This study presents a hybrid approach that integrates the Proper Generalised Decomposition (PGD) with deep learning techniques to develop real-time diagnostic and prognostic tools for clinicians, particularly in assessing stress fields in fibrotic lungs. For these tools to be clinically relevant, they must be personalised, which requires numerical techniques capable of real-time estimation of patient-specific mechanical parameters. The proposed method mitigates the curse of dimensionality inherent in parametric systems by employing a tensor decomposition. Within the Finite Element Neural Network Interpolation (FENNI) framework, based on the concept of HiDeNN, each mode of the tensor decomposition is represented by a sparse neural network with constrained weights and biases to replicate standard Finite Element Method (FEM) shape functions. This constraint enhances model interpretability and facilitates transfer learning, significantly accelerating the training process. Moreover, the model's architecture is directly determined by the number of nodes and the order of elements for interpolation, eliminating arbitrary choices and allowing mesh adaptation during the training stage. Similarly to PINNs, the physics of the problem is incorporated into the loss function during unsupervised training. The training process involves solving a minimisation problem, similar to classical model reduction. However, automatic differentiation within the neural network framework allows for greater flexibility in addressing non-linearities, particularly when linearisation is difficult. The framework, therefore, offers a flexible framework for surrogate modelling in non-linear mechanics, as demonstrated through 1D and 2D benchmark problems that validate its robustness against analytical and numerical reference solutions.