State-space modeling for control based on physics-informed neural networks

Dynamic system based on partial differential equations (PDEs), are often unsuitable for direct use in control or state estimation purposes, due to the high computational cost arising from the necessity to apply sophisticated numerical methods for a solution, such as semi-discretization, also known as spatial discretization. Hence, there is often an inevitable trade-off between accuracy and computational efficiency during the model reduction step to ensure real-time applicability. In this contribution, we propose a state-space model formulation, using so-called physics-informed neural networks. This modeling approach enables a highly efficient inclusion of complex physical system descriptions within the design of control or state estimation setups. The resulting state-space model does not require any numerical solution techniques during the state propagation, as each time step is based on the evaluation of a reasonably sized neural net that approximates the solution of the PDE. Thus, this approach is suitable for real-time applications of various complex dynamic systems that can be described by one or a set of PDEs. Besides the modeling approach itself, the contribution also provides an illustrative example of the state-space modeling method in the context of model predictive control, as well as state estimation with an extended Kalman filter. These methods will be applied to a system based on a numerical solution of the Burgers equation.