A hybrid approach with reduced order modelling and graph neural networks for non-parametric problems

Reduced Order Modelling (ROM), such as the Proper Generalized Decomposition (PGD) [1], offers accelerated solutions for partial differential equations (PDEs). However, their effectiveness often relies on parametrization, limiting applicability to fixed geometries geometries parametrizable by a couple of control points. To address this limitation, we propose integrating deep learning with ROM. Our method leverages the spatiotemporal separated variable representation of a PDE solution. We employ spatial modes from a PGD decomposition that offers a superior solution representation compared to eigenmodes of the structure [1]. A first guess of the spatial component (spatial modes) is obtained using Graph Neural Networks (GNN) [2], allowing us to handle diverse and non-parametric geometries. The temporal component (temporal modes) is derived by projecting the PDE onto spatial modes from GNN. This ensures solving the PDE at a reduced cost while maintaining physical fidelity and allows the addition of new corrections using a greedy algorithm if it appears to be necessary. As a representative example, we focus on the dimensioning of aircraft seats in fast dynamics. Using deep learning on a database, we reuse accurate alculations from various engineering projects. We tackle non-parametric challenges in variable geometries, taking into account data sparsity. The uncertainty of deep learning-based solutions impacts certification and industry acceptance. By combining deep learning with industrial solvers, we can respond to the industry’s hesitations in adopting AI-based approaches while simultaneously offering the possibility of tackling non-parametric problems.