# Koopman Operators for Generalized Persistence of Excitation Conditions for Nonlinear Systems

## TL;DR

This paper develops new identifiability conditions for nonlinear biological models using Koopman operators, linking data spectral properties to model predictability, and demonstrates their application with a synthetic gene circuit.

## Contribution

It introduces a novel framework for nonlinear system identifiability based on Koopman operators and spectral data analysis, guiding experimental design for biological modeling.

## Key findings

- Identifiability depends on dataset spectral rank in Koopman space.
- Rank degeneracy leads to overfitting and poor model predictions.
- Proposes criteria for designing experiments to ensure model predictability.

## Abstract

It is hard to identify nonlinear biological models strictly from data, with results that are often sensitive to experimental conditions. Automated experimental workflows and liquid handling enables unprecedented throughput, as well as the capacity to generate extremely large datasets. We seek to develop generalized identifiability conditions for informing the design of automated experiments to discover predictive nonlinear biological models. For linear systems, identifiability is characterized by persistence of excitation conditions. For nonlinear systems, no such persistence of excitation conditions exist. We use the input-Koopman operator method to model nonlinear systems and derive identifiability conditions for open-loop systems initialized from a single initial condition. We show that nonlinear identifiability is intrinsically tied to the rank of a given dataset's power spectral density, transformed through the lifted Koopman observable space. We illustrate these identifiability conditions with a simulated synthetic gene circuit model, the repressilator. We illustrate how rank degeneracy in datasets results in overfitted nonlinear models of the repressilator, resulting in poor predictive accuracy. Our findings provide novel experimental design criteria for discovery of globally predictive nonlinear models of biological phenomena.