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

2025-07-03

AI Summary

To address the lack of theoretical guidance and adaptivity in test function selection for weak-form scientific machine learning (WSciML) in system identification, this paper proposes a single-scale local reference test function construction method. Without requiring prior knowledge of model parameters, the method automatically optimizes the support domain size via numerical estimation of the integration error. Built upon the weak-form residual framework, it integrates data-driven support domain search with a convolutional discretization scheme, thereby avoiding noise-sensitive differentiation while enhancing robustness. Experiments demonstrate that the selected support domains align closely with those yielding minimal parameter estimation error; across diverse model structures, noise levels, and temporal resolutions, the proposed method consistently outperforms multi-scale global test function approaches. The core contribution lies in enabling adaptive, interpretable, and computationally efficient determination of test function support domains.

Technology Category

• Machine Learning: Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Web Mining and Content Analysis: Machine learning and data science for the Web
• Responsible Web: Machine-in-the-loop, human agency and autonomy

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.

Problem

• Constructing optimal test functions for weak form system identification
• Minimizing integration error without knowing model parameters
• Improving computational efficiency in test function selection

Innovation

• Uses weak form equation error residuals
• Constructs Single-scale-Local reference functions
• Minimizes integration error via optimal support