# Ludvig Eidmann: Error estimation for neural network approximations of convection problems

**Time: **
Wed 2022-06-08 14.00 - 14.30

**Location: **
KTH Lindstedsvägen 25, Room 3424

**Respondent: **
Ludvig Eidmann

**Supervisor: **
Jonas Kiessling (H-Ai AB) and Anders Szepessy

**Abstract:**

The problem of approximating a solution to the convection equation \(∂tu + f · ∇xu = h\) given data on the flux *f*, source function *h* and a final condition *g* is investigated. Specifically, two layer neural networks are used to approximate *f, h* and *g *and a solution is approximated using numerical integration. An upper bound to the expected square error of the approximated solution is derived which is dependent on the number of parameters in the approximating neural networks. The dependency of the error is investigated via numerical experiments concerning both synthetic and real world wind data. The neural networks used in the numerical experiments are trained first by the algorithm Adaptive Metropolis-Hastings and then by the SGD-type algorithm Adam. The rate of convergence of the approximation error is shown to be in line with the derived bound when approximating a solution close in time to the final condition *g*. The error is shown to decrease slower than what the derived bound suggests when approximating far away in time from the final condition g.