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

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arxiv 1801.01236 v1 pith:TYGAUROE submitted 2018-01-04 math.DS cs.NAmath.NAnlin.CDphysics.comp-phstat.ML

Maziar Raissi , Paris Perdikaris , George Em Karniadakis This is my paper · ORCID
classification math.DScs.NAmath.NAnlin.CDphysics.comp-phstat.ML
keywords nonlineardatadynamicssystemsapproachdynamicalmodelsnamely
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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.

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Cited by 11 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators

    cs.LG 2019-10 conditional novelty 8.0 of 10

    DeepONet learns nonlinear operators for differential equations via branch and trunk sub-networks, achieving high-order error convergence on small datasets.

  2. Neural Ordinary Differential Equations

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    Neural networks are redefined as continuous dynamical systems by learning the derivative of the hidden state with a neural network and integrating it with an ODE solver.

  3. Artifacts of Numerical Integration in Learning Dynamical Systems

    math.NA 2025-07 conditional novelty 7.0 of 10

    Numerical integration schemes used in optimizing models of dynamical systems from sampled data can distort learned stability properties, inducing anti-damping artifacts in originally damped systems.

  4. Continuity-Preserving Convolutional Autoencoders for Learning Continuous Latent Dynamical Models from Images

    cs.LG 2025-02 conditional novelty 7.0 of 10

    A filter-smoothness regularizer makes CNN latent codes from image frames behave like samples of a continuous trajectory, so neural ODEs fit them and predict future frames better.

  5. Universal Differential Equations for Scientific Machine Learning

    cs.LG 2020-01 unverdicted novelty 7.0 of 10

    Universal Differential Equations unify scientific models with machine learning by embedding flexible approximators into differential equations, enabling applications from biological mechanism discovery to high-dimensi...

  6. Neural Network Compression by Approximate Differential Equivalence

    cs.LG 2026-05 unverdicted novelty 6.0 of 10

    Neural networks are compressed by lumping neurons with approximately matching dynamics in a polynomial ODE encoding, yielding substantial size reduction with preserved accuracy on synthetic and regression tasks.

  7. Flow map learning in nonlinear vector autoregressive models: influence of the feature-library structure on the training error

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    NVAR models exhibit training error scaling laws tied to feature library representation of Lie-series coefficients, with delays reducing one-step error but aiding long-horizon forecasts only under sufficient nonlinearity.

  8. PnP-Corrector: A Universal Correction Framework for Coupled Spatiotemporal Forecasting

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    PnP-Corrector decouples pre-trained physics engines from a correction agent to mitigate reciprocal error amplification in coupled spatiotemporal forecasting, cutting error by 28% on a 300-day ocean-atmosphere task.

  9. Benchmarking Multi-fidelity Neural Operators on Complex PDE Problems with Non-trivial Fidelity Differences

    cs.CE 2026-08 conditional novelty 5.0 of 10

    Across four PDE benchmarks, transfer learning from low-fidelity weights is the only multi-fidelity neural operator strategy that consistently outperforms the high-fidelity-only baseline, while direct LF-input methods ...

  10. Learning Hamiltonian Dynamics with Bayesian Data Assimilation

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    An autoregressive Hamiltonian neural network coupled with an unscented Kalman filter improves long-term trajectory prediction and uncertainty quantification for unknown Hamiltonian systems.

  11. Dynamics-Encoded Deep Learning for Robust System Identification and Parameter Estimation

    cs.LG 2024-10 unverdicted novelty 4.0 of 10

    Dynamics-encoded deep learning approaches are developed for system identification and parameter estimation in dynamical systems using numerical discretization schemes.

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