mlsanityData to 7 Oct 2026

Learning Mean-Field Equations from Particle Data Using WSINDy

arXiv 2110.07756 ↗First posted 14 Oct 2021stat.ML cs.LG cs.NAcited by 29

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Abstract (arXiv)

We develop a weak-form sparse identification method for interacting particle systems (IPS) with the primary goals of reducing computational complexity for large particle number N and offering robustness to either intrinsic or extrinsic noise. In particular, we use concepts from mean-field theory of IPS in combination with the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy) to provide a fast and reliable system identification scheme for recovering the governing stochastic differential equations for an IPS when the number of particles per experiment N is on the order of several thousand and the number of experiments M is less than 100. This is in contrast to existing work showing that system identification for N less than 100 and M on the order of several thousand is feasible using strong-form methods. We prove that under some standard regularity assumptions the scheme converges with rate O(N-1/2) in the ordinary least squares setting and we demonstrate the convergence rate numerically on several systems in one and two spatial dimensions. Our examples include a canonical problem from homogenization theory (as a first step towards learning coarse-grained models), the dynamics of an attractive-repulsive swarm, and the IPS description of the parabolic-elliptic Keller-Segel model for chemotaxis.

Key facts. Cited by 29 papers in the map (data through 7 Oct 2026). Builds on 11 of its 63 references that are in the map. First posted on arXiv 14 Oct 2021.

Builds on

11 of 63 references.

References · 11

  1. 2020Weak SINDy For Partial Differential Equationscited by 154
  2. 2017Sparse learning of stochastic dynamic equationscited by 76
  3. 2020Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networkscited by 52
  4. 2018On the Convergence of the SINDy Algorithmcited by 49
  5. 2019Learning interaction kernels in heterogeneous systems of agents from multiple trajectoriescited by 25
  6. 2020Learning interaction kernels in stochastic systems of interacting particles from multiple trajectoriescited by 24
  7. 2020Learning interaction kernels in mean-field equations of 1st-order systems of interacting particlescited by 21
  8. 2019Low-Rank Tucker Approximation of a Tensor From Streaming Datacited by 15
  9. 2020Maximum likelihood estimation of potential energy in interacting particle systems from single-trajectory datacited by 10
  10. 2021Parameter Estimation for the McKean-Vlasov Stochastic Differential Equationcited by 8
  11. 2021Extracting Governing Laws from Sample Path Data of Non-Gaussian Stochastic Dynamical Systemscited by 7

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 5; 2023: 6; 2024: 8; 2025: 3; 2026: 7.

29 papers.

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  1. 2023Direct Estimation of Parameters in ODE Models Using WENDy: Weak-form Estimation of Nonlinear Dynamicscited by 20
  2. 2023Weak-Form Latent Space Dynamics Identificationcited by 20
  3. 2023Benchmarking sparse system identification with low-dimensional chaoscited by 19
  4. 2022Asymptotic consistency of the WSINDy algorithm in the limit of continuum datacited by 14
  5. 2022Online Weak-form Sparse Identification of Partial Differential Equationscited by 13
  6. 2024Physics-informed active learning with simultaneous weak-form latent space dynamics identificationcited by 13
  7. 2022Learning Anisotropic Interaction Rules from Individual Trajectories in a Heterogeneous Cellular Populationcited by 11
  8. 2023Coarse-Graining Hamiltonian Systems Using WSINDycited by 11
  9. 2024The Weak Form Is Stronger Than You Thinkcited by 11
  10. 2024Weak-Form Inference for Hybrid Dynamical Systems in Ecologycited by 7
  11. 2024Universal Approximation of Mean-Field Models via Transformerscited by 7
  12. 2023GINN-LP: A Growing Interpretable Neural Network for Discovering Multivariate Laurent Polynomial Equationscited by 6
17 more
  1. 2025Weak Form Scientific Machine Learning: Test Function Construction for System Identificationcited by 5
  2. 2022Weak Collocation Regression method: fast reveal hidden stochastic dynamics from high-dimensional aggregate datacited by 4
  3. 2024Neural McKean-Vlasov Processes: Distributional Dependence in Diffusion Processescited by 4
  4. 2025Weak Form Learning for Mean-Field Partial Differential Equations: an Application to Insect Movementcited by 3
  5. 2023Learning Collective Behaviors from Observationcited by 2
  6. 2024Physics-guided weak-form discovery of reduced-order models for trapped ultracold hydrodynamicscited by 2
  7. 2024Ensemble WSINDy for Data Driven Discovery of Governing Equations from Laser-based Full-field Measurementscited by 2
  8. 2026Learning interacting particle systems from unlabeled datacited by 2
  9. 2026Learning effective models from network dynamics data with multiple initial conditions using weak form SINDycited by 2
  10. 2022A Data-Adaptive Prior for Bayesian Learning of Kernels in Operatorscited by 1
  11. 2024Weak Collocation Regression for Inferring Stochastic Dynamics with L'evy Noisecited by 1
  12. 2025Moment Estimate and Variational Approach for Learning Generalized Diffusion with Non-gradient Structurescited by 1
  13. 2026Learning Multi-type heterogeneous interacting particle systemscited by 1
  14. 2026MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Datacited by 1
  15. 2026Structure-Aware Variational Learning of a Class of Generalized Diffusionscited by 1
  16. 2026Multi-Agent System Identification with Nonlinear Sheaf Diffusioncited by 0
  17. 2026Learning Interaction Kernels from Collective Steady Statescited by 0