We expand on the framework by Korda and Mezic (2020) to construct eigenfunctions directly from data by exploiting the eigenfunction PDE, guaranteeing closure and eliminating the need for a prior data dictionary.
The constructed models are extended to forced systems through the multi-step prediction error of a linear state-space model.

By identifying a relationship between ESPRIT and DMD applied to Hankel matrices, we simplify the original optimisation problem and significantly reducing the required model order.
A detailed numerical investigation of both autonomous dynamics and forced dynamics follows. For autonomous dynamics we report a VAF up to 90 % for a toy model and the Van der Pol oscillator, whilst the original work is unable to reconstruct the underlying dynamics on longer time scales. We were unable to reproduce accurate multi-step predictions under the influence of forcing.

We extend the constructed models by inclusion of monomial terms into the dynamics. This can be interpreted as a linear model with nonlinear output, approximated by a polynomial. Results on the Koopman generator and the inclusion of monomial terms suggest the construction of a bilinear model. A multi-step prediction is formulated, simplified and solved, expanding the predictive capabilities of the model. Whilst the inclusion of monomial terms improved the prediction of autonomous dynamics, the bilinear models failed to converge for the Duffing oscillator and Van der Pol oscillator.

We perform a further study on the constructed eigenfunctions by designing a new neural network architecture, aimed at learning Koopman eigenfunctions. The network architecture accurately recovers the autonomous dynamics of the system. The learned eigenfunctions suggest that the constructed eigenfunctions can be severely limited by the choice of the initial condition set Γ, opening the door for future research.

The concept of entanglement is one of the distinguishing features in quantum mechanics. Information about one particle can determine the state of another particle. This information and entanglement in subsystems is quantified by the entanglement entropy. The entanglement entropy has become an important research topic in recent years. Entanglement entropy is expected to grow with the boundary size for systems described by Hamiltonians with local interactions, described by the area law. For one dimension there exists a bound quantifying this area law S = O (2^1/ΔE) in terms of the energy gap ΔE between the ground-state energy and the first excited-state energy. It is an open problem whether the area law holds in higher spatial dimensions. Therefore it is of great interest to study the entanglement entropy in systems that violate the area law. In this thesis spin chain models with local interactions are studied for arbitrary spin. Hamiltonians with a unique ground state are constructed by mapping spin chains onto (coloured) Dyck and (coloured) Motzkin paths. We show that these models express logarithmic or power law violations of the area law for the bipartite entanglement entropy. These results are presented in the table below. The known bound of the area law in one dimensional systems is dependent on the energy gap of these models. The energy gap of the Motzkin-path model is investigated by constructing the an orthogonal excited state with small energy. In doing so we associate the Hamiltonian with the Laplacian of a graph. We present an alternative proof for the bound on the energy gap of the colourless Motzkin-path model than S. Bravyi et al and R. Movassagh et al. We show that the energy gap ΔE scales in terms of the chain length n as ΔE = O(n^-2) in the limit n →∞. Therefore ΔE → 0 in this limit, resulting in area-law violations of the entanglement entropy while maintaining the known bound on the entanglement entropy.