Combining Finite Element Methods and Neural Networks to Solve Elliptic Problems on 2D Geometries
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The broad goal of this work is the real-time creation of digital twins of organs (such as the liver). To this end, we present a preliminary study combining two approaches in the context of PDE solving: the classical finite element method (FEM) and more recent techniques based on neural networks. Indeed, in recent years, Physics-Informed Neural Networks (PINNs) have become particularly interesting for quickly solving such problems, especially in large dimensions. However, their lack of precision is a major drawback in this context, hence the interest in combining them with FEM, for which error estimators are already known. This combination will make it possible to correct and certify the prediction of neural networks in order to obtain a fast and accurate solution. The complete pipeline proposed here then consists of modifying the classical approximation spaces in FEM by taking the information of a prior, chosen here as the prediction of a PINNs. This preliminary work focuses on parametric problems with two-dimensional geometries, with the aim of moving towards medical simulations. Current results show that pre-processing the problem using neural networks can achieve fixed error targets with coarser meshes than in standard finite element methods, thus saving time in computing the solution. Error estimates have been proven showing that enriched spaces outperform classical ones by a factor that depends only on the quality of the prior.
