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The elapsed time equation is an age-structured model that describes the dynamics of interconnected spiking neurons through the elapsed time since the last discharge, leading to many interesting questions on the evolution of the system from a mathematical and biological point of view. In this work, we first deal with the case when transmission after a spike is instantaneous and the case when there exists a distributed delay that depends on the previous history of the system, which is a more realistic assumption. Then we revisit the well-posedness in order to make a numerical analysis by adapting the classical upwind scheme through a fixed-point approach. We improve the previous results on well-posedness by relaxing some hypotheses on the non-linearity for instantaneous transmission, including the strongly excitatory case, while for the numerical analysis we prove that the approximation given by the explicit upwind scheme converges to the solution of the non-linear problem through BV-estimates. We also show some numerical simulations to compare the behavior of the system in the case of instantaneous transmission with the case of distributed delay under different parameters, leading to solutions with different asymptotic profiles.

Image classification with neural networks (NNs) is widely used in industrial processes, situations where the model likely encounters unknown objects during deployment, i.e., out-of-distribution (OOD) data. Worryingly, NNs tend to make confident yet incorrect predictions when confronted with OOD data. To increase the models' reliability, they should quantify the uncertainty in their own predictions, communicating when the output should (not) be trusted. Deep ensembles, composed of multiple independent NNs, have been shown to perform strongly but are computationally expensive. Recent research has proposed more efficient NN ensembles, namely the snapshot, batch, and multi-input multi-output ensemble. This study investigates the predictive and uncertainty performance of efficient NN ensembles in the context of image classification for industrial processes. It is the first to provide a comprehensive comparison and it proposes a novel Diversity Quality metric to quantify the ensembles' performance on the in-distribution and OOD sets in one single metric. The results highlight the batch ensemble as a cost-effective and competitive alternative to the deep ensemble. It matches the deep ensemble in both uncertainty and accuracy while exhibiting considerable savings in training time, test time, and memory storage.

Understanding adversarial examples is crucial for improving the model's robustness, as they introduce imperceptible perturbations that deceive models. Effective adversarial examples, therefore, offer the potential to train more robust models by removing their singularities. We propose NODE-AdvGAN, a novel approach that treats adversarial generation as a continuous process and employs a Neural Ordinary Differential Equation (NODE) for simulating the dynamics of the generator. By mimicking the iterative nature of traditional gradient-based methods, NODE-AdvGAN generates smoother and more precise perturbations that preserve high perceptual similarity when added to benign images. We also propose a new training strategy, NODE-AdvGAN-T, which enhances transferability in black-box attacks by effectively tuning noise parameters during training. Experiments demonstrate that NODE-AdvGAN and NODE-AdvGAN-T generate more effective adversarial examples that achieve higher attack success rates while preserving better perceptual quality than traditional GAN-based methods.

A digital twin is a virtual representation that accurately replicates its physical counterpart, fostering bi-directional real-time data exchange throughout the entire process lifecycle. For Laser Directed Energy Deposition of Wire (DED-LB/w) additive manufacturing processes, digital twins may help to control the residual stress design in build parts. This study focuses on providing faster-than-real-time and highly accurate surrogate models for the formation of residual stresses by employing neural ordinary differential equations. The approach enables accurate prediction of temperatures and altered structural properties like stress tensor components. The developed surrogates can ultimately facilitate on-the-fly re-optimization of the ongoing manufacturing process to achieve desired structural outcomes. Consequently, this building block contributes significantly to realizing digital twins and the first-time-right paradigm in additive manufacturing.

We propose two methods for the unsupervised detection of communities in undirected multiplex networks. These networks consist of multiple layers that record different relationships between the same entities or incorporate data from different sources. Both methods are formulated as gradient flows of suitable energy functionals: the first (MPBTV) builds on the minimization of a balanced total variation functional, which we show to be equivalent to multiplex modularity maximization, while the second (DGFM3) directly maximizes multiplex modularity. The resulting non-linear matrix-valued ordinary differential equations (ODEs) are solved efficiently by a graph Merriman--Bence--Osher (MBO) scheme. Key to the efficiency is the approximate integration of the discrete linear differential operators by truncated eigendecompositions in the matrix exponential function. Numerical experiments on several real-world multiplex networks show that our methods are competitive with the state of the art with respect to various metrics. Their major benefit is a significant reduction of computational complexity leading to runtimes that are orders of magnitude faster for large multiplex networks.

