Search for: All records

Recent works have shown that physics-inspired architectures allow the training of deep graph neural networks (GNNs) without oversmoothing. The role of these physics is unclear, however, with successful examples of both reversible (e.g., Hamiltonian) and irreversible (e.g., diffusion) phenomena producing comparable results despite diametrically opposed mechanisms, and further complications arising due to empirical departures from mathematical theory. This work presents a series of novel GNN architectures based upon structure preserving bracket-based dynamical systems, which are provably guaranteed to either conserve energy or generate positive dissipation with increasing depth. It is shown that the theoretically principled framework employed here allows for inherently explainable constructions, which contextualize departures from theory in current architectures and better elucidate the roles of reversibility and irreversibility in network performance. Code is available at the Github repository https://github.com/natrask/BracketGraphs. 

The unsigned p-Willmore functional introduced in the work of Mondino [2011] generalizes important geometric functionals, which measure the area and Willmore energy of immersed surfaces. Presently, techniques from the work of Dziuk [2008] are adapted to compute the first variation of this functional as a weak-form system of equations, which are subsequently used to develop a model for the p-Willmore flow of closed surfaces in R 3 . This model is amenable to constraints on surface area and enclosed volume and is shown to decrease the p-Willmore energy monotonically. In addition, a penalty-based regularization procedure is formulated to prevent artificial mesh degeneration along the flow; inspired by a conformality condition derived in the work of Kamberov et al. [1996], this procedure encourages angle-preservation in a closed and oriented surface immersion as it evolves. Following this, a finite-element discretization of both procedures is discussed, an algorithm for running the flow is given, and an application to mesh editing is presented.