Physics-Informed Regression: Parameter Estimation in Parameter-Linear Nonlinear Dynamic Models

TL;DR

This paper introduces Physics-Informed Regression (PIR), a hybrid method for efficient parameter estimation in parameter-linear nonlinear dynamic models, demonstrating superior performance and speed over PINNs on epidemic models with real and synthetic data.

Contribution

The paper presents PIR, a novel hybrid approach leveraging regularized least squares for parameter estimation in parameter-linear models, bridging theory and data effectively.

Findings

PIR outperforms PINN in accuracy on complex epidemic models.

PIR is faster and more computationally efficient than PINN.

PIR successfully estimates time-varying parameters from real COVID-19 data.

Abstract

We present a new efficient hybrid parameter estimation method based on the idea, that if nonlinear dynamic models are stated in terms of a system of equations that is linear in terms of the parameters, then regularized ordinary least squares can be used to estimate these parameters from time series data. We introduce the term "Physics-Informed Regression" (PIR) to describe the proposed data-driven hybrid technique as a way to bridge theory and data by use of ordinary least squares to efficiently perform parameter estimation of the model coefficients of different parameter-linear models; providing examples of models based on nonlinear ordinary equations (ODE) and partial differential equations (PDE). The focus is on parameter estimation on a selection of ODE and PDE models, each illustrating performance in different model characteristics. For two relevant epidemic models of different complexity and number of parameters, PIR is tested and compared against the related technique, physics-informed neural networks (PINN), both on synthetic data generated from known target parameters and on real public Danish time series data collected during the COVID-19 pandemic in Denmark. Both methods were able to estimate the target parameters, while PIR showed to perform noticeably better, especially on a compartment model with higher complexity. Given the difference in computational speed, it is concluded that the PIR method is superior to PINN for the models considered. It is also demonstrated how PIR can be applied to estimate the time-varying parameters of a compartment model that is fitted using real Danish data from the COVID-19 pandemic obtained during a period from 2020 to 2021. The study shows how data-driven and physics-informed techniques may support reliable and fast -- possibly real-time -- parameter estimation in parameter-linear nonlinear dynamic models.