Fourier Series-Based Approximation of Time-Varying Parameters in Ordinary Differential Equations

TL;DR

This paper introduces a Fourier series-based method combined with ensemble Kalman filtering to estimate unknown, time-varying parameters in ordinary differential equations, improving parameter tracking in dynamical systems.

Contribution

The work presents a novel Fourier series-inspired approximation approach for sequentially estimating time-varying parameters in ODE models, including periodic and non-periodic cases.

Findings

Effective estimation of periodic parameters with known and unknown periods.

Accurate tracking of non-periodic time-varying parameters.

Highlighting the influence of Fourier terms on estimation accuracy.

Abstract

Many real-world systems modeled using differential equations involve unknown or uncertain parameters. Standard approaches to address parameter estimation inverse problems in this setting typically focus on estimating constants; yet some unobservable system parameters may vary with time without known evolution models. In this work, we propose a novel approximation method inspired by the Fourier series to estimate time-varying parameters in deterministic dynamical systems modeled with ordinary differential equations. Using ensemble Kalman filtering in conjunction with Fourier series-based approximation models, we detail two possible implementation schemes for sequentially updating the time-varying parameter estimates given noisy observations of the system states. We demonstrate the capabilities of the proposed approach in estimating periodic parameters, both when the period is known and unknown, as well as non-periodic time-varying parameters of different forms with several computed examples using a forced harmonic oscillator. Results emphasize the importance of the frequencies and number of approximation model terms on the time-varying parameter estimates and corresponding dynamical system predictions.