Modeling interacting dynamical systems requires capturing spatial interactions alongside long-range temporal dependencies. Graph neural networks (GNNs) provide a natural representation but typically rely on autoregressive rollouts and treat spatial and temporal dynamics separately, leading to error accumulation over long horizons. Existing approaches also focus on local interactions and short temporal contexts, limiting their ability to capture multi-hop dependencies and global structure. We introduce the Graph Mamba Operator (GraMO), a latent-space simulator that integrates state-space models with graph-based interaction learning. In contrast to prior work that sequences nodes or applies spatial and temporal updates in separate stages, GraMO couples graph-based interactions and temporal state updates within a single recurrence. The update is linear in the latent state, with input-dependent coefficients that adapt across regimes. We evaluate GraMO on N-body systems, motion capture, and robotics datasets, achieving the lowest error across benchmarks and the largest gains in long-horizon prediction.
Lattice Hamiltonian systems underpin models across condensed matter, nonlinear optics, and biophysics, yet learning their dynamics from data is obstructed by two unknowns: the interaction topology and whether node dynamics are homogeneous. Existing graph-based approaches either assume the graph is given or, as in α-separable graph Hamiltonian network, infer it only for separable Hamiltonians with homogeneous node dynamics. We introduce the Hamiltonian Graph Inference Network (HGIN), which jointly recovers the interaction graph and predicts long-time trajectories from state data alone, for both separable and non-separable Hamiltonians and under heterogeneous node dynamics. HGIN couples a structure-learning module -- a learnable weighted adjacency matrix trained under a Hamilton's-equations loss -- with a trajectory-prediction module that partitions edges into physically distinct subgraphs via k-means clustering, assigning each subgraph its own encoder and thereby breaking the parameter-sharing bottleneck of conventional GNNs. On three benchmarks -- a Klein--Gordon lattice with long-range interactions and two discrete nonlinear Schrödinger lattices (homogeneous and heterogeneous) -- HGIN reduces long-time energy prediction error and trajectory prediction error by six to thirteen orders of magnitude relative to baselines. A symmetry argument on the Hamiltonian loss further shows that the learned weights encode the parity of the underlying pair potential, yielding an interpretable readout of the system's interaction structure.
Continuous-time dynamic graphs (CTDGs) provide a richer framework to capture fine-grained temporal patterns in evolving relational data. Long-range information propagation is a key challenge while learning representations, wherein it is important to retain and update information over long temporal horizons. Existing approaches restrict models to capture one-hop or local temporal neighborhoods and fail to capture multi-hop or global structural patterns. To mitigate this, we derive a parameter-efficient state-space modeling framework for continuous-time dynamic graphs (CTDG-SSM) from first principles. We first introduce continuous-time Topology-Aware higher order polynomial projection operator (CTT-HiPPO), a novel memory-based reformulation of HiPPO to jointly encode temporal dynamics and graph structure. The solution from CTT-HiPPO is obtained by projecting the classical HiPPO solution through a polynomial of the Laplacian matrix, yielding topology-aware memory updates that admit an equivalent state-space formulation for CTDGs (CTDG-SSM). Then a computationally efficient discrete formulation is obtained using the zero-order hold approach for model implementation. Across benchmarks on dynamic link prediction, dynamic node classification, and sequence classification, CTDG-SSM achieves state-of-the-art performance. Notably, it achieves large performance gains on datasets that require long range temporal (LRT) and spatial reasoning.
In this work, we present COGENT, a continuous graph emulator with Neural Ordinary Differential Equations for long-term physical forecasting on irregular geospatial meshes. COGENT encodes a finite history of system states and associated forcing fields and external forcings with a graph-based history encoder, producing node-wise context vectors that capture both local spatial interactions and temporal evolution. These context vectors initialize and condition a latent Neural Ordinary Differential Equation whose dynamics are driven by interpolated future forcings and explicit relative rollout time. By modeling the forecast trajectory as a continuous latent dynamical system, COGENT can generate predictions at arbitrary future times rather than being restricted to a fixed temporal discretization. A residual decoder maps the resulting latent trajectories back to future physical states, enabling direct multi-step forecasting without repeatedly feeding predicted states back into the model. This formulation combines graph-based spatial representation, history-conditioned latent dynamics, and continuous-time rollout in a unified framework for mesh-based physical simulation emulation. In order to stabilize training with long-horizon supervision, we also propose effective rollout-horizon sampling and a progressive rollout-horizon scheduling strategy. We evaluate COGENT on transient ice-sheet simulations generated by the Ice-sheet and Sea-level System Model, demonstrating improved long-range stability over autoregressive graph baselines. These results suggest that continuous graph Neural ODEs provide a promising methodology for scalable physical forecasting on irregular geospatial meshes, particularly in applications that require stable long-horizon predictions and the ability to query system states at arbitrary times.