Graph neural networks informed locally by thermodynamics

Computational simulation is a discipline that has been around for 80 years or so and that has emerged as a cornerstone tool across various scientific disciplines, facilitating the prediction of physical phenomena and enabling engineers to refine designs before costly experimental setups are pursued. Traditionally, these simulations have relied heavily on mathematical formulations, often expressed through partial differential equations (PDEs), to model complex systems in fields such as structural mechanics or fluid dynamics [1]. However, with the advent of the information era--the so-called fourth paradigm of science [2]--, a shift towards data-driven approaches, particularly deep learning algorithms, has garnered attention due to their ability to address the limitations of traditional methods, including handling nonlinear dynamics under real-time restrictions [3]. Deep learning algorithms, while powerful, are often computationally demanding and require extensive datasets, posing challenges in terms of scalability and generalization [4]. To address these challenges, recent research has explored novel architectures, such as geometric deep learning, which leverage problem structures to enhance performance and reduce data consumption [5-8]. This paradigm shift, which imposes specific constraints related to problem symmetries, has opened new avenues for learning from irregular and unstructured data representations, such as graphs [9]. At the same time, traditional mesh-based representations have long been favoured in modelling complex physical systems, offering adaptability and accuracy across various domains, from aerodynamics [10] to structural mechanics [11]. Despite their advantages, mesh representations have received relatively little attention in the realm of machine learning, where grid-based approaches dominate due to their compatibility with convolutional neural network (CNN) architectures [12]. Nonetheless, recent efforts have explored the potential of adaptive mesh representations in predicting the dynamics of physical systems, showcasing their ability to allocate computational resources optimally and adaptively change discretization during simulations [13].