# Learning Mean-Field Equations from Particle Data Using WSINDy

arXiv: https://arxiv.org/abs/2110.07756 · HTML: https://mlsanity.com/paper/2110.07756/

## Key facts

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

## Builds on (most cited first, 11 of 11)

- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Sparse learning of stochastic dynamic equations](https://mlsanity.com/paper/1712.02432/) (2017)
- [Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks](https://mlsanity.com/paper/2008.10653/) (2020)
- [On the Convergence of the SINDy Algorithm](https://mlsanity.com/paper/1805.06445/) (2018)
- [Learning interaction kernels in heterogeneous systems of agents from multiple trajectories](https://mlsanity.com/paper/1910.04832/) (2019)
- [Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories](https://mlsanity.com/paper/2007.15174/) (2020)
- [Learning interaction kernels in mean-field equations of 1st-order systems of interacting particles](https://mlsanity.com/paper/2010.15694/) (2020)
- [Low-Rank Tucker Approximation of a Tensor From Streaming Data](https://mlsanity.com/paper/1904.10951/) (2019)
- [Maximum likelihood estimation of potential energy in interacting particle systems from single-trajectory data](https://mlsanity.com/paper/2007.11048/) (2020)
- [Parameter Estimation for the McKean-Vlasov Stochastic Differential Equation](https://mlsanity.com/paper/2106.13751/) (2021)
- [Extracting Governing Laws from Sample Path Data of Non-Gaussian Stochastic Dynamical Systems](https://mlsanity.com/paper/2107.10127/) (2021)

## Built on by (most cited first, 29 of 29)

- [Direct Estimation of Parameters in ODE Models Using WENDy: Weak-form Estimation of Nonlinear Dynamics](https://mlsanity.com/paper/2302.13271/) (2023)
- [Weak-Form Latent Space Dynamics Identification](https://mlsanity.com/paper/2311.12880/) (2023)
- [Benchmarking sparse system identification with low-dimensional chaos](https://mlsanity.com/paper/2302.10787/) (2023)
- [Asymptotic consistency of the WSINDy algorithm in the limit of continuum data](https://mlsanity.com/paper/2211.16000/) (2022)
- [Online Weak-form Sparse Identification of Partial Differential Equations](https://mlsanity.com/paper/2203.03979/) (2022)
- [Physics-informed active learning with simultaneous weak-form latent space dynamics identification](https://mlsanity.com/paper/2407.00337/) (2024)
- [Learning Anisotropic Interaction Rules from Individual Trajectories in a Heterogeneous Cellular Population](https://mlsanity.com/paper/2204.14141/) (2022)
- [Coarse-Graining Hamiltonian Systems Using WSINDy](https://mlsanity.com/paper/2310.05879/) (2023)
- [The Weak Form Is Stronger Than You Think](https://mlsanity.com/paper/2409.06751/) (2024)
- [GINN-LP: A Growing Interpretable Neural Network for Discovering Multivariate Laurent Polynomial Equations](https://mlsanity.com/paper/2312.10913/) (2023)
- [Weak-Form Inference for Hybrid Dynamical Systems in Ecology](https://mlsanity.com/paper/2405.20591/) (2024)
- [Universal Approximation of Mean-Field Models via Transformers](https://mlsanity.com/paper/2410.16295/) (2024)
- [Weak Form Scientific Machine Learning: Test Function Construction for System Identification](https://mlsanity.com/paper/2507.03206/) (2025)
- [Weak Collocation Regression method: fast reveal hidden stochastic dynamics from high-dimensional aggregate data](https://mlsanity.com/paper/2209.02628/) (2022)
- [Neural McKean-Vlasov Processes: Distributional Dependence in Diffusion Processes](https://mlsanity.com/paper/2404.09402/) (2024)
- [Weak Form Learning for Mean-Field Partial Differential Equations: an Application to Insect Movement](https://mlsanity.com/paper/2510.07786/) (2025)
- [Learning Collective Behaviors from Observation](https://mlsanity.com/paper/2311.00875/) (2023)
- [Physics-guided weak-form discovery of reduced-order models for trapped ultracold hydrodynamics](https://mlsanity.com/paper/2406.07519/) (2024)
- [Ensemble WSINDy for Data Driven Discovery of Governing Equations from Laser-based Full-field Measurements](https://mlsanity.com/paper/2409.20510/) (2024)
- [Learning interacting particle systems from unlabeled data](https://mlsanity.com/paper/2604.02581/) (2026)
- [Learning effective models from network dynamics data with multiple initial conditions using weak form SINDy](https://mlsanity.com/paper/2605.30432/) (2026)
- [A Data-Adaptive Prior for Bayesian Learning of Kernels in Operators](https://mlsanity.com/paper/2212.14163/) (2022)
- [Weak Collocation Regression for Inferring Stochastic Dynamics with L'evy Noise](https://mlsanity.com/paper/2403.08292/) (2024)
- [Moment Estimate and Variational Approach for Learning Generalized Diffusion with Non-gradient Structures](https://mlsanity.com/paper/2508.01854/) (2025)
- [Learning Multi-type heterogeneous interacting particle systems](https://mlsanity.com/paper/2602.03954/) (2026)
- [MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data](https://mlsanity.com/paper/2604.00333/) (2026)
- [Structure-Aware Variational Learning of a Class of Generalized Diffusions](https://mlsanity.com/paper/2604.20188/) (2026)
- [Multi-Agent System Identification with Nonlinear Sheaf Diffusion](https://mlsanity.com/paper/2605.11204/) (2026)
- [Learning Interaction Kernels from Collective Steady States](https://mlsanity.com/paper/2609.12004/) (2026)

## 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.