Feedforward neural nets as discretization schemes for ODEs and DAEs

Because of their neurophysical origin neural nets can be studied for classification tasks, approximation properties or iterative algorithms. They can be interpreted as a distributed or massively parallel computer, where each unit accumulates a scalar product and computers an one-dimensional nonlinear activation function. A supervised learning strategy defines a nonlinear least-squares problem, which is solved by gradient techniques like backpropagation. Interpreting the weights in a net as state variables, feedforward neural nets can be designed as numerical discretization schemes for ODEs or DAEs. We present the net architecture for the implicit Euler scheme and solve some test examples numerically. The net approach is especially of interest for the overdetermined index-3 approach of DAEs from multibody system dynamics. In general, these nets define a parallel shooting-type algorithm. Its merits are in real-time applications, since a hardware realization of the net is possible.