Equivariant network architectures are a well-established tool for predicting invariant or equivariant quantities. However, almost all learning problems considered in this context feature a global symmetry, i.e. each point of the underlying space is transformed with the same group element, as opposed to a local ``gauge'' symmetry, where each point is transformed with a different group element, exponentially enlarging the size of the symmetry group. Gauge equivariant networks have so far mainly been applied to problems in quantum chromodynamics. Here, we introduce a novel application domain for gauge-equivariant networks in the theory of topological condensed matter physics. We use gauge equivariant networks to predict topological invariants (Chern numbers) of multiband topological insulators. The gauge symmetry of the network guarantees that the predicted quantity is a topological invariant. We introduce a novel gauge equivariant normalization layer to stabilize the training and prove a universal approximation theorem for our setup. We train on samples with trivial Chern number only but show that our models generalize to samples with non-trivial Chern number. We provide various ablations of our setup. Our code is available at https://github.com/sitronsea/GENet/tree/main.

Fine-tuning large language models can improve task specific performance, although a general understanding of what the fine-tuned model has learned, forgotten and how to trust its predictions is still missing. We derive principled uncertainty quantification for fine-tuned LLMs with posterior approximations using computationally efficient low-rank adaptation ensembles. We analyze three common multiple-choice datasets using low-rank adaptation ensembles based on Mistral-7b, and draw quantitative and qualitative conclusions on their perceived complexity and model efficacy on the different target domains during and after fine-tuning. In particular, backed by the numerical experiments, we hypothesise about signals from entropic uncertainty measures for data domains that are inherently difficult for a given architecture to learn.