# Adaptive deep density approximation for Fokker-Planck equations

arXiv: https://arxiv.org/abs/2103.11181 · HTML: https://mlsanity.com/paper/2103.11181/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 58 references that are in the map.
- First posted on arXiv 20 Mar 2021.

## Builds on (most cited first, 26 of 26)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
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- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [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)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [A Survey of Deep Active Learning](https://mlsanity.com/paper/2009.00236/) (2020)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2012.02895/) (2020)
- [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)
- [D3M: A deep domain decomposition method for partial differential equations](https://mlsanity.com/paper/1909.12236/) (2019)
- [PFNN: A Penalty-Free Neural Network Method for Solving a Class of Second-Order Boundary-Value Problems on Complex Geometries](https://mlsanity.com/paper/2004.06490/) (2020)
- [Monge-Amp`ere Flow for Generative Modeling](https://mlsanity.com/paper/1809.10188/) (2018)
- [Numerical Aspects for Approximating Governing Equations Using Data](https://mlsanity.com/paper/1809.09170/) (2018)
- [Structure-preserving Method for Reconstructing Unknown Hamiltonian Systems from Trajectory Data](https://mlsanity.com/paper/1905.10396/) (2019)
- [Coupling the reduced-order model and the generative model for an importance sampling estimator](https://mlsanity.com/paper/1901.07977/) (2019)

## Built on by (most cited first, 28 of 28)

- [DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2112.14038/) (2021)
- [On Computing the Hyperparameter of Extreme Learning Machines: Algorithm and Application to Computational PDEs, and Comparison with Classical and High-Order Finite Elements](https://mlsanity.com/paper/2110.14121/) (2021)
- [Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck Equations](https://mlsanity.com/paper/2402.07465/) (2024)
- [Adversarial Adaptive Sampling: Unify PINN and Optimal Transport for the Approximation of PDEs](https://mlsanity.com/paper/2305.18702/) (2023)
- [Solving time dependent Fokker-Planck equations via temporal normalizing flow](https://mlsanity.com/paper/2112.14012/) (2021)
- [Numerical Computation of Partial Differential Equations by Hidden-Layer Concatenated Extreme Learning Machine](https://mlsanity.com/paper/2204.11375/) (2022)
- [Bounded KRnet and its applications to density estimation and approximation](https://mlsanity.com/paper/2305.09063/) (2023)
- [Deep adaptive sampling for surrogate modeling without labeled data](https://mlsanity.com/paper/2402.11283/) (2024)
- [Augmented KRnet for density estimation and approximation](https://mlsanity.com/paper/2105.12866/) (2021)
- [Adaptive deep density approximation for fractional Fokker-Planck equations](https://mlsanity.com/paper/2210.14402/) (2022)
- [Adaptive deep density approximation for stochastic dynamical systems](https://mlsanity.com/paper/2405.02810/) (2024)
- [Weak Generative Sampler to Efficiently Sample Invariant Distribution of Stochastic Differential Equation](https://mlsanity.com/paper/2405.19256/) (2024)
- [Estimating Committor Functions via Deep Adaptive Sampling on Rare Transition Paths](https://mlsanity.com/paper/2501.15522/) (2025)
- [FlowKac: An Efficient Neural Fokker-Planck solver using Temporal Normalizing Flows and the Feynman-Kac Formula](https://mlsanity.com/paper/2503.11427/) (2025)
- [A Natural Primal-Dual Hybrid Gradient Method for Adversarial Neural Network Training on Solving Partial Differential Equations](https://mlsanity.com/paper/2411.06278/) (2024)
- [Generative Path-Finding Method for Wasserstein Gradient Flow](https://mlsanity.com/paper/2604.11519/) (2026)
- [Dimension-reduced KRnet maps for high-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2303.00573/) (2023)
- [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)
- [PinnDE: Physics-Informed Neural Networks for Solving Differential Equations](https://mlsanity.com/paper/2408.10011/) (2024)
- [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)
- [Consensus-based adaptive sampling and approximation for high-dimensional energy landscapes](https://mlsanity.com/paper/2311.05009/) (2023)
- [DeepSPoC: A Deep Learning Based Sequential Propagation of Chaos](https://mlsanity.com/paper/2408.16403/) (2024)
- [A Discrete Neural Operator with Adaptive Sampling for Surrogate Modeling of Parametric Transient Darcy Flows in Porous Media](https://mlsanity.com/paper/2512.03113/) (2025)
- [Adaptive Sampling for Hydrodynamic Stability](https://mlsanity.com/paper/2512.13532/) (2025)
- [Deep Adaptive Dimension Reduction for Bayesian Inference in Inverse Problems](https://mlsanity.com/paper/2605.29373/) (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 paper we present an adaptive deep density approximation strategy based on KRnet (ADDA-KR) for solving the steady-state Fokker-Planck (F-P) equations. F-P equations are usually high-dimensional and defined on an unbounded domain, which limits the application of traditional grid based numerical methods. With the Knothe-Rosenblatt rearrangement, our newly proposed flow-based generative model, called KRnet, provides a family of probability density functions to serve as effective solution candidates for the Fokker-Planck equations, which has a weaker dependence on dimensionality than traditional computational approaches and can efficiently estimate general high-dimensional density functions. To obtain effective stochastic collocation points for the approximation of the F-P equation, we develop an adaptive sampling procedure, where samples are generated iteratively using the approximate density function at each iteration. We present a general framework of ADDA-KR, validate its accuracy and demonstrate its efficiency with numerical experiments.