1. Classification of System Regimes and Learning of Controller Selectors

The control of nonlinear systems is a challenging task in particular in high dimensions. The approach of "gain scheduling" provides a general solution by combining linear control laws adapted to the current regime of the system. A numerical realization, however, is only possible for a moderate number of reference regimes or, in other words, for a parametrization of the dynamics by a moderate number of scheduling variables. The purpose of our research is to develop a scheduling dimension reduction method using a clustering model based on neural networks. The idea is as follows. The general nonlinear system is embedded in the class of so-called linear parameter varying (LPV) systems. In order to manage a nonlinear term efficiently, affine linear LPV systems are used and low dimensional scheduling variables are identified on the base of clusters in the system states.

• Keywords: autoencoders; clustering; model order reduction; linear parameter varying systems; explainable AI

2. Neural Networks for solving PDEs

In recent years, physics-informed neural networks (PINNs) have been widely used to solve partial differential equations alongside numerical methods because PINNs can be trained without observations and deal with continuous-time problems directly. In contrast, optimizing the parameters of such models is difficult, and individual training sessions must be performed to predict the evolutions of each different initial condition. The research goal is to develop neural network architectures which can alleviate these problems. 

• Keywords: data-driven models; physics-informed neural networks; partial differential equations