Comparative Analysis of Graph Neural Networks for Dynamics Prediction

Thesis Abstract

Graph neural networks (GNNs) have emerged as powerful tools for modeling complex dynamical systems by exploiting relational structures in data. Despite rapid performance improvements, a systematic comparative understanding of how different GNN architectures (e.g., Graph Convolutional Networks, Graph Attention Networks, and Message Passing Neural Networks) fare across diverse dynamical regimes remains underexplored. This study addresses the problem by evaluating the predictive accuracy, stability, and generalization of leading GNN architectures for dynamics prediction in multi-agent and physical systems, where interdependencies evolve over time. The aim is to delineate the conditions under which specific GNN variants outperform others and to identify architectural principles that enhance long-horizon forecasting and robustness to noise. The objectives are to (i) benchmark five representative GNN models on standardized dynamical datasets, (ii) assess sensitivity to graph construction choices, (iii) analyze temporal evolution through sequence-to-sequence and autoregressive setups, (iv) evaluate robustness to observation noise and missing edges, and (v) synthesize design guidelines grounded in theoretical and empirical evidence. The study employs a comparative, cross-sectional research design using both synthetic and real-world dynamical datasets. Populations consist of systems described by continuous-time and discrete-time dynamics, including coupled Kuramoto oscillator networks, N-body gravitational simulations, and traffic flow datasets. A sample of 3000 time-series trajectories is generated from synthetic models with varying coupling strengths and topologies, complemented by 12 real-world networks of social-transport and biological interactions. Data collection instruments comprise open-source simulation engines (e.g., Numba-accelerated integrators for Kuramoto and N-body systems) and publicly available traffic and mobility datasets, alongside preprocessed adjacency matrices and feature vectors capturing state variables and derived quantities such as velocity, acceleration, and phase. Instruments are validated through replication of known dynamical invariants and alignment with ground-truth trajectories. Analytical methods encompass a two-tier evaluation framework. First, predictive performance is quantified using root mean squared error (RMSE), mean absolute error (MAE), and log-likelihood for probabilistic forecasts, with a focus on one-step and multi-step horizons (horizon lengths of 5, 10, and 20 steps). Second, model interpretability and robustness are examined through ablation studies, edge-dropping experiments, and noise injection (Gaussian and impulsive disturbances). Statistical comparisons across architectures employ repeated-measures ANOVA and pairwise post-hoc tests with Holm correction. Temporal dynamics are analyzed via sequence-to-sequence modeling and autoregressive decoding with teacher forcing; stability is assessed using Lyapunov-exponent-inspired metrics and forecast drift analyses. Theoretical underpinnings draw on the dynamical systems perspective and graph signal processing, with the theoretical framework guided by two core theories (i) the Dynamical Graph Representation Theorem, which formalizes how graph topology modulates state evolution under graph-based updates, and (ii) the Attention-Driven Information Flow theory, which explains how attention mechanisms modulate information propagation in temporal graphs. Empirical findings are triangulated with these theories to interpret performance differentials. Key expected findings include that Graph Attention Networks (GAT) with residual connections exhibit superior long-horizon predictive accuracy in densely connected networks, while Graph Convolutional Networks (GCN) with adaptive edge weights excel in sparse, highly dynamic topologies. Message Passing Neural Networks (MPNN) are anticipated to offer robust performance across a range of noise levels, given their flexible aggregation schemes. The study aims to demonstrate that architecture-environment congruence—specifically, the alignment of message-passing depth with intrinsic dynamical timescales—significantly influences forecast quality and stability. The contribution to knowledge includes a comprehensive, systematized comparison of GNN variants for dynamics prediction, principled guidance on graph construction and temporal modeling choices, and evidence-based design heuristics for practitioners deploying GNNs in dynamical regimes with limited data and noise. The principal conclusion is that no single GNN architecture universally outperforms others across all dynamical contexts; instead, performance hinges on topology, dynamical timescales, and data quality. Recommendations advocate matched architecture selection to specific dynamical properties, incorporation of residual and attention mechanisms to balance depth and stability, and adoption of robust evaluation protocols that emphasize horizon-aware performance and noise robustness. Further research directions propose adaptive architecture search for dynamical graphs and exploration of hybrid models that combine the strengths of GCN, GAT, and MPNN paradigms.