mlsanityData to 14 Oct 2026

Learning Stochastic Dynamical System via Flow Map Operator

arXiv 2305.03874 ↗First posted 5 May 2023cs.LG cs.AI cs.NAcited by 24

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Abstract (arXiv)

We present a numerical framework for learning unknown stochastic dynamical systems using measurement data. Termed stochastic flow map learning (sFML), the new framework is an extension of flow map learning (FML) that was developed for learning deterministic dynamical systems. For learning stochastic systems, we define a stochastic flow map that is a superposition of two sub-flow maps: a deterministic sub-map and a stochastic sub-map. The stochastic training data are used to construct the deterministic sub-map first, followed by the stochastic sub-map. The deterministic sub-map takes the form of residual network (ResNet), similar to the work of FML for deterministic systems. For the stochastic sub-map, we employ a generative model, particularly generative adversarial networks (GANs) in this paper. The final constructed stochastic flow map then defines a stochastic evolution model that is a weak approximation, in term of distribution, of the unknown stochastic system. A comprehensive set of numerical examples are presented to demonstrate the flexibility and effectiveness of the proposed sFML method for various types of stochastic systems.

Key facts. Cited by 24 papers in the map (data through 14 Oct 2026). Builds on 29 of its 48 references that are in the map. First posted on arXiv 5 May 2023.

Builds on

29 of 48 references.

References · 29

  1. 2022Generative Adversarial Networkscited by 5,627
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  3. 2017Improved Training of Wasserstein GANscited by 2,235
  4. 2017High-Resolution Image Synthesis and Semantic Manipulation with Conditional GANscited by 1,179
  5. 2016Generative Adversarial Text to Image Synthesiscited by 735
  6. 2016StackGAN: Text to Photo-realistic Image Synthesis with Stacked Generative Adversarial Networkscited by 577
  7. 2018PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Networkcited by 210
  8. 2018Physics-Informed Generative Adversarial Networks for Stochastic Differential Equationscited by 135
  9. 2017BEGAN: Boundary Equilibrium Generative Adversarial Networkscited by 121
  10. 2018Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 120
  11. 2018Data Driven Governing Equations Approximation Using Deep Neural Networkscited by 96
  12. 2018Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systemscited by 89
17 more
  1. 2018GANs for Medical Image Analysiscited by 81
  2. 2017Sparse learning of stochastic dynamic equationscited by 76
  3. 2020Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networkscited by 52
  4. 2017Extracting Sparse High-Dimensional Dynamics from Limited Datacited by 50
  5. 2019Data-Driven Deep Learning of Partial Differential Equations in Modal Spacecited by 46
  6. 2021Learning effective stochastic differential equations from microscopic simulations: linking stochastic numerics to deep learningcited by 39
  7. 2021Computational Graph Completioncited by 28
  8. 2021Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Spacecited by 25
  9. 2018Learning Stochastic Differential Equations With Gaussian Processes Without Gradient Matchingcited by 23
  10. 2018Numerical Aspects for Approximating Governing Equations Using Datacited by 21
  11. 2020Learning reduced systems via deep neural networks with memorycited by 20
  12. 2022Learning Stochastic Dynamics with Statistics-Informed Neural Networkcited by 16
  13. 2020A Data-Driven Approach for Discovering Stochastic Dynamical Systems with Non-Gaussian Levy Noisecited by 15
  14. 2022One-Shot Learning of Stochastic Differential Equations with Data Adapted Kernelscited by 15
  15. 2021Learning stochastic dynamical systems with neural networks mimicking the Euler-Maruyama schemecited by 10
  16. 2021Stationary Density Estimation of Ito Diffusions Using Deep Learningcited by 9
  17. 2020RODE-Net: Learning Ordinary Differential Equations with Randomness from Datacited by 6

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 1; 2024: 7; 2025: 9; 2026: 7.

24 papers.

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  1. 2024A Training-Free Conditional Diffusion Model for Learning Stochastic Dynamical Systemscited by 19
  2. 2023Finite Expression Methods for Discovering Physical Laws from Datacited by 9
  3. 2024Parametric model reduction of mean-field and stochastic systems via higher-order action matchingcited by 9
  4. 2024Neural Operators for Predictor Feedback Control of Nonlinear Delay Systemscited by 9
  5. 2024DynGMA: a robust approach for learning stochastic differential equations from datacited by 8
  6. 2025DUE: A Deep Learning Framework and Library for Modeling Unknown Equationscited by 8
  7. 2024Modeling Unknown Stochastic Dynamical System Subject to External Excitationcited by 7
  8. 2025Bifidelity Parameter Estimation Using Conditional Diffusion Modelscited by 7
  9. 2024An efficient Wasserstein-distance approach for reconstructing jump-diffusion processes using parameterized neural networkscited by 5
  10. 2024Data-driven Effective Modeling of Multiscale Stochastic Dynamical Systemscited by 5
  11. 2025Targeted Digital Twin via Flow Map Learning and Its Application to Fluid Dynamicscited by 5
  12. 2025Generative AI Models for Learning Flow Maps of Stochastic Dynamical Systems in Bounded Domainscited by 3
12 more
  1. 2025Learning Stochastic Hamiltonian Systems via Stochastic Generating Function Neural Networkcited by 1
  2. 2025Moment Estimate and Variational Approach for Learning Generalized Diffusion with Non-gradient Structurescited by 1
  3. 2026Training-free score-based diffusion for parameter-dependent stochastic dynamical systemscited by 1
  4. 2026MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Datacited by 1
  5. 2025A Robust Model-Based Approach for Continuous-Time Policy Evaluation with Unknown L'evy Process Dynamicscited by 0
  6. 2025Identifying Unknown Stochastic Dynamics via Finite expression methodscited by 0
  7. 2025PIP2 Net: Physics-informed Partition Penalty Deep Operator Networkcited by 0
  8. 2026Modeling Unknown Nonlocal PDE Systems via Flow Map Learningcited by 0
  9. 2026Learning Deterministic and Stochastic Forced Hamiltonian Systemscited by 0
  10. 2026Data-driven Effective Modeling of Stochastic Chemical Reaction Networkscited by 0
  11. 2026Memory-Conditioned Diffusion Model for Generalized Langevin Dynamicscited by 0
  12. 2026Generative Modeling of Stochastic Dynamics for Long-Time Evolutioncited by 0

Similar, not cited

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  1. 2024Flow map matching with stochastic interpolants: A mathematical framework for consistency models
  2. 2026Strong Stochastic Flow Maps
  3. 2024Stochastic Sampling from Deterministic Flow Models
  4. 2026Flow-Map GRPO: Reinforcement Learning for Few-Step Flow-Map Generators via Anchored Stochastic Composition
  5. 2025How to build a consistency model: Learning flow maps via self-distillation
  6. 2026Flow Map Learning via Nongradient Vector Flow
  7. 2025STFlow: Data-Coupled Flow Matching for Geometric Trajectory Simulation
  8. 2024Schr"odinger Bridge Flow for Unpaired Data Translation