TG-PhyNN: An Enhanced Physically-Aware Graph Neural Network framework for forecasting Spatio-Temporal Data

TL;DR

TG-PhyNN tackles the challenge of forecasting spatio-temporal processes on graphs by embedding physical laws into GNN training through a physics-informed loss $L = \lambda_1 Loss_{data} + \lambda_2 Loss_{Phy}$, and by using a two-step prediction strategy to estimate discrete temporal and spatial derivatives with $\frac{df}{dt} \approx \frac{f(t+\delta t) - f(t)}{\delta t}$ and $\frac{\partial f}{\partial x_i} \approx \frac{1}{N_i} \sum_{j \in \mathcal{N}(i)} (f(i) - f(j))$. The framework demonstrates improved forecasting accuracy over traditional baselines (e.g., GRU, LSTM, GAT, GCN) on real-world datasets such as PedalMe, England Covid, and Chickenpox by leveraging PDE-informed constraints, while offering scalable and flexible integration with sparse GNNs. Results indicate strong gains for several TG-PhyNN variants, though some dataset-specific failures (notably with TG-PhyNN-enhanced GCN on certain tasks) motivate future hybrid approaches that can handle non-periodic, stochastic, and chaotic dynamics. The work highlights the practical potential of physics-informed graph modeling for real-world forecasting and points to future enhancements via Fourier Neural Operators or wavelet-transform-based architectures to better capture multi-scale and seasonal effects.

Abstract

Accurately forecasting dynamic processes on graphs, such as traffic flow or disease spread, remains a challenge. While Graph Neural Networks (GNNs) excel at modeling and forecasting spatio-temporal data, they often lack the ability to directly incorporate underlying physical laws. This work presents TG-PhyNN, a novel Temporal Graph Physics-Informed Neural Network framework. TG-PhyNN leverages the power of GNNs for graph-based modeling while simultaneously incorporating physical constraints as a guiding principle during training. This is achieved through a two-step prediction strategy that enables the calculation of physical equation derivatives within the GNN architecture. Our findings demonstrate that TG-PhyNN significantly outperforms traditional forecasting models (e.g., GRU, LSTM, GAT) on real-world spatio-temporal datasets like PedalMe (traffic flow), COVID-19 spread, and Chickenpox outbreaks. These datasets are all governed by well-defined physical principles, which TG-PhyNN effectively exploits to offer more reliable and accurate forecasts in various domains where physical processes govern the dynamics of data. This paves the way for improved forecasting in areas like traffic flow prediction, disease outbreak prediction, and potentially other fields where physics plays a crucial role.

Figures (2)

• Figure 1: Overview of TG-PhyNN Architecture. A GNN is trained to forecast the graph at timesteps t+1 and t+2, which are used to compute the derivatives forming the physical loss. These same predictions are compared to the real values, constituting the labeled loss. The total loss is a linear combination of both losses, which is backpropagated and minimized by the optimizer to enhance the forecasting capabilities.
• Figure 2: Comparison of average ranking performance across various datasets for base models and models enhanced with TG-PhyNN. The graph illustrates the performance enhancement achieved by incorporating TG-PhyNN into different models such as LSTM, GAT, GRU, GCN, and EvolveGCNH. Each model's performance is ranked based on its predictive accuracy based on both MAE and MSE combining all three datasets, with lower values indicating superior performance.