Geodynamical processes are important to understand and assess the evolution of the Earth system as well as its natural resources. Given the wide range of characteristic spatial and temporal scales of geodynamic processes, their analysis routinely relies on computer-assisted numerical simulations. To provide reliable predictions such simulations need to consider a wide range of potential input parameters, material properties as they vary in space and time, in order to address associated uncertainties. To obtain any quantifiable measure of these uncertainties is challenging both because of the high computational cost of the forward simulation and because data is typically limited to direct observations at the near surface and for the present day state. To account for both of these challenges, we present how to construct efficient and reliable surrogate models that are applicable to a wide range of geodynamic problems using a physics-based machine learning method. In this study, we apply our approach to the case study of the Alpine region, as a natural example for a complex geodynamic setting where several subduction slabs as imaged by tomographic methods interact below a heterogeneous lithosphere. We specifically develop surrogates for two sets of observables, topography and surface velocity, to provide models that can be used in probabilistic frameworks to validate the underlying model structure and parametrization. We additionally construct models for the deeper crustal and mantle domains of the model, to improve the system understanding. For this last family of models, we highlight different construction methods to develop models to either allow evaluations in the entirety of the 3D model or only at specific depth intervals.