A neural network framework for reduced order modelling of non-linear hyperbolic equations in computational fluid dynamics

Reduced Order Modelling (ROM) found widespread application in numerical modelling at both industrial and academic level; its popularity is due primarily to the capability of those algorithms to retrieve the fundamental dynamics of a differential problem by reduction of the dimensionality of the parametric manifold that characterises the solution. In Computational Fluid Dynamics these methods are of particular importance since the parametric dependence of the models that are treated numerically is embedded in the engineering design process itself (e.g. shape optimisation). There is a particular class of problems however that pose a challenge to the effectiveness of ROM over the full-order simulations and those are the hyperbolic partial differential equations; there is in fact a sort of contradiction when it comes to time-dependent, parametric, transport equation and that is while at full order these models are easily handled, at least in CFD, with accurate, stable and robust methods, the low-rank representation is not straight-forward and difficult to retrieve with desired accuracy. Many efforts have been devised to overcome the difficulties brought by these models in the construction of the low dimensional manifold and in particular one was proposed in 2018 by Reiss et. al. called the sPOD (shifted Proper Orthogonal Decomposition) which features great scalability. In this work we devised a statistical learning framework that extends the sPOD to non-linear hyperbolic PDEs; we approached the problem of detecting the correct transformation of the full-order solution in a non-intrusive, data-driven fashion by implementing a neural network architecture within the ROM itself thereby generalising the procedure of shifting to the initial condition to any sort of transport fields and in particular to non-linear ones that are the solution of the Navier-Stokes equations. Once built and tested against simple, $2-$dimensional linear test cases of hyperbolic models, the resulting algorithm, called NNsPOD (Neural Network shifted Proper Orthogonal Decomposition) has been applied to a particularly challenging non-linear problem in CFD, the multiphase problem, in which a passive scalar field (the phase volume fraction) is transported across the computational grid by a field that is itself a solution of the incompressible Navier-Stokes. The results proved that NNsPOD reaches the goal of generalisation of ROM by shifted proper orthogonal decomposition for non-linear hyperbolic PDEs.