Continuous neural networks applied to identify a class of uncertain parabolic partial differential equations

There are a lot of examples in science and engineering that may be described using a set of partial differential equations (PDEs). Those PDEs are obtained applying a process of mathematical modeling using complex physical, chemical, etc. laws. Nevertheless, there are many sources of uncertainties around the aforementioned mathematical representation. It is well known that neural networks can approximate a large set of continuous functions defined on a compact set. If the continuous mathematical model is incomplete or partially known, the methodology based on Differential Neural Network (DNN) provides an effective tool to solve problems on control theory such as identification, state estimation, trajectories tracking, etc. In this paper, a strategy based on DNN for the no parametric identification of a mathematical model described by parabolic partial differential equations is proposed. The identification solution allows finding an exact expression for the weights' dynamics. The weights adaptive laws ensure the "practical stability" of DNN trajectories. To verify the qualitative behavior of the suggested methodology, a no parametric modeling problem for a couple of distributed parameter plants is analyzed: the plug-flow reactor model and the anaerobic digestion system. The results obtained in the numerical simulations confirm the identification capability of the suggested methodology.