  Relying on recent research results on Neural ODEs, this paper presents a
methodology for the design of state observers for nonlinear systems based on
Neural ODEs, learning Luenberger-like observers and their nonlinear extension
(Kazantzis-Kravaris-Luenberger (KKL) observers) for systems with
partially-known nonlinear dynamics and fully unknown nonlinear dynamics,
respectively. In particular, for tuneable KKL observers, the relationship
between the design of the observer and its trade-off between convergence speed
and robustness is analysed and used as a basis for improving the robustness of
the learning-based observer in training. We illustrate the advantages of this
approach in numerical simulations.
