# Solving time dependent Fokker-Planck equations via temporal normalizing flow

arXiv: https://arxiv.org/abs/2112.14012 · HTML: https://mlsanity.com/paper/2112.14012/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 43 references that are in the map.
- First posted on arXiv 28 Dec 2021.

## Builds on (most cited first, 27 of 27)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [WaveGlow: A Flow-based Generative Network for Speech Synthesis](https://mlsanity.com/paper/1811.00002/) (2018)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics](https://mlsanity.com/paper/1707.09571/) (2017)
- [Neural Importance Sampling](https://mlsanity.com/paper/1808.03856/) (2018)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Discovering physical concepts with neural networks](https://mlsanity.com/paper/1807.10300/) (2018)
- [Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks](https://mlsanity.com/paper/2008.10653/) (2020)
- [Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow models](https://mlsanity.com/paper/2108.12956/) (2021)
- [On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs](https://mlsanity.com/paper/2004.01806/) (2020)
- [Adaptive deep density approximation for Fokker-Planck equations](https://mlsanity.com/paper/2103.11181/) (2021)
- [Machine Learning and Computational Mathematics](https://mlsanity.com/paper/2009.14596/) (2020)
- [Learning the temporal evolution of multivariate densities via normalizing flows](https://mlsanity.com/paper/2107.13735/) (2021)
- [An augmented Lagrangian deep learning method for variational problems with essential boundary conditions](https://mlsanity.com/paper/2106.14348/) (2021)
- [Temporal Normalizing Flows](https://mlsanity.com/paper/1912.09092/) (2019)

## Built on by (most cited first, 17 of 17)

- [Failure-informed adaptive sampling for PINNs](https://mlsanity.com/paper/2210.00279/) (2022)
- [Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck Equations](https://mlsanity.com/paper/2402.07465/) (2024)
- [Bounded KRnet and its applications to density estimation and approximation](https://mlsanity.com/paper/2305.09063/) (2023)
- [Adaptive deep density approximation for fractional Fokker-Planck equations](https://mlsanity.com/paper/2210.14402/) (2022)
- [PI-VEGAN: Physics Informed Variational Embedding Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/2307.11289/) (2023)
- [Adaptive deep density approximation for stochastic dynamical systems](https://mlsanity.com/paper/2405.02810/) (2024)
- [FlowKac: An Efficient Neural Fokker-Planck solver using Temporal Normalizing Flows and the Feynman-Kac Formula](https://mlsanity.com/paper/2503.11427/) (2025)
- [Generative modeling of time-dependent densities via optimal transport and projection pursuit](https://mlsanity.com/paper/2304.09663/) (2023)
- [A hybrid FEM-PINN method for time-dependent partial differential equations](https://mlsanity.com/paper/2409.02810/) (2024)
- [Neural Galerkin Normalizing Flow for Transition Probability Density Functions of Diffusion Models](https://mlsanity.com/paper/2603.18907/) (2026)
- [Amortized Filtering and Smoothing with Conditional Normalizing Flows](https://mlsanity.com/paper/2604.07169/) (2026)
- [PINF: Continuous Normalizing Flows for Physics-Constrained Deep Learning](https://mlsanity.com/paper/2309.15139/) (2023)
- [Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Levy Equations](https://mlsanity.com/paper/2406.11676/) (2024)
- [Integral regularization PINNs for evolution equations](https://mlsanity.com/paper/2503.23729/) (2025)
- [Adaptive Probability Flow Residual Minimization for High-Dimensional Fokker-Planck Equations](https://mlsanity.com/paper/2512.19196/) (2025)
- [Neural Galerkin Normalizing Flows for Bayesian Inference of Diffusions with Inaccessible Boundaries](https://mlsanity.com/paper/2606.04324/) (2026)
- [A transition-density-based operator learning method for Fokker-Planck equations with various initial conditions](https://mlsanity.com/paper/2606.09434/) (2026)

## 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.