Learning continuous models for continuous physics

TL;DR

This paper introduces a convergence test based on numerical analysis to verify if machine learning models accurately learn continuous dynamics in dynamical systems, improving their scientific utility.

Contribution

It develops a novel convergence test that assesses whether ML models capture true continuous dynamics, bridging numerical analysis and machine learning validation.

Findings

Models passing the test better interpolate and extrapolate dynamics.

The test helps identify models that fail to capture underlying continuous behavior.

Couples numerical analysis with ML training/testing for scientific validation.

Abstract

Dynamical systems that evolve continuously over time are ubiquitous throughout science and engineering. Machine learning (ML) provides data-driven approaches to model and predict the dynamics of such systems. A core issue with this approach is that ML models are typically trained on discrete data, using ML methodologies that are not aware of underlying continuity properties. This results in models that often do not capture any underlying continuous dynamics -- either of the system of interest, or indeed of any related system. To address this challenge, we develop a convergence test based on numerical analysis theory. Our test verifies whether a model has learned a function that accurately approximates an underlying continuous dynamics. Models that fail this test fail to capture relevant dynamics, rendering them of limited utility for many scientific prediction tasks; while models that pass this test enable both better interpolation and better extrapolation in multiple ways. Our results illustrate how principled numerical analysis methods can be coupled with existing ML training/testing methodologies to validate models for science and engineering applications.