Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A method of moments estimator for interacting particle systems and their mean field limit

Grigorios A. Pavliotis, Andrea Zanoni

TL;DR

The paper addresses parameter inference for McKean–Vlasov SDEs arising as mean-field limits of interacting particle systems, using a method of moments based on invariant-measure moments and the quadratic variation from a single observed particle path. By converting stationaryFokker–Planck moment conditions into a low-dimensional linear system and augmenting with a quadratic-variation constraint, the authors obtain a practical estimator whose consistency is established under ergodicity and propagation of chaos, with a convergence rate of order $O(1/\sqrt{T}+1/\sqrt{N})$. The approach is validated through extensive simulations, including the mean-field OU process, bistable potentials, multiplicative noise, and a two-dimensional Fitzhugh–Nagumo neuron model, demonstrating robustness even when the mean-field limit has multiple equilibria and when only discrete observations are available. The method is easy to implement and scalable, relying on ergodic moment estimates and a standard linear solver, with clear potential for extensions to nonparametric settings and higher-dimensional systems. Overall, it provides a simple, effective tool for inferring parameters in nonlinear, law-dependent SDEs with polynomial structure in practical, data-limited scenarios.

Abstract

We study the problem of learning unknown parameters in stochastic interacting particle systems with polynomial drift, interaction and diffusion functions from the path of one single particle in the system. Our estimator is obtained by solving a linear system which is constructed by imposing appropriate conditions on the moments of the invariant distribution of the mean field limit and on the quadratic variation of the process. Our approach is easy to implement as it only requires the approximation of the moments via the ergodic theorem and the solution of a low-dimensional linear system. Moreover, we prove that our estimator is asymptotically unbiased in the limits of infinite data and infinite number of particles (mean field limit). In addition, we present several numerical experiments that validate the theoretical analysis and show the effectiveness of our methodology to accurately infer parameters in systems of interacting particles.

A method of moments estimator for interacting particle systems and their mean field limit

TL;DR

The paper addresses parameter inference for McKean–Vlasov SDEs arising as mean-field limits of interacting particle systems, using a method of moments based on invariant-measure moments and the quadratic variation from a single observed particle path. By converting stationaryFokker–Planck moment conditions into a low-dimensional linear system and augmenting with a quadratic-variation constraint, the authors obtain a practical estimator whose consistency is established under ergodicity and propagation of chaos, with a convergence rate of order . The approach is validated through extensive simulations, including the mean-field OU process, bistable potentials, multiplicative noise, and a two-dimensional Fitzhugh–Nagumo neuron model, demonstrating robustness even when the mean-field limit has multiple equilibria and when only discrete observations are available. The method is easy to implement and scalable, relying on ergodic moment estimates and a standard linear solver, with clear potential for extensions to nonparametric settings and higher-dimensional systems. Overall, it provides a simple, effective tool for inferring parameters in nonlinear, law-dependent SDEs with polynomial structure in practical, data-limited scenarios.

Abstract

We study the problem of learning unknown parameters in stochastic interacting particle systems with polynomial drift, interaction and diffusion functions from the path of one single particle in the system. Our estimator is obtained by solving a linear system which is constructed by imposing appropriate conditions on the moments of the invariant distribution of the mean field limit and on the quadratic variation of the process. Our approach is easy to implement as it only requires the approximation of the moments via the ergodic theorem and the solution of a low-dimensional linear system. Moreover, we prove that our estimator is asymptotically unbiased in the limits of infinite data and infinite number of particles (mean field limit). In addition, we present several numerical experiments that validate the theoretical analysis and show the effectiveness of our methodology to accurately infer parameters in systems of interacting particles.
Paper Structure (13 sections, 3 theorems, 84 equations, 6 figures, 1 algorithm)

This paper contains 13 sections, 3 theorems, 84 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Method of moments
  4. The mean field Ornstein--Uhlenbeck process
  5. Convergence analysis
  6. Numerical experiments
  7. Sensitivity analysis
  8. Rate of convergence
  9. Comparison with other estimators in case of discrete observations
  10. Bistable potential
  11. Multiplicative noise
  12. Interacting Fitzhugh--Nagumo Neurons
  13. Conclusion

Key Result

Lemma 4.3

Let as:analysisas:propagation_chaos hold and let $\nu = \mu$. Then, for all $r \in \mathbb{N}$ and for all $q \in [1,2)$ there exists a constant $C>0$ independent of $T$ and $N$ such that

Figures (6)

  • Figure 1: Sensitivity analysis for the mean field Ornstein--Uhlenbeck process with respect to the number $M$ of moments equations. Left: error of the estimator $\widehat{\theta}_{T,N}^M$. Right: condition number of the matrix $\mathcal{A}$.
  • Figure 2: Rate of convergence of the estimator $\widehat{\theta}_{T,N}^M$ towards the exact value $\theta^*$ with respect to the final time $T$ (left) and the number of interacting particles $N$ (right).
  • Figure 3: Comparison between the estimator $\widehat{\theta}_{T,N}^M$ with the MLE and the eigenfunction estimator in case of discrete-time observations for different values of the sampling rate $\Delta$. Left: estimation obtained with one particle. Right: average of the estimations obtained with all the particles in the system.
  • Figure 4: Inference of the drift, interaction and diffusion coefficients for the cases of bistable drift and quadratic interaction (top) and quadratic drift and bistable interaction (bottom) with constant diffusion. Diagonal: histogram of the estimations of each component obtained from all particles. Off-diagonal: scatter plot of the estimations obtained from all particles for two components at a time. Black and red stars/lines represent the average of the estimations and the exact value, respectively.
  • Figure 5: Scatter plot for the inference of the two-dimensional diffusion coefficient for the case of simultaneous additive and multiplicative noise.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Remark 2.1
  • Example 2.3
  • Remark 2.4
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 4.2
  • Lemma 4.3
  • proof
  • Remark 4.4
  • ...and 5 more