Predicting Euler Characteristics and Constructing Topological Structure Using Machine Learning Techniques

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.

We present a demonstration of image classification using an extreme learning machine (ELM) based on a unique simulated magnetic tunnel junction (MTJ) delayed in time. As the ground state of the MTJ is a magnetic vortex, we refer to it as a vortex-based spin-torque oscillator (STVO). The dynamics of the magnetic vortex is simulated with a model called the data-driven Thiele equation approach (DD-TEA). This allows to avoid the constraints associated with repeated experimental manipulation for hyperparameters search and benchmarking. We showcase the versatility of our implementation by using it successfully for classification tasks on the MNIST, EMNIST-letters and Fashion MNIST datasets. Through simulations, we show that within an ELM with a sufficient number of parameters, the performance reached using the STVO dynamics as a source of nonlinearity is equivalent to the ones obtained with classical software activation functions such as the reLU and the sigmoid. While achieving state-of-the-art accuracy levels on the MNIST dataset, our model's performance on EMNIST-letters and Fashion MNIST is lower due to the simplicity of the network architecture and the increased complexity of the data. We expect that the DD-TEA framework will enable the exploration of deeper and more complex STVO-based architectures, ultimately leading to improved classification accuracy.

Metallic magnets exhibit complex spin dynamics governed by electronically generated interactions. Predictive simulations of such dynamics typically require repeated solutions of an underlying electronic problem throughout the time evolution, creating a major computational bottleneck. Here we introduce a graph neural network (GNN) magnetic force-field framework that learns the effective magnetic energy functional governing itinerant spin dynamics directly from electronic calculations. Conceptually analogous to machine-learned interatomic potentials, the proposed framework enables efficient evaluation of spin torques while capturing the nonlinear and spatially extended interactions generated by itinerant electrons. We benchmark the method on representative metallic magnetic systems exhibiting collinear, noncollinear, and noncoplanar magnetic order. The learned force fields accurately reproduce electronically generated spin torques and yield nonequilibrium spin dynamics in excellent agreement with direct electronic simulations. Our results establish graph neural networks as a powerful framework for machine-learned magnetic force fields, providing a pathway toward predictive large-scale simulations of nonequilibrium magnetism across multiple length and time scales.