Parameter identification from single trajectory data: from linear to nonlinear

TL;DR

This paper investigates parameter identification from single trajectory data in differential equations, extending previous theoretical results to the nonlinear Lotka-Volterra system through numerical analysis and introducing the $P_n$-diagram for visualization.

Contribution

It extends the framework for parameter inference to nonlinear systems, specifically the Lotka-Volterra model, and introduces the $P_n$-diagram for analyzing data features and non-uniqueness.

Findings

Core features extend to nonlinear Lotka-Volterra systems.

The $P_n$-diagram effectively visualizes parameter features.

Non-uniqueness manifests as multi-layered structures in $P_2$-diagrams.

Abstract

Our recent work lays out a general framework for inferring information about the parameters and associated dynamics of a differential equation model from a discrete set of data points collected from the system being modeled. Rigorous mathematical results have justified this approach and have identified some common features that arise for certain classes of integrable models. In this work we present a thorough numerical investigation that shows that several of these core features extend to a paradigmatic linear-in-parameters model, the Lotka-Volterra (LV) system, which we consider in the conservative case as well as under the addition of terms that perturb the system away from this regime. A central construct for this analysis is a concise representation of parameter features in the data space that we call the $P_n$-diagram, which is particularly useful for visualization of results for low-dimensional (small $n$) systems. Our work also exposes some new properties related to non-uniqueness that arise for these LV systems, with non-uniqueness manifesting as a multi-layered structure in the associated $P_2$-diagrams.