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A Fourier-based inference method for learning interaction kernels in particle systems

Grigorios A. Pavliotis, Andrea Zanoni

TL;DR

This paper introduces a Fourier-based, semiparametric framework to infer the interaction kernel $W'$ in one-dimensional mean-field particle systems from a single observed trajectory. By expanding $V'$ and $W'$ in an orthonormal basis with respect to the invariant measure $\rho$ and deriving a linear system from the stationary Fokker–Planck equation, the method yields a low-dimensional estimator for the Fourier coefficients of $W'$, computed from empirical moments. The authors provide convergence analyses for the orthogonal polynomials and the kernel estimator, showing how estimation error scales with observation time $T$, system size $N$, and truncation level $K$, and they demonstrate the approach with numerical experiments on mean-field Ornstein–Uhlenbeck dynamics and a variety of kernels. The framework is data-efficient (requiring only a single trajectory) and adaptable to discrete-time data, but its performance hinges on the choice of $K$ and the conditioning of the resulting linear systems, motivating future work on adaptive selection and higher-dimensional extensions.

Abstract

We consider the problem of inferring the interaction kernel of stochastic interacting particle systems from observations of a single particle. We adopt a semi-parametric approach and represent the interaction kernel in terms of a generalized Fourier series. The basis functions in this expansion are tailored to the problem at hand and are chosen to be orthogonal polynomials with respect to the invariant measure of the mean-field dynamics. The generalized Fourier coefficients are obtained as the solution of an appropriate linear system whose coefficients depend on the moments of the invariant measure, and which are approximated from the particle trajectory that we observe. We quantify the approximation error in the Lebesgue space weighted by the invariant measure and study the asymptotic properties of the estimator in the joint limit as the observation interval and the number of particles tend to infinity, i.e. the joint large time-mean field limit. We also explore the regime where an increasing number of generalized Fourier coefficients is needed to represent the interaction kernel. Our theoretical results are supported by extensive numerical simulations.

A Fourier-based inference method for learning interaction kernels in particle systems

TL;DR

This paper introduces a Fourier-based, semiparametric framework to infer the interaction kernel in one-dimensional mean-field particle systems from a single observed trajectory. By expanding and in an orthonormal basis with respect to the invariant measure and deriving a linear system from the stationary Fokker–Planck equation, the method yields a low-dimensional estimator for the Fourier coefficients of , computed from empirical moments. The authors provide convergence analyses for the orthogonal polynomials and the kernel estimator, showing how estimation error scales with observation time , system size , and truncation level , and they demonstrate the approach with numerical experiments on mean-field Ornstein–Uhlenbeck dynamics and a variety of kernels. The framework is data-efficient (requiring only a single trajectory) and adaptable to discrete-time data, but its performance hinges on the choice of and the conditioning of the resulting linear systems, motivating future work on adaptive selection and higher-dimensional extensions.

Abstract

We consider the problem of inferring the interaction kernel of stochastic interacting particle systems from observations of a single particle. We adopt a semi-parametric approach and represent the interaction kernel in terms of a generalized Fourier series. The basis functions in this expansion are tailored to the problem at hand and are chosen to be orthogonal polynomials with respect to the invariant measure of the mean-field dynamics. The generalized Fourier coefficients are obtained as the solution of an appropriate linear system whose coefficients depend on the moments of the invariant measure, and which are approximated from the particle trajectory that we observe. We quantify the approximation error in the Lebesgue space weighted by the invariant measure and study the asymptotic properties of the estimator in the joint limit as the observation interval and the number of particles tend to infinity, i.e. the joint large time-mean field limit. We also explore the regime where an increasing number of generalized Fourier coefficients is needed to represent the interaction kernel. Our theoretical results are supported by extensive numerical simulations.
Paper Structure (12 sections, 9 theorems, 95 equations, 6 figures, 1 algorithm)

This paper contains 12 sections, 9 theorems, 95 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Outline.
  3. Orthonormal polynomials w.r.t. the invariant measure
  4. Convergence analysis for the approximated orthogonal polynomials
  5. Generalized method of moments
  6. Inference of the drift term
  7. Convergence analysis for the interaction kernel estimator
  8. Numerical experiments
  9. Mean-field Ornstein--Uhlenbeck process
  10. Discrete-time, low-frequency observations
  11. Inference of interaction kernels
  12. Conclusion

Key Result

Lemma 2.2

Let $\lambda_{kj}$ be defined in equation eq:lambda_def, and let $\widetilde{\lambda}_{kj}$ be its approximation. Under as:unique, for all $k,j \ge 0$ and for all $q \in [1,2)$ there exists a constant $C = C(k) > 0$, independent of $T$ and $N$, such that

Figures (6)

  • Figure 1: Schematic illustration summarizing the key mathematical objects and their relationships.
  • Figure 2: Comparison between the first four (excluding the constant function) exact ($\psi_k$) and approximated ($\widetilde{\psi}_k$) orthogonal polynomials with respect to the invariant measure $\rho$ of the mean-field Ornstein--Uhlenbeck process. Top: we fix $T = 10\,000$ and vary $N = 5, 50, 500$. Bottom: we fix $N = 500$ and vary $T = 100, 1\,000, 10\,000$.
  • Figure 3: Comparison between the theoretical and empirical rate of convergence of the first four (excluding the constant function) orthogonal polynomials with respect to the invariant measure $\rho$ of the mean-field Ornstein--Uhlenbeck process in $L^2(\rho)$, for both the number of particles $N$ (left) and the final time $T$ (right).
  • Figure 4: Comparison between the true interaction kernel $W'$ and the estimators $(\widehat{W}')_{T,N}^{(K)}$ in two different cases ($T = 1\,000, N = 50$ and $T = 10\,000, N = 500$), for different numbers of Fourier coefficients $K = 1, \dots, 8$ for the Ornstein-Uhlenbeck interaction kernel.
  • Figure 5: Comparison between the true interaction kernel $W_0'$ from equation \ref{['eq:W0']} and the estimator $(\widehat{W}_0')_{T,N}^{(K)}$, for the case of discrete-time observations with different sampling rates $\Delta = 1, 2, 4, 8$.
  • ...and 1 more figures

Theorems & Definitions (26)

  • Remark 1.2
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Proposition 2.7
  • ...and 16 more