mlsanityData to 7 Oct 2026

Adaptive deep density approximation for Fokker-Planck equations

arXiv 2103.11181 ↗First posted 20 Mar 2021stat.ML cs.LGcited by 28

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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.

Key facts. Cited by 28 papers in the map (data through 7 Oct 2026). Builds on 26 of its 58 references that are in the map. First posted on arXiv 20 Mar 2021.

Builds on

26 of 58 references.

References · 26

  1. 2015Deep Residual Learning for Image Recognitioncited by 51,809
  2. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  3. 2015Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shiftcited by 6,979
  4. 2013Auto-Encoding Variational Bayescited by 3,980
  5. 2018Neural Ordinary Differential Equationscited by 3,733
  6. 2016Density estimation using Real NVPcited by 2,029
  7. 2018Glow: Generative Flow with Invertible 1x1 Convolutionscited by 1,539
  8. 2016Optimization Methods for Large-Scale Machine Learningcited by 1,140
  9. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  10. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  11. 2020A Survey of Deep Active Learningcited by 547
  12. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 362
14 more
  1. 2017The Expressive Power of Neural Networks: A View from the Widthcited by 318
  2. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 261
  3. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
  4. 2018Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantificationcited by 199
  5. 2020PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domaincited by 173
  6. 2017Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equationscited by 158
  7. 2020Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equationscited by 78
  8. 2020Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networkscited by 52
  9. 2019D3M: A deep domain decomposition method for partial differential equationscited by 44
  10. 2020PFNN: A Penalty-Free Neural Network Method for Solving a Class of Second-Order Boundary-Value Problems on Complex Geometriescited by 38
  11. 2018Monge-Amp`ere Flow for Generative Modelingcited by 33
  12. 2018Numerical Aspects for Approximating Governing Equations Using Datacited by 21
  13. 2019Structure-preserving Method for Reconstructing Unknown Hamiltonian Systems from Trajectory Datacited by 18
  14. 2019Coupling the reduced-order model and the generative model for an importance sampling estimatorcited by 3

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Citing papers per year: 2021: 4; 2022: 2; 2023: 5; 2024: 8; 2025: 5; 2026: 4.

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  1. 2021DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equationscited by 66
  2. 2021On Computing the Hyperparameter of Extreme Learning Machines: Algorithm and Application to Computational PDEs, and Comparison with Classical and High-Order Finite Elementscited by 25
  3. 2024Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck Equationscited by 22
  4. 2023Adversarial Adaptive Sampling: Unify PINN and Optimal Transport for the Approximation of PDEscited by 18
  5. 2021Solving time dependent Fokker-Planck equations via temporal normalizing flowcited by 17
  6. 2022Numerical Computation of Partial Differential Equations by Hidden-Layer Concatenated Extreme Learning Machinecited by 9
  7. 2023Bounded KRnet and its applications to density estimation and approximationcited by 7
  8. 2024Deep adaptive sampling for surrogate modeling without labeled datacited by 5
  9. 2021Augmented KRnet for density estimation and approximationcited by 4
  10. 2022Adaptive deep density approximation for fractional Fokker-Planck equationscited by 4
  11. 2024Adaptive deep density approximation for stochastic dynamical systemscited by 4
  12. 2024Weak Generative Sampler to Efficiently Sample Invariant Distribution of Stochastic Differential Equationcited by 4
16 more
  1. 2025Estimating Committor Functions via Deep Adaptive Sampling on Rare Transition Pathscited by 3
  2. 2025FlowKac: An Efficient Neural Fokker-Planck solver using Temporal Normalizing Flows and the Feynman-Kac Formulacited by 3
  3. 2024A Natural Primal-Dual Hybrid Gradient Method for Adversarial Neural Network Training on Solving Partial Differential Equationscited by 2
  4. 2023Dimension-reduced KRnet maps for high-dimensional Bayesian inverse problemscited by 1
  5. 2023PINF: Continuous Normalizing Flows for Physics-Constrained Deep Learningcited by 1
  6. 2024Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Levy Equationscited by 1
  7. 2024PinnDE: Physics-Informed Neural Networks for Solving Differential Equationscited by 1
  8. 2025Adaptive Probability Flow Residual Minimization for High-Dimensional Fokker-Planck Equationscited by 1
  9. 2026Generative Path-Finding Method for Wasserstein Gradient Flowcited by 1
  10. 2026Neural Galerkin Normalizing Flows for Bayesian Inference of Diffusions with Inaccessible Boundariescited by 1
  11. 2023Consensus-based adaptive sampling and approximation for high-dimensional energy landscapescited by 0
  12. 2024DeepSPoC: A Deep Learning Based Sequential Propagation of Chaoscited by 0
  13. 2025A Discrete Neural Operator with Adaptive Sampling for Surrogate Modeling of Parametric Transient Darcy Flows in Porous Mediacited by 0
  14. 2025Adaptive Sampling for Hydrodynamic Stabilitycited by 0
  15. 2026Deep Adaptive Dimension Reduction for Bayesian Inference in Inverse Problemscited by 0
  16. 2026A transition-density-based operator learning method for Fokker-Planck equations with various initial conditionscited by 0