mlsanityData to 7 Oct 2026

Solving time dependent Fokker-Planck equations via temporal normalizing flow

arXiv 2112.14012 ↗First posted 28 Dec 2021cs.LGcited by 17

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Abstract (arXiv)

In this work, we propose an adaptive learning approach based on temporal normalizing flows for solving time-dependent Fokker-Planck (TFP) equations. It is well known that solutions of such equations are probability density functions, and thus our approach relies on modelling the target solutions with the temporal normalizing flows. The temporal normalizing flow is then trained based on the TFP loss function, without requiring any labeled data. Being a machine learning scheme, the proposed approach is mesh-free and can be easily applied to high dimensional problems. We present a variety of test problems to show the effectiveness of the learning approach.

Key facts. Cited by 17 papers in the map (data through 7 Oct 2026). Builds on 27 of its 43 references that are in the map. First posted on arXiv 28 Dec 2021.

Builds on

27 of 43 references.

References · 27

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2013Auto-Encoding Variational Bayescited by 3,980
  3. 2015Variational Inference with Normalizing Flowscited by 2,161
  4. 2016Density estimation using Real NVPcited by 2,029
  5. 2018Glow: Generative Flow with Invertible 1x1 Convolutionscited by 1,539
  6. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,315
  7. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  8. 2016Improving Variational Inference with Inverse Autoregressive Flowcited by 636
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  10. 2017Solving high-dimensional partial differential equations using deep learningcited by 549
  11. 2019Machine Learning for Fluid Mechanicscited by 380
  12. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 329
15 more
  1. 2018WaveGlow: A Flow-based Generative Network for Speech Synthesiscited by 308
  2. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 261
  3. 2017Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanicscited by 213
  4. 2018Neural Importance Samplingcited by 154
  5. 2018Physics-Informed Generative Adversarial Networks for Stochastic Differential Equationscited by 133
  6. 2018Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 120
  7. 2018Discovering physical concepts with neural networkscited by 117
  8. 2020Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networkscited by 52
  9. 2021Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow modelscited by 34
  10. 2020On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEscited by 31
  11. 2021Adaptive deep density approximation for Fokker-Planck equationscited by 28
  12. 2020Machine Learning and Computational Mathematicscited by 22
  13. 2021Learning the temporal evolution of multivariate densities via normalizing flowscited by 18
  14. 2021An augmented Lagrangian deep learning method for variational problems with essential boundary conditionscited by 11
  15. 2019Temporal Normalizing Flowscited by 8

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 2; 2023: 4; 2024: 4; 2025: 3; 2026: 4.

17 papers.

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  1. 2022Failure-informed adaptive sampling for PINNscited by 64
  2. 2024Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck Equationscited by 22
  3. 2023Bounded KRnet and its applications to density estimation and approximationcited by 7
  4. 2022Adaptive deep density approximation for fractional Fokker-Planck equationscited by 4
  5. 2023PI-VEGAN: Physics Informed Variational Embedding Generative Adversarial Networks for Stochastic Differential Equationscited by 4
  6. 2024Adaptive deep density approximation for stochastic dynamical systemscited by 4
  7. 2025FlowKac: An Efficient Neural Fokker-Planck solver using Temporal Normalizing Flows and the Feynman-Kac Formulacited by 3
  8. 2023Generative modeling of time-dependent densities via optimal transport and projection pursuitcited by 2
  9. 2024A hybrid FEM-PINN method for time-dependent partial differential equationscited by 2
  10. 2026Neural Galerkin Normalizing Flow for Transition Probability Density Functions of Diffusion Modelscited by 2
  11. 2026Amortized Filtering and Smoothing with Conditional Normalizing Flowscited by 2
  12. 2023PINF: Continuous Normalizing Flows for Physics-Constrained Deep Learningcited by 1
5 more
  1. 2024Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Levy Equationscited by 1
  2. 2025Integral regularization PINNs for evolution equationscited by 1
  3. 2025Adaptive Probability Flow Residual Minimization for High-Dimensional Fokker-Planck Equationscited by 1
  4. 2026Neural Galerkin Normalizing Flows for Bayesian Inference of Diffusions with Inaccessible Boundariescited by 1
  5. 2026A transition-density-based operator learning method for Fokker-Planck equations with various initial conditionscited by 0