  In data-driven modelling of complex dynamic processes, it is often desirable
to combine different classes of models to enhance performance. Examples include
coupled models of different fidelities, or hybrid models based on physical
knowledge and data-driven strategies. A key limitation of the broad adoption of
model combination in applications is intrusiveness: training combined models
typically requires significant modifications to the learning algorithm
implementations, which may often be already well-developed and optimized for
individual model spaces. In this work, we propose an iterative, non-intrusive
methodology to combine two model spaces to learn dynamics from data. We show
that this can be understood, at least in the linear setting, as finding the
optimal solution in the direct sum of the two hypothesis spaces, while
leveraging only the projection operators in each individual space. Hence, the
proposed algorithm can be viewed as iterative projections, for which we can
obtain estimates on its convergence properties. To highlight the extensive
applicability of our framework, we conduct numerical experiments in various
problem settings, with particular emphasis on various hybrid models based on
the Koopman operator approach.
