mlsanityData to 7 Oct 2026

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

arXiv 2507.08749 ↗First posted 11 Jul 2025cs.LGcited by 6

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Abstract (arXiv)

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.

Key facts. Cited by 6 papers in the map (data through 7 Oct 2026). Builds on 38 of its 128 references that are in the map. First posted on arXiv 11 Jul 2025.

Builds on

38 of 128 references.

References · 38

  1. 2015Deep Residual Learning for Image Recognitioncited by 51,809
  2. 2020Denoising Diffusion Probabilistic Modelscited by 20,851
  3. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
  4. 2018Neural Ordinary Differential Equationscited by 3,733
  5. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  6. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  7. 2017Deep learning for universal linear embeddings of nonlinear dynamicscited by 491
  8. 2021Modern Koopman Theory for Dynamical Systemscited by 269
  9. 2022A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networkscited by 233
  10. 2018Deep learning to represent sub-grid processes in climate modelscited by 156
  11. 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 128
  12. 2021KalmanNet: Neural Network Aided Kalman Filtering for Partially Known Dynamicscited by 128
26 more
  1. 2017Learning Active Learning from Datacited by 100
  2. 2023Machine learning with data assimilation and uncertainty quantification for dynamical systems: a reviewcited by 74
  3. 2019Physics-Informed Probabilistic Learning of Linear Embeddings of Non-linear Dynamics With Guaranteed Stabilitycited by 53
  4. 2021Koopman Operator Dynamical Models: Learning, Analysis and Controlcited by 42
  5. 2020Combining data assimilation and machine learning to infer unresolved scale parametrisationcited by 40
  6. 2019Enforcing Statistical Constraints in Generative Adversarial Networks for Modeling Chaotic Dynamical Systemscited by 33
  7. 2021Auto-differentiable Ensemble Kalman Filterscited by 28
  8. 2019Data-Driven POD-Galerkin Reduced Order Model for Turbulent Flowscited by 21
  9. 2024Generative Learning of the Solution of Parametric Partial Differential Equations Using Guided Diffusion Models and Virtual Observationscited by 16
  10. 2023Neural Dynamical Operator: Continuous Spatial-Temporal Model with Gradient-Based and Derivative-Free Optimization Methodscited by 14
  11. 2024MODNO: Multi Operator Learning With Distributed Neural Operatorscited by 14
  12. 2023Learning About Structural Errors in Models of Complex Dynamical Systemscited by 13
  13. 2022Deep learning-enhanced ensemble-based data assimilation for high-dimensional nonlinear dynamical systemscited by 12
  14. 2023Reduced-Order Autodifferentiable Ensemble Kalman Filterscited by 12
  15. 2024Data-Driven Stochastic Closure Modeling via Conditional Diffusion Model and Neural Operatorcited by 12
  16. 2020Data Assimilation Networkscited by 10
  17. 2022A Causality-Based Learning Approach for Discovering the Underlying Dynamics of Complex Systems from Partial Observations with Stochastic Parameterizationcited by 8
  18. 2023CEBoosting: Online Sparse Identification of Dynamical Systems with Regime Switching by Causation Entropy Boostingcited by 7
  19. 2024CGKN: A Deep Learning Framework for Modeling Complex Dynamical Systems and Efficient Data Assimilationcited by 7
  20. 2021State, global and local parameter estimation using local ensemble Kalman filters: applications to online machine learning of chaotic dynamicscited by 6
  21. 2024CGNSDE: Conditional Gaussian Neural Stochastic Differential Equation for Modeling Complex Systems and Data Assimilationcited by 6
  22. 2024Physics-informed neural networks for parameter learning of wildfire spreadingcited by 6
  23. 2025Active Learning of Model Discrepancy with Bayesian Experimental Designcited by 5
  24. 2017Efficient Statistically Accurate Algorithms for the Fokker-Planck Equation in Large Dimensionscited by 4
  25. 2023Active-Learning-Driven Surrogate Modeling for Efficient Simulation of Parametric Nonlinear Systemscited by 4
  26. 2024Minimum Reduced-Order Models via Causal Inferencecited by 2

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 2; 2026: 4.

6 papers.

  1. 2025Uncertainty Propagation Networks for Neural Ordinary Differential Equationscited by 1
  2. 2026Pathwise Learning of Stochastic Dynamical Systems with Partial Observationscited by 1
  3. 2026Synergizing Transport-Based Generative Models and Latent Geometry for Stochastic Closure Modelingcited by 1
  4. 2026A deep learning framework for jointly solving transient Fokker-Planck equations with arbitrary parameters and initial distributionscited by 1
  5. 2025DeepKoopFormer: A Koopman Enhanced Transformer Based Architecture for Time Series Forecastingcited by 0
  6. 2026Omega: Operator-based Mixture Ensemble for Generative Assimilationcited by 0