Physics-encoded Spatio-temporal Regression

TL;DR

This paper introduces a physics-encoded regression model for spatio-temporal data that leverages PDE structures for interpretability, efficiency, and optimal convergence, demonstrated through simulations and real data application.

Contribution

It presents a novel physics-encoded regression approach that incorporates PDE structures, improving interpretability and efficiency over penalty-based methods.

Findings

Achieves minimax optimal convergence rate.

Demonstrates superior performance in simulations.

Successfully applied to fluorescence recovery data.

Abstract

Physics-informed methods have gained a great success in analyzing data with partial differential equation (PDE) constraints, which are ubiquitous when modeling dynamical systems. Different from the common penalty-based approach, this work promotes adherence to the underlying physical mechanism that facilitates statistical procedures. The motivating application concerns modeling fluorescence recovery after photobleaching, which is used for characterization of diffusion processes. We propose a physics-encoded regression model for handling spatio-temporally distributed data, which enables principled interpretability, parsimonious computation and efficient estimation by exploiting the structure of solutions of a governing evolution equation. The rate of convergence attaining the minimax optimality is theoretically demonstrated, generalizing the result obtained for the spatial regression. We conduct simulation studies to assess the performance of our proposed estimator and illustrate its usage in the aforementioned real data example.