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Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems

arXiv 1801.01236 ↗First posted 4 Jan 2018math.DS cs.NA math.NAcited by 89

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Abstract (arXiv)

The process of transforming observed data into predictive mathematical models of the physical world has always been paramount in science and engineering. Although data is currently being collected at an ever-increasing pace, devising meaningful models out of such observations in an automated fashion still remains an open problem. In this work, we put forth a machine learning approach for identifying nonlinear dynamical systems from data. Specifically, we blend classical tools from numerical analysis, namely the multi-step time-stepping schemes, with powerful nonlinear function approximators, namely deep neural networks, to distill the mechanisms that govern the evolution of a given data-set. We test the effectiveness of our approach for several benchmark problems involving the identification of complex, nonlinear and chaotic dynamics, and we demonstrate how this allows us to accurately learn the dynamics, forecast future states, and identify basins of attraction. In particular, we study the Lorenz system, the fluid flow behind a cylinder, the Hopf bifurcation, and the Glycoltic oscillator model as an example of complicated nonlinear dynamics typical of biological systems.

Key facts. Cited by 89 papers in the map (data through 14 Oct 2026). First posted on arXiv 4 Jan 2018.

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Citing papers per year: 2020: 1; 2021: 33; 2022: 22; 2023: 13; 2024: 8; 2025: 6; 2026: 6.

89 papers.

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  1. 2021Interpretable Machine Learning: Fundamental Principles and 10 Grand Challengescited by 367
  2. 2021Granger Causality: A Review and Recent Advancescited by 63
  3. 2022Bayesian Physics-Informed Neural Networks for real-world nonlinear dynamical systemscited by 54
  4. 2022Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systemscited by 50
  5. 2021GFINNs: GENERIC Formalism Informed Neural Networks for Deterministic and Stochastic Dynamical Systemscited by 49
  6. 2021Physics-Guided Deep Learning for Dynamical Systems: A Surveycited by 44
  7. 2021Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Mapscited by 41
  8. 2021Neural Networks with Physics-Informed Architectures and Constraints for Dynamical Systems Modelingcited by 41
  9. 2021HyperPINN: Learning parameterized differential equations with physics-informed hypernetworkscited by 39
  10. 2021Hamiltonian-based Neural ODE Networks on the SE(3) Manifold For Dynamics Learning and Controlcited by 38
  11. 2021A Framework for Machine Learning of Model Error in Dynamical Systemscited by 38
  12. 2021Discovery of Nonlinear Dynamical Systems using a Runge-Kutta Inspired Dictionary-based Sparse Regression Approachcited by 36
38 more
  1. 2022Tractable Dendritic RNNs for Reconstructing Nonlinear Dynamical Systemscited by 36
  2. 2024Out-of-Domain Generalization in Dynamical Systems Reconstructioncited by 33
  3. 2021SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEscited by 32
  4. 2021Constructing Neural Network-Based Models for Simulating Dynamical Systemscited by 31
  5. 2021Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equationscited by 30
  6. 2021Learning elliptic partial differential equations with randomized linear algebracited by 29
  7. 2022Learning Sparse Nonlinear Dynamics via Mixed-Integer Optimizationcited by 29
  8. 2021DAE-PINN: A Physics-Informed Neural Network Model for Simulating Differential-Algebraic Equations with Application to Power Networkscited by 28
  9. 2022Learning continuous models for continuous physicscited by 28
  10. 2021Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Spacecited by 25
  11. 2023Learning Stochastic Dynamical System via Flow Map Operatorcited by 24
  12. 2022On Numerical Integration in Neural Ordinary Differential Equationscited by 21
  13. 2023Flow Map Learning for Unknown Dynamical Systems: Overview, Implementation, and Benchmarkscited by 18
  14. 2021The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimationcited by 14
  15. 2022DeepGraphONet: A Deep Graph Operator Network to Learn and Zero-shot Transfer the Dynamic Response of Networked Systemscited by 14
  16. 2022Homotopy-based training of NeuralODEs for accurate dynamics discoverycited by 14
  17. 2023Energy-Dissipative Evolutionary Deep Operator Neural Networkscited by 14
  18. 2021Gaussian processes meet NeuralODEs: A Bayesian framework for learning the dynamics of partially observed systems from scarce and noisy datacited by 13
  19. 2023On Approximating the Dynamic Response of Synchronous Generators via Operator Learning: A Step Towards Building Deep Operator-based Power Grid Simulatorscited by 13
  20. 2023Deep-OSG: Deep Learning of Operators in Semigroupcited by 11
  21. 2023MINN: Learning the dynamics of differential-algebraic equations and application to battery modelingcited by 10
  22. 2021Constrained Block Nonlinear Neural Dynamical Modelscited by 9
  23. 2021DeepParticle: learning invariant measure by a deep neural network minimizing Wasserstein distance on data generated from an interacting particle methodcited by 9
  24. 2022Physical Modeling using Recurrent Neural Networks with Fast Convolutional Layerscited by 9
  25. 2022Deep Learning of Chaotic Systems from Partially-Observed Datacited by 9
  26. 2023AI-Lorenz: A physics-data-driven framework for black-box and gray-box identification of chaotic systems with symbolic regressioncited by 9
  27. 2024Physics-Informed Neural Networks and Extensionscited by 8
  28. 2025DUE: A Deep Learning Framework and Library for Modeling Unknown Equationscited by 8
  29. 2021Learning Interaction Kernels for Agent Systems on Riemannian Manifoldscited by 7
  30. 2021Learning Dynamics from Noisy Measurements using Deep Learning with a Runge-Kutta Constraintcited by 7
  31. 2024Modeling Unknown Stochastic Dynamical System Subject to External Excitationcited by 7
  32. 2022Continual Learning of Dynamical Systems with Competitive Federated Reservoir Computingcited by 6
  33. 2023MetaPhysiCa: OOD Robustness in Physics-informed Machine Learningcited by 6
  34. 2023Implementation and (Inverse Modified) Error Analysis for implicitly-templated ODE-netscited by 6
  35. 2022Robust Modeling of Unknown Dynamical Systems via Ensemble Averaged Learningcited by 5
  36. 2022Neural ODEs with Irregular and Noisy Datacited by 5
  37. 2024Sparse discovery of differential equations based on multi-fidelity Gaussian processcited by 5
  38. 2024Data-driven Effective Modeling of Multiscale Stochastic Dynamical Systemscited by 5

50 most cited of 89 All 89 (JSON)