Uwe Jaekel

Martin Brückmann, Babette Dellen, Uwe Jaekel

Discovering the governing equations of a dynamical system from observed trajectories provides deeper insight into its structure than mere prediction of future states. We present a data-driven approach to model discovery based on complex-valued product-unit networks, in which each unit represents a complex monomial and the network output is a sparse linear combination of such monomials. In contrast to established library-based methods such as SINDy, our approach does not require a predefined set of candidate functions: the relevant monomials, including those with fractional or negative exponents, are learned directly from data. Across four chaotic benchmark systems (Lorenz63, Lorenz84, the Four-Wing attractor, and a fractional variant of Lorenz63), we recover the exact governing equations in 90% of trials for the first three systems, and in 70-90% of trials for the fractional case, using at least 3000 training points. Applied to real-world human-gait accelerometer signals, the model produced stable trajectories with bounded prediction errors, corresponding to an RMSE of approximately 12-14% of the signal amplitude range over a test horizon three times longer than the training interval, demonstrating its potential for high-dimensional systems in which analytic equations are unavailable.

Osamah Sufyan, Martin Brückmann, Ralph Wickenhöfer et al.

In CT angiography, the accurate segmentation of abdominal aortic aneurysms (AAAs) is difficult due to large anatomical variability, low-contrast vessel boundaries, and the close proximity of organs whose intensities resemble vascular structures, often leading to false positives. To address these challenges, we propose an anatomy-aware segmentation framework that integrates organ exclusion masks derived from TotalSegmentator into the training process. These masks encode explicit anatomical priors by identifying non-vascular organsand penalizing aneurysm predictions within these regions, thereby guiding the U-Net to focus on the aorta and its pathological dilation while suppressing anatomically implausible predictions. Despite being trained on a relatively small dataset, the anatomy-aware model achieves high accuracy, substantially reduces false positives, and improves boundary consistency compared to a standard U-Net baseline. The results demonstrate that incorporating anatomical knowledge through exclusion masks provides an efficient mechanism to enhance robustness and generalization, enabling reliable AAA segmentation even with limited training data.

Ziyuan Li, Osamah Sufyan, Uwe Jaekel et al.

Many dense prediction networks rely on additive feature transformations and model higher-order feature interactions only implicitly. Product units provide an explicit mechanism for multiplicative feature modeling, but their logarithmic--exponential formulation can cause numerical instability, which has limited their use in deep dense prediction networks. In this work, we propose Product-Unit U-Net (PU-UNet), a residual U-Net that integrates stable product-unit residual blocks into rich low-resolution stages for medical image segmentation. The proposed formulation combines smooth positivity mapping with log-domain clipping, enabling stable multiplicative feature learning with negligible computational overhead. On ISIC 2018, Kvasir-SEG, and BUSI, PU-UNet achieves Dice scores of 0.942, 0.959, and up to 0.925, respectively. Compared with a matched Residual U-Net baseline, PU-UNet consistently improves Dice and IoU while keeping parameters, FLOPs, and inference latency nearly unchanged, and reduces the image-level false-positive rate on normal BUSI cases from 0.077 to zero. Ablation studies suggest that the gains are associated with product-unit interactions, are strongest under low-resolution placement, and benefit from the proposed stabilization design. These results suggest that stable product-unit residual learning can be an effective way to enhance U-Net-style segmentation networks with explicit multiplicative interactions.

Ziyuan Li, Uwe Jaekel, Babette Dellen

Understanding nonlinear feature interactions is crucial in science and engineering, yet standard multilayer perceptrons (MLPs) often capture such interactions only implicitly, leading to entangled representations that can impair robustness and interpretability. We investigate product-unit residual networks (PURe) that integrate multiplicative product units with residual connections to explicitly model cross-feature couplings while stabilizing optimization. We conduct a systematic evaluation on an interaction-driven synthetic benchmark and two real-world datasets, assessing predictive accuracy, robustness to Gaussian feature noise, and performance under limited training data, and we compare real- and complex-valued variants under a matched parameter budget. Beyond accuracy, SHapley Additive exPlanations (SHAP)-based interaction analyses show that PURe learns more concentrated and structurally coherent interaction patterns than MLP baselines. Overall, PURe achieves competitive or improved performance, better robustness and sample efficiency in low-data regimes, and enhanced interaction-level interpretability.

Ziyuan Li, Uwe Jaekel, Babette Dellen

We propose a deep product-unit residual neural network (PURe) that integrates product units into residual blocks to improve the expressiveness and parameter efficiency of deep convolutional networks. Unlike standard summation neurons, product units enable multiplicative feature interactions, potentially offering a more powerful representation of complex patterns. PURe replaces conventional convolutional layers with 2D product units in the second layer of each residual block, eliminating nonlinear activation functions to preserve structural information. We validate PURe on three benchmark datasets. On Galaxy10 DECaLS, PURe34 achieves the highest test accuracy of 84.89%, surpassing the much deeper ResNet152, while converging nearly five times faster and demonstrating strong robustness to Poisson noise. On ImageNet, PURe architectures outperform standard ResNet models at similar depths, with PURe34 achieving a top-1 accuracy of 80.27% and top-5 accuracy of 95.78%, surpassing deeper ResNet variants (ResNet50, ResNet101) while utilizing significantly fewer parameters and computational resources. On CIFAR-10, PURe consistently outperforms ResNet variants across varying depths, with PURe272 reaching 95.01% test accuracy, comparable to ResNet1001 but at less than half the model size. These results demonstrate that PURe achieves a favorable balance between accuracy, efficiency, and robustness. Compared to traditional residual networks, PURe not only achieves competitive classification performance with faster convergence and fewer parameters, but also demonstrates greater robustness to noise. Its effectiveness across diverse datasets highlights the potential of product-unit-based architectures for scalable and reliable deep learning in computer vision.

Babette Dellen, Uwe Jaekel, Paulo S. A. Freitas et al.

Accurate estimation of nuclear masses and their prediction beyond the experimentally explored domains of the nuclear landscape are crucial to an understanding of the fundamental origin of nuclear properties and to many applications of nuclear science, most notably in quantifying the $r$-process of stellar nucleosynthesis. Neural networks have been applied with some success to the prediction of nuclear masses, but they are known to have shortcomings in application to extrapolation tasks. In this work, we propose and explore a novel type of neural network for mass prediction in which the usual neuron-like processing units are replaced by complex-valued product units that permit multiplicative couplings of inputs to be learned from the input data. This generalized network model is tested on both interpolation and extrapolation data sets drawn from the Atomic Mass Evaluation. Its performance is compared with that of several neural-network architectures, substantiating its suitability for nuclear mass prediction. Additionally, a prediction-uncertainty measure for such complex-valued networks is proposed that serves to identify regions of expected low prediction error.