## 1 Introduction  
The problem of system state reconstruction from local and partial state measurements has important impacts on the ability of controlling the system state and monitoring system health and performance [1, 2, 3]. This becomes particularly involved in the framework of spatially continuously distributed dynamical systems, which are typically modeled using partial differential equations (pdes). The so called associated state estimation or observer design problem has been studied over the last decades and already reached a state of maturity in particular for linear systems (see, e.g., [4, 5]). Additionally, for some classes of nonlinear or semilinear systems, design approaches have been obtained, which differ in design complexity and computational load in their implementation. A particular approach, which is simple and can be implemented with small computational load is the pointwise measurement injection (or pointwise innovation) observer design, which has been employed for solving the problem for some types of semilinear parabolic problems [6, 7, 8, 9, 10]. This approach basically extends the idea of a reduced order observer design [1, 2] to the considered class of infinite-dimensional systems. The reduced order property is to be understood in relation to the number of internal states of the observer in comparison to the number of states in the system to be monitored. Basically the idea consists in providing a non-redundant state estimation scheme, which does not need to estimate the measured output, but only non-measured state variables. Considering pde models leads to the additional problem that for implementing any observer in a real-time monitoring scheme, the associated set of pdes has to be solved numerically. This is done using some kind of discretization in space and time, so that a finite-dimensional discrete-time system representation is obtained, which represents a digital twin of the original pde system. The number of internal state variables of this approximation model is typically very high (from about hundred to several thousands of variables, e.g., for finite-element approximations) and thus requires model-order reduction approaches for real-time applicability. For this purpose, typically balanced truncation or proper orthogonal decomposition approaches or similar are employed (see, e.g., [11] and references therein). Besides these, in recent years a focus was led to approaches using dynamic mode decomposition (DMD) [11; 12] and its extension to DMDc to include external (control) inputs [13]. The usage of DMD (or DMDc) is particularly motivated by the Koopman theory, according to which nonlinear finite-dimensional systems can be embedded into an infinite-dimensional space on which the linear Kooppman operator provides a complete description of the time evolution of the system observables. The (Koopman) eigenfunctions or the associated Koopman operator form a basis of this space and can be used in principle to determine a finite-dimensional approximation of the dominant modes of this operator (associated to the dominant eigenvalues of the operator). As this embedding can in most cases not be carried out analytically (see, e.g., the discussion in [3; 11; 12]) one can employ DMD (or DMDc) as a data-based identification approach using snapshots of the state observables as training data. This idea has been combined with the use of autoencoders (AEs) to learn from the training data a low-dimensional basis of the associated dominant Koopman eigenfunction subspace in which a linear dynamics can then be identified using classical DMDc [14]. The use of autoencoders and Koopman theory for automatic learning of finite-dimensional system dynamics is also mainstream in contemporary control systems applications in general (see, e.g., [15; 16]). Having finite-dimensional approximations of the infinite-dimensional pde models, one can also directly address the observer design problem in the finite-dimensional setup.