mlsanityData to 7 Oct 2026

Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow models

arXiv 2108.12956 ↗First posted 30 Aug 2021cs.LG cs.NA math.NAcited by 34

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Abstract (arXiv)

We introduce in this work the normalizing field flows (NFF) for learning random fields from scattered measurements. More precisely, we construct a bijective transformation (a normalizing flow characterizing by neural networks) between a Gaussian random field with the Karhunen-Lo`eve (KL) expansion structure and the target stochastic field, where the KL expansion coefficients and the invertible networks are trained by maximizing the sum of the log-likelihood on scattered measurements. This NFF model can be used to solve data-driven forward, inverse, and mixed forward/inverse stochastic partial differential equations in a unified framework. We demonstrate the capability of the proposed NFF model for learning Non Gaussian processes and different types of stochastic partial differential equations.

Key facts. Cited by 34 papers in the map (data through 7 Oct 2026). Builds on 16 of its 35 references that are in the map. First posted on arXiv 30 Aug 2021.

Builds on

16 of 35 references.

References · 16

  1. 2016Density estimation using Real NVPcited by 2,029
  2. 2018Glow: Generative Flow with Invertible 1x1 Convolutionscited by 1,539
  3. 2014NICE: Non-linear Independent Components Estimationcited by 1,052
  4. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 362
  5. 2019Flow++: Improving Flow-Based Generative Models with Variational Dequantization and Architecture Designcited by 258
  6. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
  7. 2017Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equationscited by 208
  8. 2018Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantificationcited by 199
  9. 2017Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equationscited by 188
  10. 2017Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equationscited by 158
  11. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 144
  12. 2020BayesFlow: Learning complex stochastic models with invertible neural networkscited by 139
4 more
  1. 2017Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equationscited by 72
  2. 2018FloWaveNet : A Generative Flow for Raw Audiocited by 54
  3. 2020Solving inverse problems using conditional invertible neural networkscited by 37
  4. 2019Universal audio synthesizer control with normalizing flowscited by 16

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 1; 2022: 10; 2023: 7; 2024: 9; 2025: 4; 2026: 3.

34 papers.

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  1. 2022Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisonscited by 169
  2. 2022Failure-informed adaptive sampling for PINNscited by 64
  3. 2023L-HYDRA: Multi-Head Physics-Informed Neural Networkscited by 29
  4. 2025Consistent Sampling and Simulation: Molecular Dynamics with Energy-Based Diffusion Modelscited by 23
  5. 2024Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck Equationscited by 22
  6. 2022A survey of unsupervised learning methods for high-dimensional uncertainty quantification in black-box-type problemscited by 20
  7. 2022Fully probabilistic deep models for forward and inverse problems in parametric PDEscited by 19
  8. 2023IB-UQ: Information bottleneck based uncertainty quantification for neural function regression and neural operator learningcited by 18
  9. 2021Solving time dependent Fokker-Planck equations via temporal normalizing flowcited by 17
  10. 2022Taming Hyperparameter Tuning in Continuous Normalizing Flows Using the JKO Schemecited by 17
  11. 2023Diffusion-Model-Assisted Supervised Learning of Generative Models for Density Estimationcited by 13
  12. 2022TrafficFlowGAN: Physics-informed Flow based Generative Adversarial Network for Uncertainty Quantificationcited by 11
22 more
  1. 2023Physics-Informed Deep Learning For Traffic State Estimation: A Survey and the Outlookcited by 11
  2. 2023Bounded KRnet and its applications to density estimation and approximationcited by 7
  3. 2024Modeling Unknown Stochastic Dynamical System Subject to External Excitationcited by 7
  4. 2023Efficient Bayesian inference using physics-informed invertible neural networks for inverse problemscited by 6
  5. 2022Adaptive deep density approximation for fractional Fokker-Planck equationscited by 4
  6. 2023PI-VEGAN: Physics Informed Variational Embedding Generative Adversarial Networks for Stochastic Differential Equationscited by 4
  7. 2024Resolution invariant deep operator network for PDEs with complex geometriescited by 4
  8. 2022Less Emphasis on Difficult Layer Regions: Curriculum Learning for Singularly Perturbed Convection-Diffusion-Reaction Problemscited by 3
  9. 2024Conditional Pseudo-Reversible Normalizing Flow for Surrogate Modeling in Quantifying Uncertainty Propagationcited by 3
  10. 2024A Primer on Variational Inference for Physics-Informed Deep Generative Modellingcited by 3
  11. 2024Towards Gaussian Process for operator learning: an uncertainty aware resolution independent operator learning algorithm for computational mechanicscited by 3
  12. 2024TRADE: Transfer of Distributions between External Conditions with Normalizing Flowscited by 3
  13. 2025Flow-based Bayesian filtering for high-dimensional nonlinear stochastic dynamical systemscited by 3
  14. 2022MC-Nonlocal-PINNs: handling nonlocal operators in PINNs via Monte Carlo samplingcited by 1
  15. 2024Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Levy Equationscited by 1
  16. 2024PinnDE: Physics-Informed Neural Networks for Solving Differential Equationscited by 1
  17. 2025Learning Stochastic Hamiltonian Systems via Stochastic Generating Function Neural Networkcited by 1
  18. 2026GenVoid: Uncertainty-Aware Learning of Subsurface Material Defects with an Experimentally Validated Physics-Informed Generative Modelcited by 1
  19. 2022Bayesian deep learning framework for uncertainty quantification in high dimensionscited by 0
  20. 2025Scalable physics-informed deep generative model for solving forward and inverse stochastic differential equationscited by 0
  21. 2026Neural Flow Operators can Approximate any Operator: Abstract Frameworks and Universal Approximationscited by 0
  22. 2026Diagnosing the conditional-mean barrier in scientific machine-learning surrogatescited by 0