Taylor mapping method for solving and learning of dynamic processes

In this paper, we discuss the so-called Taylor map approach to solve systems of ordinary differential equations. We demonstrate its capabilities of solving the corresponding inverse problems including parameter identification. The method applies even if the underlying ordinary differential equation is not explicitly known. This procedure is interpreted in terms of Polynomial Neural Networks. Physical knowledge is incorporated into the neural network since its architecture is designed directly on top of the Taylor map approach. This does not only improve the data efficiency but also reduce the effort of the included training procedure. We prove an asymptotic convergence result of polynomial neural networks. On this basis, we demonstrate validation examples to highlight the capabilities of this method in practice.