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M. Ploner:
"A Neural network approach for differential equations in biomedical applications";
Supervisor: A. Körner, S. Winkler; Institut für Analysis und Scientific Computing, 2021; final examination: 2020-12-16.

Artificial neural networks are state of the art and used in a broad variety of scientific disciplines, such as natural sciences, economics or in the field of big data. From image and speech recognition to weather forecasts and economic models, networks inspired by neurons have a significant impact. This thesis focuses on the numerical solution of differential equations using artificial neural networks. Following a general introduction, the first part of this thesis deals with the cost function of the respective neural network, which has to be minimized. For this purpose, the iteration steps, training steps and activation functions of the network are varied and compared. Furthermore, the approximation errors of analytically solvable differential equations are determined, considering not only the training interval, but also a numerical approximation of the solution is made outside of the interval. In the second part, different differential equations are studied and compared with other numerical methods. The error to the respective analytical solution is used as a reference for the approximation capability. A focus is given on applications in biomedical sciences and their numerical solutions. The Bateman function describes the relation between the concentration of a drug in the blood plasma after administration with time, using ordinary differential equations of first order and a compartment model for deduction. Another example for a differential equation is given by the logistic tumor growth. Furthermore, the harmonic oscillation is used to approximate the blood pressure during a heart muscle contraction within one second. This oscillation differential equation is also compared with neural network solutions. Finally, the used methods are analysed and conclusions are made, following an outlook to further possible approaches.

Modelling and Simulation; Artificial Neural Networks; Ordinary Differential Equations