Estimation and approximation in multidimensional dynamics

TL;DR

This paper introduces frequentist and Bayesian methods for jointly estimating parameters and state functions in PDEs, including strategies for incorporating differential conditions, validated through simulations and real data.

Contribution

It presents novel joint estimation techniques for PDE parameters and states, with strategies for integrating differential conditions, advancing data-driven dynamic system modeling.

Findings

Effective joint estimation methods demonstrated on simulated data

Strategies successfully incorporate initial and boundary conditions

Methods validated with real-world data applications

Abstract

Differential equations (DEs) are commonly used to describe dynamic systems evolving in one (ordinary differential equations or ODEs) or in more than one dimensions (partial differential equations or PDEs). In real data applications the parameters involved in the DE models are usually unknown and need to be estimated from the available measurements together with the state function. In this paper, we present frequentist and Bayesian approaches for the joint estimation of the parameters and of the state functions involved in PDEs. We also propose two strategies to include differential (initial and/or boundary) conditions in the estimation procedure. We evaluate the performances of the proposed strategy on simulated and real data applications.