Weak Form Scientific Machine Learning: Test Function Construction for System Identification

TL;DR

This paper introduces a novel data-driven method for constructing test functions in Weak form Scientific Machine Learning, improving noise robustness and computational efficiency in system identification without requiring model parameters.

Contribution

It proposes a new approach for constructing reference test functions that minimizes integration error, enhancing WSciML performance and efficiency.

Findings

Supports align with regions of minimal parameter error

Method outperforms previous multi-scale global strategies

Demonstrates robustness across models and noise levels

Abstract

Weak form Scientific Machine Learning (WSciML) is a recently developed framework for data-driven modeling and scientific discovery. It leverages the weak form of equation error residuals to provide enhanced noise robustness in system identification via convolving model equations with test functions, reformulating the problem to avoid direct differentiation of data. The performance, however, relies on wisely choosing a set of compactly supported test functions. In this work, we mathematically motivate a novel data-driven method for constructing Single-scale-Local reference functions for creating the set of test functions. Our approach numerically approximates the integration error introduced by the quadrature and identifies the support size for which the error is minimal, without requiring access to the model parameter values. Through numerical experiments across various models, noise levels, and temporal resolutions, we demonstrate that the selected supports consistently align with regions of minimal parameter estimation error. We also compare the proposed method against the strategy for constructing Multi-scale-Global (and orthogonal) test functions introduced in our prior work, demonstrating the improved computational efficiency.