turkmath.org

When approximating numerically a mathematical model one often faces the need for many solves. This happens for example in optimisation, or when solutions need to be given depending on a parameter. In these situations, fine approximations are not affordable for all cases, but perhaps just a few, and coarse approximations need to be employed in most simulations. The idea of the methodology to be presented is to improve coarse solutions from the knowledge of fine solutions. This is in general convenient, and in some cases even necessary. Examples of the latter are those in which the coarse solution is unstable, or fails to satisfy basic physical principles (for example, equilibrium). What we propose in this talk is to introduce a correction of the coarse model, depending on the coarse solution, designed to obtain a coarse solution as close as possible to the (projection of the) fine solution for the situations (that we call configurations) in which the fine solution is known. This corrective term is designed using an Artificial Neural Network, having as training set the collection of fine solutions for the configurations in which these are available. We have applied this concept to different problems: a) In Reduced Order Modelling (ROM), where the coarse model is built from a reduced basis and the training set are the collection of snapshots of departure, b) In the coarsening of finite element meshes in space, in which a fine solution is known in a fine mesh but then this mesh is coarsened to continue the simulation, c) In increasing the time step size in transient problem, in particular in wave propagation. Nevertheless, many other applications can be devised for the general concept proposed.