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Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks

arXiv 2008.10653 ↗First posted 24 Aug 2020physics.comp-ph cs.LG stat.MLcited by 52

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Abstract (arXiv)

The Fokker-Planck (FP) equation governing the evolution of the probability density function (PDF) is applicable to many disciplines but it requires specification of the coefficients for each case, which can be functions of space-time and not just constants, hence requiring the development of a data-driven modeling approach. When the data available is directly on the PDF, then there exist methods for inverse problems that can be employed to infer the coefficients and thus determine the FP equation and subsequently obtain its solution. Herein, we address a more realistic scenario, where only sparse data are given on the particles' positions at a few time instants, which are not sufficient to accurately construct directly the PDF even at those times from existing methods, e.g., kernel estimation algorithms. To this end, we develop a general framework based on physics-informed neural networks (PINNs) that introduces a new loss function using the Kullback-Leibler divergence to connect the stochastic samples with the FP equation, to simultaneously learn the equation and infer the multi-dimensional PDF at all times. In particular, we consider two types of inverse problems, type I where the FP equation is known but the initial PDF is unknown, and type II in which, in addition to unknown initial PDF, the drift and diffusion terms are also unknown. In both cases, we investigate problems with either Brownian or Levy noise or a combination of both. We demonstrate the new PINN framework in detail in the one-dimensional case (1D) but we also provide results for up to 5D demonstrating that we can infer both the FP equation and} dynamics simultaneously at all times with high accuracy using only very few discrete observations of the particles.

Key facts. Cited by 52 papers in the map (data through 14 Oct 2026). First posted on arXiv 24 Aug 2020.

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Citing papers per year: 2021: 9; 2022: 7; 2023: 6; 2024: 9; 2025: 8; 2026: 13.

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  1. 2021Towards a mathematical theory of trajectory inferencecited by 31
  2. 2021Learning Mean-Field Equations from Particle Data Using WSINDycited by 29
  3. 2023AI-Aristotle: A Physics-Informed framework for Systems Biology Gray-Box Identificationcited by 29
  4. 2021Adaptive deep density approximation for Fokker-Planck equationscited by 28
  5. 2023Learning Stochastic Dynamical System via Flow Map Operatorcited by 24
  6. 2023Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networkscited by 23
  7. 2024Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck Equationscited by 22
  8. 2024A Training-Free Conditional Diffusion Model for Learning Stochastic Dynamical Systemscited by 19
  9. 2021Learning the temporal evolution of multivariate densities via normalizing flowscited by 18
  10. 2021Solving time dependent Fokker-Planck equations via temporal normalizing flowcited by 17
  11. 2021Neural network stochastic differential equation models with applications to financial data forecastingcited by 14
  12. 2022Revisiting PINNs: Generative Adversarial Physics-informed Neural Networks and Point-weighting Methodcited by 10
38 more
  1. 2021Extracting stochastic dynamical systems with α-stable L'evy noise from datacited by 8
  2. 2022Temperature Field Inversion of Heat-Source Systems via Physics-Informed Neural Networkscited by 8
  3. 2024DynGMA: a robust approach for learning stochastic differential equations from datacited by 8
  4. 2022An end-to-end deep learning approach for extracting stochastic dynamical systems with α-stable L'evy noisecited by 7
  5. 2024Modeling Unknown Stochastic Dynamical System Subject to External Excitationcited by 7
  6. 2021Extracting Stochastic Governing Laws by Nonlocal Kramers-Moyal Formulascited by 6
  7. 2025Convolution-weighting method for the physics-informed neural network: A Primal-Dual Optimization Perspectivecited by 6
  8. 2024Tackling the Curse of Dimensionality in Fractional and Tempered Fractional PDEs with Physics-Informed Neural Networkscited by 5
  9. 2024Data-driven Effective Modeling of Multiscale Stochastic Dynamical Systemscited by 5
  10. 2021Computing the Invariant Distribution of Randomly Perturbed Dynamical Systems Using Deep Learningcited by 4
  11. 2022Weak Collocation Regression method: fast reveal hidden stochastic dynamics from high-dimensional aggregate datacited by 4
  12. 2022Adaptive deep density approximation for fractional Fokker-Planck equationscited by 4
  13. 2024Neural McKean-Vlasov Processes: Distributional Dependence in Diffusion Processescited by 4
  14. 2025Generative AI Models for Learning Flow Maps of Stochastic Dynamical Systems in Bounded Domainscited by 3
  15. 2022An Optimal Control Method to Compute the Most Likely Transition Path for Stochastic Dynamical Systems with Jumpscited by 2
  16. 2023Reservoir Computing with Error Correction: Long-term Behaviors of Stochastic Dynamical Systemscited by 2
  17. 2022Drift Identification for L'evy alpha-Stable Stochastic Systemscited by 1
  18. 2023Maximum-likelihood Estimators in Physics-Informed Neural Networks for High-dimensional Inverse Problemscited by 1
  19. 2024Weak Collocation Regression for Inferring Stochastic Dynamics with L'evy Noisecited by 1
  20. 2024Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Levy Equationscited by 1
  21. 2025Moment Estimate and Variational Approach for Learning Generalized Diffusion with Non-gradient Structurescited by 1
  22. 2025Adaptive Probability Flow Residual Minimization for High-Dimensional Fokker-Planck Equationscited by 1
  23. 2026Training-free score-based diffusion for parameter-dependent stochastic dynamical systemscited by 1
  24. 2026Lightweight Geometric Adaptation for Training Physics-Informed Neural Networkscited by 1
  25. 2026Structure-Aware Variational Learning of a Class of Generalized Diffusionscited by 1
  26. 2023Physics-Informed Solution of The Stationary Fokker-Plank Equation for a Class of Nonlinear Dynamical Systems: An Evaluation Studycited by 0
  27. 2025A Robust Model-Based Approach for Continuous-Time Policy Evaluation with Unknown L'evy Process Dynamicscited by 0
  28. 2025Time-dependent density estimation using binary classifierscited by 0
  29. 2025Moment Estimates and DeepRitz Methods on Learning Diffusion Systems with Non-gradient Driftscited by 0
  30. 2025AutoBalance: An Automatic Balancing Framework for Training Physics-Informed Neural Networkscited by 0
  31. 2026Learning collision operators from plasma phase space data using differentiable simulatorscited by 0
  32. 2026Nonlocal Kramers-Moyal formulas and data-driven discovery of stochastic dynamical systems with multiplicative L'evy noisecited by 0
  33. 2026On the Unique Recovery of Transport Maps and Vector Fields from Finite Measure-Valued Datacited by 0
  34. 2026Variational Smoothing and Inference for SDEs from Sparse Data with Dynamic Neural Flowscited by 0
  35. 2026SPLIT-PINN: Separable Probability Learning Technique via Physics-Informed Neural Networks for High-Dimensional Probabilistic Modelingcited by 0
  36. 2026A transition-density-based operator learning method for Fokker-Planck equations with various initial conditionscited by 0
  37. 2026Identifiability Limits of Physics-Informed Inference for Spatial Stochastic Dynamics from Static Snapshotscited by 0
  38. 2026From Non-Convex Self-Concordant Regularization to Scalable Quasi-Newton Training of PINNscited by 0

50 most cited of 52 All 52 (JSON)