Data-Driven Continuum Dynamics via Transport-Teleport Duality

TL;DR

This paper introduces a novel mathematical transform that simplifies the learning of transport phenomena by modeling the dynamics as discrete disappearance and reappearance events, reducing data and model complexity.

Contribution

It proposes a new transform-based approach that inherently incorporates conservation laws, enabling effective data-driven learning of transport dynamics without explicit governing equations.

Findings

Few observational data suffice to learn real-world transport dynamics.

The method reduces model complexity and training data requirements.

Applicable across various fields involving conserved quantities.

Abstract

In recent years, machine learning methods have been widely used to study physical systems that are challenging to solve with governing equations. Physicists and engineers are framing the data-driven paradigm as an alternative approach to physical sciences. In this paradigm change, the deep learning approach is playing a pivotal role. However, most learning architectures do not inherently incorporate conservation laws in the form of continuity equations, and they require dense data to learn the dynamics of conserved quantities. In this study, we introduce a clever mathematical transform to represent the classical dynamics as a point-wise process of disappearance and reappearance of a quantity, which dramatically reduces model complexity and training data for machine learning of transport phenomena. We demonstrate that just a few observational data and a simple learning model can be enough to learn the dynamics of real-world objects. The approach does not require the explicit use of governing equations and only depends on observation data. Because the continuity equation is a general equation that any conserved quantity should obey, the applicability may range from physical to social and medical sciences or any field where data are conserved quantities.