mlsanityData to 7 Oct 2026

Adaptive deep density approximation for stochastic dynamical systems

arXiv 2405.02810 ↗First posted 5 May 2024math.NA cs.NA stat.MLcited by 4

Not on the map. Why?

Abstract (arXiv)

In this paper we consider adaptive deep neural network approximation for stochastic dynamical systems. Based on the Liouville equation associated with the stochastic dynamical systems, a new temporal KRnet (tKRnet) is proposed to approximate the probability density functions (PDFs) of the state variables. The tKRnet gives an explicit density model for the solution of the Liouville equation, which alleviates the curse of dimensionality issue that limits the application of traditional grid based numerical methods. To efficiently train the tKRnet, an adaptive procedure is developed to generate collocation points for the corresponding residual loss function, where samples are generated iteratively using the approximate density function at each iteration. A temporal decomposition technique is also employed to improve the long-time integration. Theoretical analysis of our proposed method is provided, and numerical examples are presented to demonstrate its performance.

Key facts. Cited by 4 papers in the map (data through 7 Oct 2026). Builds on 29 of its 65 references that are in the map. First posted on arXiv 5 May 2024.

Builds on

29 of 65 references.

References · 29

  1. 2016SGDR: Stochastic Gradient Descent with Warm Restartscited by 4,993
  2. 2018Neural Ordinary Differential Equationscited by 3,733
  3. 2015Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classificationcited by 3,590
  4. 2016Gaussian Error Linear Units (GELUs)cited by 3,431
  5. 2016Density estimation using Real NVPcited by 2,029
  6. 2018Glow: Generative Flow with Invertible 1x1 Convolutionscited by 1,539
  7. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,315
  8. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  9. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  10. 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 279
  11. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 261
  12. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
17 more
  1. 2018Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantificationcited by 199
  2. 2020PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domaincited by 173
  3. 2019PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEscited by 116
  4. 2021Long-time integration of parametric evolution equations with physics-informed DeepONetscited by 87
  5. 2020Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equationscited by 78
  6. 2021DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equationscited by 66
  7. 2022Matching Normalizing Flows and Probability Paths on Manifoldscited by 57
  8. 2023A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositionscited by 48
  9. 2019Data-Driven Deep Learning of Partial Differential Equations in Modal Spacecited by 46
  10. 2019D3M: A deep domain decomposition method for partial differential equationscited by 44
  11. 2020PFNN: A Penalty-Free Neural Network Method for Solving a Class of Second-Order Boundary-Value Problems on Complex Geometriescited by 38
  12. 2022Universal approximation property of invertible neural networkscited by 30
  13. 2021Adaptive deep density approximation for Fokker-Planck equationscited by 28
  14. 2021Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Spacecited by 25
  15. 2021Solving time dependent Fokker-Planck equations via temporal normalizing flowcited by 17
  16. 2018Deep Multiscale Model Learningcited by 8
  17. 2021Approximation capabilities of measure-preserving neural networkscited by 4

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 1; 2026: 3.

4 papers.

  1. 2026Amortized Filtering and Smoothing with Conditional Normalizing Flowscited by 2
  2. 2025Adaptive Probability Flow Residual Minimization for High-Dimensional Fokker-Planck Equationscited by 1
  3. 2026A deep learning framework for jointly solving transient Fokker-Planck equations with arbitrary parameters and initial distributionscited by 1
  4. 2026A transition-density-based operator learning method for Fokker-Planck equations with various initial conditionscited by 0