Adaptive Estimation of the Pennes' Bio-Heat Equation - II: a NN-based implementation for real-time applications

This is the companion paper of a two-part work on the observation of the heat transfer phenomenon in biological tissues. In particular, we are interested in real-time estimation of the temperature in the interior of a spatial domain of interest using measurements at its boundary. The prevailing model for heat transfer in biological tissues, pioneered by Pennes [1], relies on a parabolic reaction-diffusion partial differential equation. However, neither the observation problem has been fully explored nor have the available solutions proved suitable for real-time applications. In the companion paper [2], we propose the design of an observer whose formal properties, however, cannot be easily reflected in its practical performance, due to computational issues arising with the use of common numerical solvers. The difficulties are mostly related to the integration of a system of coupled PDEs/ODE, required by the algorithm. In this paper, we propose an alternative implementation of the observer that makes use of deep neural networks for predicting the PDEs state, thus avoiding the online integration. Preliminary results show that this approach is very effective in solving the considered problem and is amenable to extension to other classes of PDEs and to higher dimensions.