Daniel Franzen

Isaac Labrie-Boulay, Thomas Brian Winkler, Daniel Franzen et al.

One of the most important magnetic spin structure is the topologically stabilised skyrmion quasi-particle. Its interesting physical properties make them candidates for memory and efficient neuromorphic computation schemes. For the device operation, detection of the position, shape, and size of skyrmions is required and magnetic imaging is typically employed. A frequently used technique is magneto-optical Kerr microscopy where depending on the samples material composition, temperature, material growing procedures, etc., the measurements suffer from noise, low-contrast, intensity gradients, or other optical artifacts. Conventional image analysis packages require manual treatment, and a more automatic solution is required. We report a convolutional neural network specifically designed for segmentation problems to detect the position and shape of skyrmions in our measurements. The network is tuned using selected techniques to optimize predictions and in particular the number of detected classes is found to govern the performance. The results of this study shows that a well-trained network is a viable method of automating data pre-processing in magnetic microscopy. The approach is easily extendable to other spin structures and other magnetic imaging methods.

Jean Philip Filling, Daniel Franzen, Michael Wand

We introduce a tensor-channel equivariant graph neural network for direct prediction of molecular polarizability tensors. Building on the efficient PaiNN architecture, we augment the hidden representation with explicit symmetric rank-2 tensor channels aligned with the decomposition of polarizability into isotropic and anisotropic components. In contrast to approaches that construct tensor outputs only at readout, our model propagates tensor structure throughout message passing using geometrically motivated tensor bases. This yields a target-aligned architecture for tensor-valued molecular prediction. On optimized QM7-X geometries, the proposed model achieves lower full-tensor and anisotropic error than both a PaiNN-style readout baseline and a dielectric MACE baseline under matched training conditions and at nearly identical parameter count. In this controlled setting, it also outperforms MACE while remaining substantially faster at inference. Ablation studies show that the gain does not arise from increased capacity alone, but from the combination of explicit tensor propagation and a traceless target parameterization matched to the anisotropic part of the polarizability tensor. Among the tensor bases considered, the strongest results are obtained from interactions between learned directional features, indicating that these are particularly effective for modeling molecular polarizability. Rotational equivariance tests further confirm that all compared models are numerically equivariant, so the observed improvements are attributable to better learning of the target tensor itself. Overall, our results show that for structured tensor-valued targets, propagating target-aligned tensor features can outperform both readout-only tensor construction and a more general higher-order equivariant model in the present training setting.

Daniel Franzen, Jean Philip Filling, Michael Wand

Group-convolutional neural networks (GCNNs) are among the most important methods for introducing symmetry as an inductive bias in deep learning: In each linear layer, GCNNs sample a transformation group $G$ densely and correlate data and filters in different poses (with suitable anti-aliasing for steerable GCNNs) to maintain equivariance with respect to $G$. Unfortunately, applying filters to many data items resulting from this sampling is expensive (even for translations alone, i.e., in ordinary CNNs), and costs grow exponentially with increasing degrees of freedom (such as translations and rotations in 3D), which often hinders practical applications. In this paper, we propose sampling in feature space, i.e., replacing geometrically dense samples with representative samples selected by feature similarity. This decouples geometric resolution from memory and processing costs during training and inference, providing a novel way to trade off computational effort and accuracy. Our main empirical finding is that a coarse feature-space sampling already preserves classification accuracy remarkably well, which permits precomputation based on geometric similarity, accelerating the training of equivariant 3D classifiers substantially.

Daniel Franzen, Jan Disselhoff, David Hartmann

The Abstraction and Reasoning Corpus (ARC-AGI) poses a significant challenge for large language models (LLMs), exposing limitations in their abstract reasoning abilities. In this work, we leverage task-specific data augmentations throughout the training, generation, and scoring phases, and employ a depth-first search algorithm to generate diverse, high-probability candidate solutions. Furthermore, we utilize the LLM not only as a generator but also as a scorer, using its output probabilities to select the most promising solutions. Our method achieves a score of 71.6% (286.5/400 solved tasks) on the public ARC-AGI evaluation set, demonstrating state-of-the-art performance among publicly available approaches. While concurrent closed-source work has reported higher scores, our method distinguishes itself through its transparency, reproducibility, and remarkably low inference cost, averaging only around 2ct per task on readily available hardware (we assume a price of 36ct/hour for a Nvidia 4090 GPU).

Daniel Franzen, Michael Wand

Invariance under symmetry is an important problem in machine learning. Our paper looks specifically at equivariant neural networks where transformations of inputs yield homomorphic transformations of outputs. Here, steerable CNNs have emerged as the standard solution. An inherent problem of steerable representations is that general nonlinear layers break equivariance, thus restricting architectural choices. Our paper applies harmonic distortion analysis to illuminate the effect of nonlinearities on Fourier representations of SO(2). We develop a novel FFT-based algorithm for computing representations of non-linearly transformed activations while maintaining band-limitation. It yields exact equivariance for polynomial (approximations of) nonlinearities, as well as approximate solutions with tunable accuracy for general functions. We apply the approach to build a fully E(3)-equivariant network for sampled 3D surface data. In experiments with 2D and 3D data, we obtain results that compare favorably to the state-of-the-art in terms of accuracy while permitting continuous symmetry and exact equivariance.