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

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.

Knowledge-Based Convolutional Neural Network for the Simulation and Prediction of Two-Phase Darcy Flows

Physics-informed neural networks (PINNs) have gained significant prominence as a powerful tool in the field of scientific computing and simulations. Their ability to seamlessly integrate physical principles into deep learning architectures has revolutionized the approaches to solving complex problems in physics and engineering. However, a persistent challenge faced by mainstream PINNs lies in their handling of discontinuous input data, leading to inaccuracies in predictions. This study addresses these challenges by incorporating the discretized forms of the governing equations into the PINN framework. We propose to combine the power of neural networks with the dynamics imposed by the discretized differential equations. By discretizing the governing equations, the PINN learns to account for the discontinuities and accurately capture the underlying relationships between inputs and outputs, improving the accuracy compared to traditional interpolation techniques. Moreover, by leveraging the power of neural networks, the computational cost associated with numerical simulations is substantially reduced. We evaluate our model on a large-scale dataset for the prediction of pressure and saturation fields demonstrating high accuracies compared to non-physically aware models.

Knowledge-based Deep Learning for Modeling Chaotic Systems

Deep Learning has received increased attention due to its unbeatable success in many fields, such as computer vision, natural language processing, recommendation systems, and most recently in simulating multiphysics problems and predicting nonlinear dynamical systems. However, modeling and forecasting the dynamics of chaotic systems remains an open research problem since training deep learning models requires big data, which is not always available in many cases. Such deep learners can be trained from additional information obtained from simulated results and by enforcing the physical laws of the chaotic systems. This paper considers extreme events and their dynamics and proposes elegant models based on deep neural networks, called knowledge-based deep learning (KDL). Our proposed KDL can learn the complex patterns governing chaotic systems by jointly training on real and simulated data directly from the dynamics and their differential equations. This knowledge is transferred to model and forecast real-world chaotic events exhibiting extreme behavior. We validate the efficiency of our model by assessing it on three real-world benchmark datasets: El Nino sea surface temperature, San Juan Dengue viral infection, and Bj{\o}rn{\o}ya daily precipitation, all governed by extreme events' dynamics. Using prior knowledge of extreme events and physics-based loss functions to lead the neural network learning, we ensure physically consistent, generalizable, and accurate forecasting, even in a small data regime.