Modeling stochastic dynamics of manufacturing processes with manifold signals: A harmonic analysis approach with NP-ODEs

Abstract

Modern manufacturing increasingly leverages intricate signals that reflect the complex geometries of produced parts. These signals hold vital insights into the quality and working status of the underlying manufacturing systems. While conventional deep learning approaches excel with structured data like time series processing or image vision tasks, they can struggle to model these geometrically complex signals, which are inherently in a non-Euclidean manifold domain and exhibit stochastic dynamical evolution behaviors over time. In this paper, we present a novel methodology that leverages the principles of harmonic analysis with the potential of continuous-time dynamics modeling and stochastic process representation. Specifically, our contributions are twofold: (1) we present a general and flexible framework for modeling the stochastic dynamics of manifold signals within manufacturing processes. This framework uniquely synergizes Neural ODEs and Neural Processes (NPs), enhanced by a neural numerical integration scheme for computational efficiency; (2) we develop a rigorous and tailored representation approach rooted in harmonic analysis, enabling the construction of learnable wavelet filters for multiscale pattern analysis on manifold signals, with Geometric Deep Learning (GDL) principles ensuring compatibility with modern deep learning architectures. We comprehensively validate our methodology through simulations and a real-world automotive case study. Results demonstrate its effectiveness in modeling the complex stochastic dynamics inherent in manifold signals found in practical manufacturing settings, highlighting its potential for industrial applications.