Evaluating the forward solution operator for parametric partial differential equations (PDEs) typically involves solving large linear systems, making the process computationally expensive. Operator surrogates aim to provide a computationally efficient approximation to these parameter-to-solution operators. This is particularly beneficial in applications requiring many evaluations of the forward map, such as uncertainty quantification (UQ) or optimization.
Mathematically, operator surrogates provide approximations of mappings between infinite dimensional spaces. We analyze an encoder-decoder framework, where the input and output function spaces are parametrized using coefficient sequences of admissable representation systems. The goal then becomes to approximate a coefficient-to-coefficient map, which can be realized using various approximation tools. In the present work we focus on low-rank tensor representation using the tensor-train (TT) format. This format allows for efficient storage, evaluation and basic linear algebra as long as the tensor ranks remain low. Its structure also allows for efficient quadrature, useful e.g.\ in UQ applications. We show approximation rates with regards to the storage complexity of such TT surrogates for certain holomorphic maps by providing rank bounds. Additionally, we present benchmark results comparing the TT surrogate to a range of other operator surrogate methods applied to a parametric diffusion problem with varying input smoothness.

Koopman theory turns nonlinear dynamics into a linear spectral problem. In computation, however, everything depends on a hard finite-dimensional choice: the observables must be expressive, nearly invariant under the dynamics, and, ideally, compatible with composition. Deep Koopman methods learn flexible coordinates, whereas structure-preserving methods enforce operator identities on fixed dictionaries. We combine these ideas by introducing Deep Embedded Multiplicative Dynamic Mode Decomposition (DeepMDMD), a method that learns a latent space and a partition of it, while enforcing the Koopman product rule as an exact algebraic constraint. Training alternates between an exact multiplicative operator update and a differentiable latent-clustering step that promotes Koopman closure. The result is a finite transition map on learned latent cells. Its nonzero spectrum lies on the unit circle, its dictionary is shaped by the dynamics rather than by ambient geometry, and forecasts are made in latent coordinates before being decoded to physical space. Across Hamiltonian, chaotic, and fluid examples, DeepMDMD learns dictionaries that are far more compact and dynamically coherent than those produced by geometric MDMD partitions. It reduces spectral pollution, reveals richer continuous-spectrum structure, and gives stable forecasts under severe noise. In high-dimensional flows, including a 158,624-dimensional cylinder wake and a noisy $\mathrm{Re}=20,000$ lid-driven cavity, it preserves coherent structures and long-time spectral statistics where state-space MDMD fails.
These results suggest a practical rule for Koopman learning: learn the coordinates, constrain the algebra.

The computation of eigenvalues and eigenvectors is a cornerstone of linear algebra, with profound applications across science and engineering. The power method and the inverse power method are classical iterative algorithms designed to find the largest and the smallest eigenvalue respectively. In recent years, many problems in fields such as machine learning, data science, and image analysis have led to non-linear analogues of eigenvalue problems.
The project generalises the notion of eigenvectors and eigenvalues to the non-linear case in a suitable way, i.e. in such a way that the key properties from the linear case are preserved. Building upon this theoretical foundation, we propose novel generalisations of the power and inverse power methods tailored for non-linear settings. The central theoretical challenge lies in proving the convergence of these iterative schemes to the stationary points of the generalised Rayleigh quotient, despite the lack of convexity in general. By leveraging the Kurdyka-Łojasiewicz (KL) property as a core analytical tool, we overcome this limitation and establish rigorous convergence results for the proposed algorithms. This is joint work with Carola-Bibiane Schönlieb, Matthias Ehrhardt and Martin Benning.