AugRelNet: Relation-Augmented Dynamics for Structure Discovery and Forecasting from Limited Data

Complex dynamics often arise from persistent interactions whose effects vary with the system state. We introduce AugRelNet for short- and medium-horizon trajectory forecasting from limited data. A shared learnable relation matrix explicitly routes source signals to targets, while evolving local features $u$ and a persistent response state $\eta$ transform them into nonlinear, history-dependent responses. Feeding these responses back into the dynamics separates shared relational organization from its changing realization. The learned relations reflect dense coupling in Lotka–Volterra systems and develop structured, neighbour-preferential organization in locally interacting systems. On 40-dimensional Lorenz–96, AugRelNet uses about 4,000 training states, roughly an order of magnitude fewer than representative prior Lorenz–96 forecasting studies. Across five seeds, every true interaction ranks above every nonedge, with a mean absolute-weight contrast of approximately $146:1$. Under the same data budget, eight-step RMSE is about 23–59× lower than MNO, DySLIM, DeepSkip, and AL-RNN, 14–55× lower than NRI and dense GraphODE, and 37% lower than PySINDy with a matched quadratic dictionary. The same relational representation extends to grids by treating sites as variables and sharing relations across spatial offsets: in discrete (Game of Life), continuous (FitzHugh–Nagumo), and stochastic (Ising) dynamics, learned weights rank physical neighbours above other candidate offsets while supporting multi-step prediction. AugRelNet learns these relations through prediction losses and regularization, without structural labels or system-specific physical losses.