mlsanityData to 14 Oct 2026

Neural Dynamical Operator: Continuous Spatial-Temporal Model with Gradient-Based and Derivative-Free Optimization Methods

arXiv 2311.11798 ↗First posted 20 Nov 2023cs.LGcited by 14

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Abstract (arXiv)

Data-driven modeling techniques have been explored in the spatial-temporal modeling of complex dynamical systems for many engineering applications. However, a systematic approach is still lacking to leverage the information from different types of data, e.g., with different spatial and temporal resolutions, and the combined use of short-term trajectories and long-term statistics. In this work, we build on the recent progress of neural operator and present a data-driven modeling framework called neural dynamical operator that is continuous in both space and time. A key feature of the neural dynamical operator is the resolution-invariance with respect to both spatial and temporal discretizations, without demanding abundant training data in different temporal resolutions. To improve the long-term performance of the calibrated model, we further propose a hybrid optimization scheme that leverages both gradient-based and derivative-free optimization methods and efficiently trains on both short-term time series and long-term statistics. We investigate the performance of the neural dynamical operator with three numerical examples, including the viscous Burgers' equation, the Navier-Stokes equations, and the Kuramoto-Sivashinsky equation. The results confirm the resolution-invariance of the proposed modeling framework and also demonstrate stable long-term simulations with only short-term time series data. In addition, we show that the proposed model can better predict long-term statistics via the hybrid optimization scheme with a combined use of short-term and long-term data.

Key facts. Cited by 14 papers in the map (data through 14 Oct 2026). Builds on 29 of its 89 references that are in the map. First posted on arXiv 20 Nov 2023.

Builds on

29 of 89 references.

References · 29

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  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
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  7. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 527
  8. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 427
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17 more
  1. 2018Neural Lander: Stable Drone Landing Control using Learned Dynamicscited by 85
  2. 2022Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systemscited by 50
  3. 2022Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learningcited by 44
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Citing papers per year: 2024: 4; 2025: 6; 2026: 4.

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  3. 2024CGKN: A Deep Learning Framework for Modeling Complex Dynamical Systems and Efficient Data Assimilationcited by 7
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  5. 2025Modeling Partially Observed Nonlinear Dynamical Systems and Efficient Data Assimilation via Discrete-Time Conditional Gaussian Koopman Networkcited by 6
  6. 2025Active Learning of Model Discrepancy with Bayesian Experimental Designcited by 5
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  9. 2025Bayesian Experimental Design for Model Discrepancy Calibration: An Auto-Differentiable Ensemble Kalman Inversion Approachcited by 2
  10. 2025An Inverse Scattering Inspired Fourier Neural Operator for Time-Dependent PDE Learningcited by 1
  11. 2026Weak-Form Evolutionary Kolmogorov-Arnold Networks for Solving Partial Differential Equationscited by 1
  12. 2026Bayesian Experimental Design for Model Discrepancy Calibration: A Rivalry between Kullback--Leibler Divergence and Wasserstein Distancecited by 0
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  2. 2026Geometry-aware Incremental Neural Operator for Long-Horizon PDE predictioncited by 0