# Flow Map Learning for Unknown Dynamical Systems: Overview, Implementation, and Benchmarks

arXiv: https://arxiv.org/abs/2307.11013 · HTML: https://mlsanity.com/paper/2307.11013/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 26 references that are in the map.
- First posted on arXiv 20 Jul 2023.

## Builds on (most cited first, 14 of 14)

- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [Extracting Sparse High-Dimensional Dynamics from Limited Data](https://mlsanity.com/paper/1707.08528/) (2017)
- [NeuPDE: Neural Network Based Ordinary and Partial Differential Equations for Modeling Time-Dependent Data](https://mlsanity.com/paper/1908.03190/) (2019)
- [Numerical Aspects for Approximating Governing Equations Using Data](https://mlsanity.com/paper/1809.09170/) (2018)
- [Learning reduced systems via deep neural networks with memory](https://mlsanity.com/paper/2003.09451/) (2020)
- [Deep Learning of Chaotic Systems from Partially-Observed Data](https://mlsanity.com/paper/2205.08384/) (2022)
- [Robust Modeling of Unknown Dynamical Systems via Ensemble Averaged Learning](https://mlsanity.com/paper/2203.03458/) (2022)

## Built on by (most cited first, 18 of 18)

- [Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learning](https://mlsanity.com/paper/2405.09285/) (2024)
- [DUE: A Deep Learning Framework and Library for Modeling Unknown Equations](https://mlsanity.com/paper/2504.10373/) (2025)
- [Modeling Unknown Stochastic Dynamical System Subject to External Excitation](https://mlsanity.com/paper/2406.15747/) (2024)
- [Data-driven Effective Modeling of Multiscale Stochastic Dynamical Systems](https://mlsanity.com/paper/2408.14821/) (2024)
- [Targeted Digital Twin via Flow Map Learning and Its Application to Fluid Dynamics](https://mlsanity.com/paper/2510.07549/) (2025)
- [Neural Entropy-stable conservative flux form neural networks for learning hyperbolic conservation laws](https://mlsanity.com/paper/2507.01795/) (2025)
- [Moment Estimate and Variational Approach for Learning Generalized Diffusion with Non-gradient Structures](https://mlsanity.com/paper/2508.01854/) (2025)
- [Learning Gradient Flow: Using Equation Discovery to Accelerate Engineering Optimization](https://mlsanity.com/paper/2602.13513/) (2026)
- [Entropy stable conservative flux form neural networks](https://mlsanity.com/paper/2411.01746/) (2024)
- [HiPreNets: High-Precision Neural Networks through Progressive Training](https://mlsanity.com/paper/2506.15064/) (2025)
- [Data-Driven Modeling and Correction of Vehicle Dynamics](https://mlsanity.com/paper/2512.00289/) (2025)
- [Enabling Real-Time Training of a Wildfire-to-Smoke Map with Multilinear Operators](https://mlsanity.com/paper/2605.04164/) (2026)
- [Flow map learning in nonlinear vector autoregressive models: influence of the feature-library structure on the training error](https://mlsanity.com/paper/2605.31438/) (2026)
- [Blind Recovery of Latent Domains via Unsupervised Symmetry Discovery](https://mlsanity.com/paper/2606.17782/) (2026)
- [Modeling Unknown Nonlocal PDE Systems via Flow Map Learning](https://mlsanity.com/paper/2608.00400/) (2026)
- [Flux-form spatiotemporal neural operators for coarse-grained dynamics of multiscale PDEs](https://mlsanity.com/paper/2608.18148/) (2026)
- [Physics-Constrained Neural Flow Maps for Long-Horizon Prediction of Spin Dynamics](https://mlsanity.com/paper/2608.22006/) (2026)
- [Data-driven Effective Modeling of Stochastic Chemical Reaction Networks](https://mlsanity.com/paper/2608.25421/) (2026)

## Abstract (arXiv)

Flow map learning (FML), in conjunction with deep neural networks (DNNs), has shown promises for data driven modeling of unknown dynamical systems. A remarkable feature of FML is that it is capable of producing accurate predictive models for partially observed systems, even when their exact mathematical models do not exist. In this paper, we present an overview of the FML framework, along with the important computational details for its successful implementation. We also present a set of well defined benchmark problems for learning unknown dynamical systems. All the numerical details of these problems are presented, along with their FML results, to ensure that the problems are accessible for cross-examination and the results are reproducible.