mlsanityData to 14 Oct 2026

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

arXiv 2307.11013 ↗First posted 20 Jul 2023cs.LG math.DS stat.MLcited by 18

Not on the map. Why?

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.

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

14 of 26 references.

References · 14

  1. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 373
  2. 2017PDE-Net: Learning PDEs from Datacited by 311
  3. 2018Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equationscited by 235
  4. 2018PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Networkcited by 210
  5. 2017Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equationscited by 161
  6. 2017Machine Learning of Linear Differential Equations using Gaussian Processescited by 140
  7. 2018Data Driven Governing Equations Approximation Using Deep Neural Networkscited by 96
  8. 2018Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systemscited by 89
  9. 2017Extracting Sparse High-Dimensional Dynamics from Limited Datacited by 50
  10. 2019NeuPDE: Neural Network Based Ordinary and Partial Differential Equations for Modeling Time-Dependent Datacited by 42
  11. 2018Numerical Aspects for Approximating Governing Equations Using Datacited by 21
  12. 2020Learning reduced systems via deep neural networks with memorycited by 20
2 more
  1. 2022Deep Learning of Chaotic Systems from Partially-Observed Datacited by 9
  2. 2022Robust Modeling of Unknown Dynamical Systems via Ensemble Averaged Learningcited by 5

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 4; 2025: 6; 2026: 8.

18 papers.

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  1. 2024Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learningcited by 17
  2. 2025DUE: A Deep Learning Framework and Library for Modeling Unknown Equationscited by 8
  3. 2024Modeling Unknown Stochastic Dynamical System Subject to External Excitationcited by 7
  4. 2024Data-driven Effective Modeling of Multiscale Stochastic Dynamical Systemscited by 5
  5. 2025Targeted Digital Twin via Flow Map Learning and Its Application to Fluid Dynamicscited by 5
  6. 2025Neural Entropy-stable conservative flux form neural networks for learning hyperbolic conservation lawscited by 3
  7. 2025Moment Estimate and Variational Approach for Learning Generalized Diffusion with Non-gradient Structurescited by 1
  8. 2026Learning Gradient Flow: Using Equation Discovery to Accelerate Engineering Optimizationcited by 1
  9. 2024Entropy stable conservative flux form neural networkscited by 0
  10. 2025HiPreNets: High-Precision Neural Networks through Progressive Trainingcited by 0
  11. 2025Data-Driven Modeling and Correction of Vehicle Dynamicscited by 0
  12. 2026Enabling Real-Time Training of a Wildfire-to-Smoke Map with Multilinear Operatorscited by 0
6 more
  1. 2026Flow map learning in nonlinear vector autoregressive models: influence of the feature-library structure on the training errorcited by 0
  2. 2026Blind Recovery of Latent Domains via Unsupervised Symmetry Discoverycited by 0
  3. 2026Modeling Unknown Nonlocal PDE Systems via Flow Map Learningcited by 0
  4. 2026Flux-form spatiotemporal neural operators for coarse-grained dynamics of multiscale PDEscited by 0
  5. 2026Physics-Constrained Neural Flow Maps for Long-Horizon Prediction of Spin Dynamicscited by 0
  6. 2026Data-driven Effective Modeling of Stochastic Chemical Reaction Networkscited by 0