In the framework of physics-informed statistical models, this work proposes a multi-domain spatial regression method with a regularization term involving the Laplace-Beltrami operator, specifically designed for data observed over surface domains. To illustrate its application, we employ the proposed method on high-dimensional resting-state fMRI signals from various subjects, with the precuneus designated as the Region of Interest for computing the Functional Connectivity maps. Here, we treat the Functional Connectivity Map of each subject as the response variable, with available data on cortical thickness serving as the regressor.