The conservation of cultural heritage represents a task in which novel technologies can provide crucial support. In this context, identifying a framework to monitor and maintain cultural assets is a challenge involving several factors in which multi-scale problems play a significant role. This work aims to introduce a comprehensive architecture to monitor, analyze, and improve the maintenance of cultural heritage. Such an integrated framework allows the user to define protocols interacting with digital replicas of the cultural assets, analyze problems inherent to their degradation, and provide reliable simulations to support experts in the field of restoration for predictive maintenance tasks. Our framework exploits a Scientific Machine Learning approach integrating the physical knowledge of the problem with data acquired through sensors. Specifically, the framework employs Physics-Informed Neural Networks to solve parametrized Partial Differential Equations (PDEs). Within the same environment, we can process both direct problems, combining physics-based and data-driven approaches, and inverse problems in a Reduced Order Models (ROMs) workflow, when there is no prior knowledge of the system's parameter. The framework takes advantage of the interconnections with the digital replica of the cultural assets to sample points in the PINN context, or to generate meshes for model order reduction. Finally, the validation of the proposed framework exploits several simulated scenarios for digital replicas subject to different parametrized PDEs.