Well-conditioned AI-assisted sub-matrix selection for numerically stable constrained form-finding of reticulated shells using geometric deep Q-learning

Abstract

The selection of well-conditioned sub-matrices is a critical concern in problems across multiple disciplines, particularly those demanding robust numerical stability. This research introduces an innovative, AI-assisted approach to sub-matrix selection, aimed at enhancing the form-finding of reticulated shell structures under the xy-constrained Force Density Method (also known as Thrust Network Analysis), using independent edge sets. The goal is to select a well-conditioned sub-matrix within a larger matrix with an inherent graph interpretation where each column represents an edge in the corresponding graph. The selection of ill-conditioned edges poses a significant challenge because it can render large segments of the parameter space numerically unstable, leading to numerical sensitivities that may impede design exploration and optimisation. By improving the selection of edges, the research assists in computing a pseudo-inverse for a critical sub-problem in structural form-finding, thereby enhancing numerical stability. Central to the selection strategy is a novel combination of deep reinforcement learning based on Deep Q-Networks and geometric deep learning based on CW Network. The proposed framework, which generalises across a trans-topological design space encompassing patterns of varying sizes and connectivity, offers a robust strategy that effectively identifies better-conditioned independent edges leading to improved optimisation routines with the potential to be extended for sub-matrix selection problems with graph interpretations in other domains.