Despite advances in vision-language understanding, implementing image segmentation within multimodal architectures remains a fundamental challenge in modern artificial intelligence systems. Existing vision-language models, which primarily rely on backbone architectures or CLIP-based embedding learning, demonstrate inherent limitations in fine-grained spatial localization and operational capabilities. This paper introduces SJTU: Spatial Judgments in multimodal models - Towards Unified segmentation through coordinate detection, a novel framework that leverages spatial coordinate understanding to bridge vision-language interaction and precise segmentation, enabling accurate target identification through natural language instructions. The framework proposes a novel approach for integrating segmentation techniques with vision-language models based on multimodal spatial inference. By leveraging normalized coordinate detection for bounding boxes and translating it into actionable segmentation outputs, we explore the possibility of integrating multimodal spatial and language representations. Based on the proposed technical approach, the framework demonstrates superior performance on various benchmark datasets as well as accurate object segmentation. Results on the COCO 2017 dataset for general object detection and Pascal VOC datasets for semantic segmentation demonstrate the generalization capabilities of the framework.

Traditional electrostatic simulation are meshed-based methods which convert partial differential equations into an algebraic system of equations and their solutions are approximated through numerical methods. These methods are time consuming and any changes in their initial or boundary conditions will require solving the numerical problem again. Newer computational methods such as the physics informed neural net (PINN) similarly require re-training when boundary conditions changes. In this work, we propose an end-to-end deep learning approach to model parameter changes to the boundary conditions. The proposed method is demonstrated on the test problem of a long air-filled capacitor structure. The proposed approach is compared to plain vanilla deep learning (NN) and PINN. It is shown that our method can significantly outperform both NN and PINN under dynamic boundary condition as well as retaining its full capability as a forward model.

We are interested in computing an approximation of the maximum flow in large (brain) connectivity networks. The maximum flow in such networks is of interest in order to better understand the routing of information in the human brain. However, the runtime of $O(|V||E|^2)$ for the classic Edmonds-Karp algorithm renders computations of the maximum flow on networks with millions of vertices infeasible, where $V$ is the set of vertices and $E$ is the set of edges. In this contribution, we propose a new Monte Carlo algorithm which is capable of computing an approximation of the maximum flow in networks with millions of vertices via subsampling. Apart from giving a point estimate of the maximum flow, our algorithm also returns valid confidence bounds for the true maximum flow. Importantly, its runtime only scales as $O(B \cdot |\tilde{V}| |\tilde{E}|^2)$, where $B$ is the number of Monte Carlo samples, $\tilde{V}$ is the set of subsampled vertices, and $\tilde{E}$ is the edge set induced by $\tilde{V}$. Choosing $B \in O(|V|)$ and $|\tilde{V}| \in O(\sqrt{|V|})$ (implying $|\tilde{E}| \in O(|V|)$) yields an algorithm with runtime $O(|V|^{3.5})$ while still guaranteeing the usual "root-n" convergence of the confidence interval of the maximum flow estimate. We evaluate our proposed algorithm with respect to both accuracy and runtime on simulated graphs as well as graphs downloaded from the Brain Networks Data Repository (//networkrepository.com).

We hypothesize that due to the greedy nature of learning in multi-modal deep neural networks, these models tend to rely on just one modality while under-fitting the other modalities. Such behavior is counter-intuitive and hurts the models' generalization, as we observe empirically. To estimate the model's dependence on each modality, we compute the gain on the accuracy when the model has access to it in addition to another modality. We refer to this gain as the conditional utilization rate. In the experiments, we consistently observe an imbalance in conditional utilization rates between modalities, across multiple tasks and architectures. Since conditional utilization rate cannot be computed efficiently during training, we introduce a proxy for it based on the pace at which the model learns from each modality, which we refer to as the conditional learning speed. We propose an algorithm to balance the conditional learning speeds between modalities during training and demonstrate that it indeed addresses the issue of greedy learning. The proposed algorithm improves the model's generalization on three datasets: Colored MNIST, Princeton ModelNet40, and NVIDIA Dynamic Hand Gesture.

Graph representation learning for hypergraphs can be used to extract patterns among higher-order interactions that are critically important in many real world problems. Current approaches designed for hypergraphs, however, are unable to handle different types of hypergraphs and are typically not generic for various learning tasks. Indeed, models that can predict variable-sized heterogeneous hyperedges have not been available. Here we develop a new self-attention based graph neural network called Hyper-SAGNN applicable to homogeneous and heterogeneous hypergraphs with variable hyperedge sizes. We perform extensive evaluations on multiple datasets, including four benchmark network datasets and two single-cell Hi-C datasets in genomics. We demonstrate that Hyper-SAGNN significantly outperforms the state-of-the-art methods on traditional tasks while also achieving great performance on a new task called outsider identification. Hyper-SAGNN will be useful for graph representation learning to uncover complex higher-order interactions in different applications.