[Paper Review] On the identifiability of interaction functions in systems of interacting particles

Zhongyang Li, Fei Lu+3 more|arXiv (Cornell University)|Dec 27, 2019

Advanced Thermodynamics and Statistical Mechanics28 references4 citations

TL;DR

This paper establishes the identifiability of interaction functions in first-order stochastic systems of interacting particles by proving that a coercivity condition—equivalent to strict positivity of an integral operator—is both sufficient and necessary (in the infinite particle limit) for unique recovery of the interaction potential from observed trajectories. The analysis relies on Müntz-type theorems to verify the strict positive definiteness of integral kernels in $ L^2 $ spaces.

ABSTRACT

We address a fundamental issue in the nonparametric inference for systems of interacting particles: the identifiability of the interaction functions. We prove that the interaction functions are identifiable for a class of first-order stochastic systems, including linear systems with general initial laws and nonlinear systems with stationary distributions. We show that a coercivity condition is sufficient for identifiability and becomes necessary when the number of particles approaches infinity. The coercivity is equivalent to the strict positivity of related integral operators, which we prove by showing that their integral kernels are strictly positive definite by using Müntz type theorems.

Motivation & Objective

• To resolve the fundamental challenge of non-identifiability in nonparametric inference of interaction functions from particle trajectories.
• To establish conditions under which the interaction function $\phi$ can be uniquely recovered from observed data in systems of interacting particles.
• To investigate the role of the coercivity condition in ensuring identifiability, particularly in the limit of large particle counts.
• To connect identifiability to the strict positivity of integral operators via positive-definite kernels and functional analytic tools.
• To provide sufficient conditions for identifiability in linear systems and a class of three-particle nonlinear systems with stationary distributions.

Proposed method

• Formulates the interaction system as a first-order stochastic gradient system with pairwise interaction depending on relative distances.
• Introduces the average-in-time distribution $\bar{\rho}_T$ of pairwise distances as the natural measure for inference in $L^2(\bar{\rho}_T)$.
• Defines identifiability via the injectivity of the integral operator associated with the interaction function.
• Establishes that the coercivity condition—equivalent to strict positivity of the integral operator—is sufficient for identifiability.
• Uses Müntz-type theorems to prove that the integral kernel is strictly positive definite under mild conditions on the interaction potential.
• Applies results from positive-definite kernels and negative definite functions to verify the coercivity condition in linear and nonlinear systems.

Experimental results

Research questions

• RQ1 Under what conditions is the interaction function $\phi$ identifiable from observed trajectories of interacting particles?
• RQ2 Is the coercivity condition both necessary and sufficient for identifiability in the limit of infinitely many particles?
• RQ3 Can the coercivity condition be verified for linear systems and nonlinear systems with stationary distributions?
• RQ4 How does the structure of the integral operator associated with $\phi$ relate to identifiability?
• RQ5 What role do positive-definite kernels and Müntz-type theorems play in establishing the strict positivity of the integral operator?

Key findings

• The coercivity condition is sufficient for identifiability of the interaction function $\phi$ in first-order stochastic systems of interacting particles.
• In the limit as the number of particles $N \to \infty$, the coercivity condition becomes necessary for identifiability.
• For linear systems with general initial laws, the coercivity condition holds, ensuring identifiability.
• For three-particle nonlinear systems with a stationary distribution, the coercivity condition holds under certain assumptions on the interaction potential $\Phi(r) = r^{2\beta}$.
• The integral operator associated with the interaction function is strictly positive definite when the kernel is strictly positive definite, which is verified using Müntz-type theorems on the half-line.
• The completeness of $\{r^{2k}\}_{k=1}^\infty$ in $L^2([0,\infty), \rho)$ under exponential moment conditions ensures that the kernel is not degenerate, supporting the coercivity condition.