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

arXiv: https://arxiv.org/abs/2605.31438 · HTML: https://mlsanity.com/paper/2605.31438/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 45 of its 110 references that are in the map.
- First posted on arXiv 29 May 2026.

## Builds on (most cited first, 30 of 45)

- [Scheduled Sampling for Sequence Prediction with Recurrent Neural Networks](https://mlsanity.com/paper/1506.03099/) (2015)
- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Next Generation Reservoir Computing](https://mlsanity.com/paper/2106.07688/) (2021)
- [Generalization in Generation: A closer look at Exposure Bias](https://mlsanity.com/paper/1910.00292/) (2019)
- [Discovering physical concepts with neural networks](https://mlsanity.com/paper/1807.10300/) (2018)
- [On Linear Identifiability of Learned Representations](https://mlsanity.com/paper/2007.00810/) (2020)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [How (not) to Train your Generative Model: Scheduled Sampling, Likelihood, Adversary?](https://mlsanity.com/paper/1511.05101/) (2015)
- [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017)
- [Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2205.14027/) (2022)
- [Generalized Teacher Forcing for Learning Chaotic Dynamics](https://mlsanity.com/paper/2306.04406/) (2023)
- [From Fourier to Koopman: Spectral Methods for Long-term Time Series Prediction](https://mlsanity.com/paper/2004.00574/) (2020)
- [Koopman Operator Dynamical Models: Learning, Analysis and Control](https://mlsanity.com/paper/2102.02522/) (2021)
- [Model scale versus domain knowledge in statistical forecasting of chaotic systems](https://mlsanity.com/paper/2303.08011/) (2023)
- [Finite-data error bounds for Koopman-based prediction and control](https://mlsanity.com/paper/2108.07102/) (2021)
- [Interpretable Polynomial Neural Ordinary Differential Equations](https://mlsanity.com/paper/2208.05072/) (2022)
- [Learning Spatiotemporal Chaos Using Next-Generation Reservoir Computing](https://mlsanity.com/paper/2203.13294/) (2022)
- [Koopman-based lifting techniques for nonlinear systems identification](https://mlsanity.com/paper/1709.02003/) (2017)
- [Flow Map Learning for Unknown Dynamical Systems: Overview, Implementation, and Benchmarks](https://mlsanity.com/paper/2307.11013/) (2023)
- [The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimation](https://mlsanity.com/paper/2103.11488/) (2021)
- [Koopman Operators in Robot Learning](https://mlsanity.com/paper/2408.04200/) (2024)
- [Convergence of weak-SINDy Surrogate Models](https://mlsanity.com/paper/2209.15573/) (2022)
- [Catch-22s of reservoir computing](https://mlsanity.com/paper/2210.10211/) (2022)
- [Deep-OSG: Deep Learning of Operators in Semigroup](https://mlsanity.com/paper/2302.03358/) (2023)
- [Extrapolating tipping points and simulating non-stationary dynamics of complex systems using efficient machine learning](https://mlsanity.com/paper/2312.06283/) (2023)
- [Identifiability and Asymptotics in Learning Homogeneous Linear ODE Systems from Discrete Observations](https://mlsanity.com/paper/2210.05955/) (2022)
- [Exploring the limits of multifunctionality across different reservoir computers](https://mlsanity.com/paper/2205.11375/) (2022)
- [Surrogate modeling with functional nonlinear autoregressive models (F-NARX)](https://mlsanity.com/paper/2410.07293/) (2024)

## Abstract (arXiv)

Time series forecasting often requires learning nonlinear and time-delayed dependencies. A paradigmatic class of forecasting models are nonlinear vector autoregressive processes (NVAR), also known as next-generation reservoir computers (NG-RCs). These models approximate the Koopman operator on the space spanned by their explicit feature library. We consider the identifiability problem for learning Markovian nonlinear dynamical systems and show that the training error as a function of time resolution follows characteristic (pre-)asymptotic scaling laws. These laws depend on whether the feature library can represent the early Lie-series coefficients of the flow map (propagator) exactly or merely approximately. For dynamical systems governed by polynomial vector fields, we demonstrate the mechanism for NVAR/NG-RC models with monomial and Fourier feature libraries. We determine the dependence of the training error on the temporal resolution, the involved nonlinear degree, and the number of delay terms. While delay terms reduce the optimal one-step training error, they improve long-horizon forecasts only when the library provides sufficient nonlinearity. Thus, small training error coexists with weak generalization as the model class is mismatched to the true data-generating process. Numerical experiments on various chaotic dynamical systems confirm the theoretical predictions.