mlsanityData to 14 Oct 2026

Learning the temporal evolution of multivariate densities via normalizing flows

arXiv 2107.13735 ↗First posted 29 Jul 2021stat.ML cs.LG math.DScited by 18

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Abstract (arXiv)

In this work, we propose a method to learn multivariate probability distributions using sample path data from stochastic differential equations. Specifically, we consider temporally evolving probability distributions (e.g., those produced by integrating local or nonlocal Fokker-Planck equations). We analyze this evolution through machine learning assisted construction of a time-dependent mapping that takes a reference distribution (say, a Gaussian) to each and every instance of our evolving distribution. If the reference distribution is the initial condition of a Fokker-Planck equation, what we learn is the time-T map of the corresponding solution. Specifically, the learned map is a multivariate normalizing flow that deforms the support of the reference density to the support of each and every density snapshot in time. We demonstrate that this approach can approximate probability density function evolutions in time from observed sampled data for systems driven by both Brownian and L'evy noise. We present examples with two- and three-dimensional, uni- and multimodal distributions to validate the method.

Key facts. Cited by 18 papers in the map (data through 14 Oct 2026). Builds on 24 of its 65 references that are in the map. First posted on arXiv 29 Jul 2021.

Builds on

24 of 65 references.

References · 24

  1. 2018Neural Ordinary Differential Equationscited by 3,779
  2. 2015Variational Inference with Normalizing Flowscited by 2,170
  3. 2016Density estimation using Real NVPcited by 2,042
  4. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,323
  5. 2017Masked Autoregressive Flow for Density Estimationcited by 684
  6. 2016Improving Variational Inference with Inverse Autoregressive Flowcited by 640
  7. 2019Neural Spline Flowscited by 596
  8. 2020Scalable Gradients for Stochastic Differential Equationscited by 271
  9. 2017Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equationscited by 191
  10. 2019Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limitcited by 176
  11. 2020Stochastic Normalizing Flowscited by 157
  12. 2019Neural Jump Stochastic Differential Equationscited by 150
12 more
  1. 2018Physics-Informed Generative Adversarial Networks for Stochastic Differential Equationscited by 135
  2. 2019Data-driven approximation of the Koopman generator: Model reduction, system identification, and controlcited by 85
  3. 2020Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networkscited by 52
  4. 2019Coarse-scale PDEs from fine-scale observations via machine learningcited by 35
  5. 2020A Data-Driven Approach for Discovering Stochastic Dynamical Systems with Non-Gaussian Levy Noisecited by 15
  6. 2017Non-parametric Estimation of Stochastic Differential Equations with Sparse Gaussian Processescited by 12
  7. 2019Towards Robust and Stable Deep Learning Algorithms for Forward Backward Stochastic Differential Equationscited by 8
  8. 2019Temporal Normalizing Flowscited by 8
  9. 2020Generative Ensemble Regression: Learning Particle Dynamics from Observations of Ensembles with Physics-Informed Deep Generative Modelscited by 6
  10. 2016Vector-Valued Optimal Mass Transportcited by 3
  11. 2016Bayesian Optical Flow with Uncertainty Quantificationcited by 2
  12. 2019Optimal Transportation of Vector-Valued Measurescited by 1

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 3; 2022: 1; 2023: 2; 2024: 7; 2025: 2; 2026: 3.

18 papers.

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  1. 2024Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck Equationscited by 22
  2. 2022Parsimonious Physics-Informed Random Projection Neural Networks for Initial-Value Problems of ODEs and index-1 DAEscited by 19
  3. 2021Solving time dependent Fokker-Planck equations via temporal normalizing flowcited by 17
  4. 2024RandONet: Shallow-Networks with Random Projections for learning linear and nonlinear operatorscited by 16
  5. 2021Extracting stochastic dynamical systems with α-stable L'evy noise from datacited by 8
  6. 2024Modeling Unknown Stochastic Dynamical System Subject to External Excitationcited by 7
  7. 2021Extracting Stochastic Governing Laws by Nonlocal Kramers-Moyal Formulascited by 6
  8. 2023Generative modeling of time-dependent densities via optimal transport and projection pursuitcited by 2
  9. 2023Reservoir Computing with Error Correction: Long-term Behaviors of Stochastic Dynamical Systemscited by 2
  10. 2024MD-NOMAD: Mixture density nonlinear manifold decoder for emulating stochastic differential equations and uncertainty propagationcited by 2
  11. 2026Neural Galerkin Normalizing Flow for Transition Probability Density Functions of Diffusion Modelscited by 2
  12. 2024Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Levy Equationscited by 1
6 more
  1. 2024Flow to Rare Events: An Application of Normalizing Flow in Temporal Importance Sampling for Automated Vehicle Validationcited by 1
  2. 2024Least Squares and Marginal Log-Likelihood Model Predictive Control using Normalizing Flowscited by 1
  3. 2025Integral regularization PINNs for evolution equationscited by 1
  4. 2026Neural Galerkin Normalizing Flows for Bayesian Inference of Diffusions with Inaccessible Boundariescited by 1
  5. 2025Time-dependent density estimation using binary classifierscited by 0
  6. 2026A transition-density-based operator learning method for Fokker-Planck equations with various initial conditionscited by 0