PIANO: PATH-INTEGRAL ADAPTIVE NEURAL ODES ON GRAPHS

Many natural and engineered spatiotemporal processes evolve through continuous interactions over a network, while the spatial range of relevant interactions can vary with the system state. Existing graph ODEs typically rely on a predetermined hop structure within each vector-field evaluation, preventing the propagation mass assigned to different spatial orders from adapting with the evolving state. We introduce PIANO (Path-Integral Adaptive Neural ODE), whose instantaneous spatial operator is a state-adaptive path-integral propagator: an expected finite-step graph-walk propagator under a state-conditioned distribution over walk lengths. A lightweight topology-aware gate further enables node-wise modulation when structural roles are informative. The adaptive operator is implemented with sparse graph operations whose cost scales linearly with maximum propagation order, edge count, and feature dimension. We show that each realized spatial operator is non-expansive, establish well-posedness of the nonlinear flow, and, for a frozen linear surrogate, characterize how the maximum propagation order and integration horizon jointly determine a sufficient finite-hop approximation radius. Controlled experiments validate the proposed adaptive propagation and topology-aware mechanisms. On ocean and severe-weather forecasting, PIANO achieves strong performance with a compact model and further extends effectively to large-scale sparse graph learning.