mlsanityData to 14 Oct 2026

Data-driven discovery of intrinsic dynamics

arXiv 2108.05928 ↗First posted 12 Aug 2021cs.LGcited by 45

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Abstract (arXiv)

Dynamical models underpin our ability to understand and predict the behavior of natural systems. Whether dynamical models are developed from first-principles derivations or from observational data, they are predicated on our choice of state variables. The choice of state variables is driven by convenience and intuition, and in the data-driven case the observed variables are often chosen to be the state variables. The dimensionality of these variables (and consequently the dynamical models) can be arbitrarily large, obscuring the underlying behavior of the system. In truth, these variables are often highly redundant and the system is driven by a much smaller set of latent intrinsic variables. In this study, we combine the mathematical theory of manifolds with the representational capacity of neural networks to develop a method that learns a system's intrinsic state variables directly from time series data, and also learns predictive models for their dynamics. What distinguishes our method is its ability to reduce data to the intrinsic dimensionality of the nonlinear manifold they live on. This ability is enabled by the concepts of charts and atlases from the theory of manifolds, whereby a manifold is represented by a collection of patches that are sewn together -- a necessary representation to attain intrinsic dimensionality. We demonstrate this approach on several high-dimensional systems with low-dimensional behavior. The resulting framework provides the ability to develop dynamical models of the lowest possible dimension, capturing the essence of a system.

Key facts. Cited by 45 papers in the map (data through 14 Oct 2026). Builds on 12 of its 79 references that are in the map. First posted on arXiv 12 Aug 2021.

Builds on

12 of 79 references.

References · 12

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2017Deep learning for universal linear embeddings of nonlinear dynamicscited by 494
  3. 2017Learning Koopman Invariant Subspaces for Dynamic Mode Decompositioncited by 158
  4. 2018Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networkscited by 155
  5. 2017Linearly-Recurrent Autoencoder Networks for Learning Dynamicscited by 145
  6. 2019Backpropagation Algorithms and Reservoir Computing in Recurrent Neural Networks for the Forecasting of Complex Spatiotemporal Dynamicscited by 134
  7. 2017Visual Interaction Networkscited by 95
  8. 2020Multiscale Simulations of Complex Systems by Learning their Effective Dynamicscited by 86
  9. 2018Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systemscited by 54
  10. 2019Deep learning to discover and predict dynamics on an inertial manifoldcited by 43
  11. 2020Implicit Rank-Minimizing Autoencodercited by 35
  12. 2019Chart Auto-Encoders for Manifold Structured Datacited by 33

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 5; 2023: 15; 2024: 10; 2025: 9; 2026: 6.

45 papers.

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  1. 2023Generative learning for nonlinear dynamicscited by 22
  2. 2022Data-driven control of spatiotemporal chaos with reduced-order neural ODE-based models and reinforcement learningcited by 17
  3. 2023Manifold Learning by Mixture Models of VAEs for Inverse Problemscited by 17
  4. 2023Autoencoders for discovering manifold dimension and coordinates in data from complex dynamical systemscited by 13
  5. 2023Interpretable statistical representations of neural population dynamics and geometrycited by 12
  6. 2023Interpretable learning of effective dynamics for multiscale systemscited by 11
  7. 2024LD-EnSF: Synergizing Latent Dynamics with Ensemble Score Filters for Fast Data Assimilation with Sparse Observationscited by 11
  8. 2022Data-driven low-dimensional dynamic model of Kolmogorov flowcited by 9
  9. 2024VENI, VINDy, VICI: a generative reduced-order modeling framework with uncertainty quantificationcited by 9
  10. 2022Semi-Supervised Manifold Learning with Complexity Decoupled Chart Autoencoderscited by 8
  11. 2022Deep learning delay coordinate dynamics for chaotic attractors from partial observable datacited by 8
  12. 2022Emerging trends in machine learning for computational fluid dynamicscited by 8
33 more
  1. 2023Dynamics of a data-driven low-dimensional model of turbulent minimal Couette flowcited by 8
  2. 2023Stochastic Latent Transformer: Efficient Modelling of Stochastically Forced Zonal Jetscited by 8
  3. 2025Discovering Symbolic Differential Equations with Symmetry Invariantscited by 6
  4. 2023Turbulence control in plane Couette flow using low-dimensional neural ODE-based models and deep reinforcement learningcited by 5
  5. 2023Adaptive learning of effective dynamics: Adaptive real-time, online modeling for complex systemscited by 4
  6. 2023Latent Dynamics Networks (LDNets): learning the intrinsic dynamics of spatio-temporal processescited by 4
  7. 2023Finding emergence in data by maximizing effective informationcited by 4
  8. 2024LES-SINDy: Laplace-Enhanced Sparse Identification of Nonlinear Dynamical Systemscited by 4
  9. 2024Conservation-informed Graph Learning for Spatiotemporal Dynamics Predictioncited by 4
  10. 2023Tasks Makyth Models: Machine Learning Assisted Surrogates for Tipping Pointscited by 3
  11. 2024Sampling-enabled scalable manifold learning unveils the discriminative cluster structure of high-dimensional datacited by 3
  12. 2025Learning solution operator of dynamical systems with diffusion maps kernel ridge regressioncited by 3
  13. 2023Learning Continuous Network Emerging Dynamics from Scarce Observations via Data-Adaptive Stochastic Processescited by 2
  14. 2023Building symmetries into data-driven manifold dynamics models for complex flows: application to two-dimensional Kolmogorov flowcited by 2
  15. 2024RefreshNet: Learning Multiscale Dynamics through Hierarchical Refreshingcited by 2
  16. 2024On instabilities in neural network-based physics simulatorscited by 2
  17. 2025Ultralow-dimensionality reduction for identifying critical transitions by spatial-temporal PCAcited by 2
  18. 2025Efficient Manifold-Constrained Neural ODE for High-Dimensional Datasetscited by 2
  19. 2026Geometric regularization of autoencoders via observed stochastic dynamicscited by 2
  20. 2024Using Neural Implicit Flow To Represent Latent Dynamics Of Canonical Systemscited by 1
  21. 2025A Weak Penalty Neural ODE for Learning Chaotic Dynamics from Noisy Time Seriescited by 1
  22. 2026Learning dynamical systems from noisy data with Weak-form Kernel Ridge Regressioncited by 1
  23. 2023Color Image Recovery Using Generalized Matrix Completion over Higher-Order Finite Dimensional Algebracited by 0
  24. 2024Learning effective good variables from physical datacited by 0
  25. 2024Vision-based Discovery of Nonlinear Dynamics for 3D Moving Targetcited by 0
  26. 2025Scientific machine learning in Hydrology: a unified perspectivecited by 0
  27. 2025Blending data and physics for reduced-order modeling of systems with spatiotemporal chaotic dynamicscited by 0
  28. 2025A Survey and Comparative Evaluation of Intrinsic Dimension Estimators under the Manifold Hypothesiscited by 0
  29. 2025Atlas Gaussian processes on restricted domains and point cloudscited by 0
  30. 2026Information-Theoretic Multi-Model Fusion for Target-Oriented Adaptive Sampling in Materials Designcited by 0
  31. 2026Symbolic Graph Networks for Robust PDE Discovery from Noisy Sparse Datacited by 0
  32. 2026A neural operator framework for data-driven discovery of stability and receptivity in physical systemscited by 0
  33. 2026Convolutional Symmetric AutoEncoders: enhancing latent stability via differential geometrycited by 0