A finite elements - neural network approach for the numerical approximation of parametric partial differential equations (PDEs) is presented. Data are generated by solving the PDE for many values of the parameter, with an adaptive finite element method. They are then used to train a neural network in a data-driven (supervised) setting. The objective is to assess the overall error of the method in both the physical and parameter domains, and possibly to ensure that it is below some preset tolerance. For this purpose, the error coming from the finite elements is assessed using a Monte-Carlo method together with a posteriori error estimators, while the error due to the neural network surrogate is approximated with a Monte-Carlo algorithm.

The method is initially tested on two model problems, namely an elliptic problem and a stationary Stefan problem. In the two cases, an a posteriori error estimator for the finite element method is first derived. In a second stage, the error estimator is used to generate training data with adaptive finite elements. This implies that the mesh is different for each parameter and allows to control the error of the numerical method that is embedded in the training set, uniformly across the parameter space. The overall error between the exact solution of the PDE and the neural network approximation is finally assessed.

The method is ultimately applied to a laser polishing process, which couples a Stefan problem together with Navier-Stokes equations and the simulation of a free surface, the shape of which is driven by surface tension. For this industrial application, two different neural networks are used in order to predict the evolution of the free surface and of the temperature during the process. This also illustrates the flexibility of the approach.