Metriplectic systems are learned from data in a way that scales quadratically in both the size of the state and the rank of the metriplectic operators. In addition to being provably energy-conserving and entropy-stable, the proposed neural metriplectic systems (NMS) approach includes approximation results that demonstrate its ability to accurately learn metriplectic dynamics from data, along with an error estimate that indicates its potential for generalization to unseen timescales when the approximation error is low. Examples are provided to illustrate performance both with full state information available and when entropic variables are unknown, confirming that the NMS approach exhibits superior accuracy and scalability without compromising on model expressivity.

Transformers, renowned for their self-attention mechanism, have achieved state-of-the-art performance across various tasks in natural language processing, computer vision, time-series modeling, etc. However, one of the challenges with deep Transformer models is the oversmoothing problem, where representations across layers converge to indistinguishable values, leading to significant performance degradation. We interpret the original self-attention as a simple graph filter and redesign it from a graph signal processing (GSP) perspective. We propose a graph-filter-based self-attention (GFSA) to learn a general yet effective one, whose complexity, however, is slightly larger than that of the original self-attention mechanism. We demonstrate that GFSA improves the performance of Transformers in various fields, including computer vision, natural language processing, graph-level tasks, speech recognition, and code classification.

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 \url{https://github.com/natrask/BracketGraphs}.

Forecasting of time-series data requires imposition of inductive biases to obtain predictive extrapolation, and recent works have imposed Hamiltonian/Lagrangian form to preserve structure for systems with \emph{reversible} dynamics. In this work we present a novel parameterization of dissipative brackets from metriplectic dynamical systems appropriate for learning \emph{irreversible} dynamics with unknown a priori model form. The process learns generalized Casimirs for energy and entropy guaranteed to be conserved and nondecreasing, respectively. Furthermore, for the case of added thermal noise, we guarantee exact preservation of a fluctuation-dissipation theorem, ensuring thermodynamic consistency. We provide benchmarks for dissipative systems demonstrating learned dynamics are more robust and generalize better than either "black-box" or penalty-based approaches.