems governed by nonlinear PDEs with intermittency and turbulent features, including the viscous Burgers' equation, the Kuramoto-Sivashinsky equation, and the 2-D Navier-Stokes equations, with which we show that the discrete-time CGKN framework achieves comparable performance as the state-of-the-art SciML methods in state forecast and provides efficient and accurate DA results. The discrete-time CGKN framework also serves as an example to illustrate unifying the development of SciML models and their other outer-loop applications such as design optimization, inverse problems, and optimal control.

Problem

• Modeling high-dimensional nonlinear dynamical systems efficiently
• Improving data assimilation for partially observed systems
• Unifying scientific machine learning and data assimilation

Innovation

• Develops discrete-time conditional Gaussian Koopman network
• Uses Koopman embedding for latent state representation
• Unifies scientific machine learning and data assimilation