Finite Element Neural Network Interpolation. Part II: Hybridisation with the Proper Generalised Decomposition for non-linear surrogate modelling

Abstract

This work introduces a hybrid approach that combines the Proper Generalised Decomposition (PGD) with deep learning techniques to provide real-time solutions for parametrised mechanics problems. By relying on a tensor decomposition, the proposed method addresses the curse of dimensionality in parametric computations, enabling efficient handling of high-dimensional problems across multiple physics and configurations. Each mode in the tensor decomposition is generated by a sparse neural network within the Finite Element Neural Network Interpolation (FENNI) framework presented in Part I, where network parameters are constrained to replicate the classical shape functions used in the Finite Element Method. This constraint enhances the interpretability of the model, facilitating transfer learning, which improves significantly the robustness and cost of the training process. The FENNI framework also enables finding the optimal spatial and parametric discretisation dynamically during training, which accounts to optimising the model's architecture on the fly. This hybrid framework offers a flexible and interpretable solution for real-time surrogate modelling. We highlight the efficiency of the FENNI-PGD approach through 1D and 2D benchmark problems, validating its performance against analytical and numerical reference solutions. The framework is illustrated through linear and non-linear elasticity problems, showing the flexibility of the method in terms of changes in physics.