Numerical simulation of industrial welding processes is of paramount importance for predicting temperature fields, residual stresses, microstructures, and other factors that impact the dimensional accuracy of parts and manufacturing quality. Classical numerical methods, specifically the Finite Element Method (FEM), have fulfilled this purpose and reliable numerical tools currently exist. However, the computational cost of these simulations can prove prohibitive for real-time applications such as Digital Twins, inverse problem resolution, or simply when dealing with design spaces too vast to explore in depth. Recently, Deep Learning techniques have emerged to accelerate these simulations, yet ensuring these models incorporate physical rigor (preventing violations of conservation principles when applied to scenarios unseen during training) remains a significant challenge. In this work, we propose the use of MeshGraphNets, deep learning architectures tailored for graph-structured data and notably effective for computational mechanics, augmented with the GENERIC formalism. This formalism introduces a structural bias that restricts the learning process to the identification of the discrete algebraic operators required to satisfy the GENERIC bracket formulation. This ensures that the predicted dynamic evolution remains thermodynamically consistent, strictly conserving energy and enforcing non-negative entropy production. Our proposal focuses exclusively on the nonlinear thermal problem, employing a Goldak double-ellipsoid heat source model. Once trained, the model can generate rollouts at inference speed for domain geometries, technological and material parameters, boundary conditions, and even source trajectories unseen during training. This opens new possibilities for integrating these simulations to accelerate complex industrial workflows.