# Data-Driven Modeling and Prediction of Non-Linearizable Dynamics via Spectral Submanifolds

arXiv: https://arxiv.org/abs/2201.04976 · HTML: https://mlsanity.com/paper/2201.04976/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 84 references that are in the map.
- First posted on arXiv 13 Jan 2022.

## Builds on (most cited first, 7 of 7)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [Data-driven discovery of Koopman eigenfunctions for control](https://mlsanity.com/paper/1707.01146/) (2017)
- [How to Compute Invariant Manifolds and their Reduced Dynamics in High-Dimensional Finite-Element Models](https://mlsanity.com/paper/2103.10264/) (2021)
- [Principal component trajectories for modeling spectrally-continuous dynamics as forced linear systems](https://mlsanity.com/paper/2005.14321/) (2020)
- [Data-driven Nonlinear Model Reduction to Spectral Submanifolds in Mechanical Systems](https://mlsanity.com/paper/2110.01929/) (2021)
- [Exact Nonlinear Model Reduction for a von Karman beam: Slow-Fast Decomposition and Spectral Submanifolds](https://mlsanity.com/paper/1703.03001/) (2017)

## Built on by (most cited first, 25 of 25)

- [Discovering Causal Relations and Equations from Data](https://mlsanity.com/paper/2305.13341/) (2023)
- [Generative learning for nonlinear dynamics](https://mlsanity.com/paper/2311.04128/) (2023)
- [Benchmarking sparse system identification with low-dimensional chaos](https://mlsanity.com/paper/2302.10787/) (2023)
- [Learning Nonlinear Projections for Reduced-Order Modeling of Dynamical Systems using Constrained Autoencoders](https://mlsanity.com/paper/2307.15288/) (2023)
- [Mori-Zwanzig latent space Koopman closure for nonlinear autoencoder](https://mlsanity.com/paper/2310.10745/) (2023)
- [Data-driven reduced order models using invariant foliations, manifolds and autoencoders](https://mlsanity.com/paper/2206.12269/) (2022)
- [Input-to-State Stable Coupled Oscillator Networks for Closed-form Model-based Control in Latent Space](https://mlsanity.com/paper/2409.08439/) (2024)
- [Latent Dynamics Networks (LDNets): learning the intrinsic dynamics of spatio-temporal processes](https://mlsanity.com/paper/2305.00094/) (2023)
- [Machine Learning Applications to Computational Plasma Physics and Reduced-Order Plasma Modeling: A Perspective](https://mlsanity.com/paper/2409.02349/) (2024)
- [On the relationship between Koopman operator approximations and neural ordinary differential equations for data-driven time-evolution predictions](https://mlsanity.com/paper/2411.12940/) (2024)
- [Distributed computing for physics-based data-driven reduced modeling at scale: Application to a rotating detonation rocket engine](https://mlsanity.com/paper/2407.09994/) (2024)
- [A discrete physics-informed training for projection-based reduced order models with neural networks](https://mlsanity.com/paper/2504.13875/) (2025)
- [Machine-learning invariant foliations in forced systems for reduced order modelling](https://mlsanity.com/paper/2403.14514/) (2024)
- [Invertible Koopman neural operator for data-driven modeling of partial differential equations](https://mlsanity.com/paper/2503.19717/) (2025)
- [RNNs perform task computations by dynamically warping neural representations](https://mlsanity.com/paper/2512.04310/) (2025)
- [Data-driven modelling of autonomous and forced dynamical systems](https://mlsanity.com/paper/2512.12432/) (2025)
- [Nonlinear model reduction for transport-dominated problems](https://mlsanity.com/paper/2602.01397/) (2026)
- [Physics-informed reduced-order modelling with equivariant spectral submanifolds](https://mlsanity.com/paper/2608.04239/) (2026)
- [Changes from Classical Statistics to Modern Statistics and Data Science](https://mlsanity.com/paper/2211.03756/) (2022)
- [Hybrid Time-Domain Behavior Model Based on Neural Differential Equations and RNNs](https://mlsanity.com/paper/2503.22313/) (2025)
- [SINDy on slow manifolds](https://mlsanity.com/paper/2507.00747/) (2025)
- [Blending data and physics for reduced-order modeling of systems with spatiotemporal chaotic dynamics](https://mlsanity.com/paper/2507.21299/) (2025)
- [Enhancing sample efficiency in reinforcement-learning-based flow control: replacing the critic with an adaptive reduced-order model](https://mlsanity.com/paper/2604.04986/) (2026)
- [A Deep Learning-based surrogate model for Severe Accidents in nuclear reactors using ASTEC](https://mlsanity.com/paper/2607.04450/) (2026)
- [Floquet Fibre Geometry and Higher-Order Reduced Coordinates for Off-Manifold Transients near Nonlinear Aeroelastic Flutter](https://mlsanity.com/paper/2609.14674/) (2026)

## Abstract (arXiv)

We develop a methodology to construct low-dimensional predictive models from data sets representing essentially nonlinear (or non-linearizable) dynamical systems with a hyperbolic linear part that are subject to external forcing with finitely many frequencies. Our data-driven, sparse, nonlinear models are obtained as extended normal forms of the reduced dynamics on low-dimensional, attracting spectral submanifolds (SSMs) of the dynamical system. We illustrate the power of data-driven SSM reduction on high-dimensional numerical data sets and experimental measurements involving beam oscillations, vortex shedding and sloshing in a water tank. We find that SSM reduction trained on unforced data also predicts nonlinear response accurately under additional external forcing.