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Quantum neural networks (QNNs) use parameterized quantum circuits with data-dependent inputs and generate outputs through the evaluation of expectation values. Calculating these expectation values necessitates repeated circuit evaluations, thus introducing fundamental finite-sampling noise even on error-free quantum computers. We reduce this noise by introducing the variance regularization, a technique for reducing the variance of the expectation value during the quantum model training. This technique requires no additional circuit evaluations if the QNN is properly constructed. Our empirical findings demonstrate the reduced variance speeds up the training and lowers the output noise as well as decreases the number of necessary evaluations of gradient circuits. This regularization method is benchmarked on the regression of multiple functions and the potential energy surface of water. We show that in our examples, it lowers the variance by an order of magnitude on average and leads to a significantly reduced noise level of the QNN. We finally demonstrate QNN training on a real quantum device and evaluate the impact of error mitigation. Here, the optimization is feasible only due to the reduced number of necessary shots in the gradient evaluation resulting from the reduced variance.

Popular artificial neural networks (ANN) optimize parameters for unidirectional value propagation, assuming some arbitrary parametrization type like Multi-Layer Perceptron (MLP) or Kolmogorov-Arnold Network (KAN). In contrast, for biological neurons e.g. "it is not uncommon for axonal propagation of action potentials to happen in both directions"~\cite{axon} - suggesting they are optimized to continuously operate in multidirectional way. Additionally, statistical dependencies a single neuron could model is not just (expected) value dependence, but entire joint distributions including also higher moments. Such more agnostic joint distribution neuron would allow for multidirectional propagation (of distributions or values) e.g. $\rho(x|y,z)$ or $\rho(y,z|x)$ by substituting to $\rho(x,y,z)$ and normalizing. There will be discussed Hierarchical Correlation Reconstruction (HCR) for such neuron model: assuming $\rho(x,y,z)=\sum_{ijk} a_{ijk} f_i(x) f_j(y) f_k(z)$ type parametrization of joint distribution in polynomial basis $f_i$, which allows for flexible, inexpensive processing including nonlinearities, direct model estimation and update, trained through standard backpropagation or novel ways for such structure up to tensor decomposition or information bottleneck approach. Using only pairwise (input-output) dependencies, its expected value prediction becomes KAN-like with trained activation functions as polynomials, can be extended by adding higher order dependencies through included products - in conscious interpretable way, allowing for multidirectional propagation of both values and probability densities.

Physics-informed neural networks (PINNs) are a powerful approach for solving problems involving differential equations, yet they often struggle to solve problems with high frequency and/or multi-scale solutions. Finite basis physics-informed neural networks (FBPINNs) improve the performance of PINNs in this regime by combining them with an overlapping domain decomposition approach. In this work, FBPINNs are extended by adding multiple levels of domain decompositions to their solution ansatz, inspired by classical multilevel Schwarz domain decomposition methods (DDMs). Analogous to typical tests for classical DDMs, we assess how the accuracy of PINNs, FBPINNs and multilevel FBPINNs scale with respect to computational effort and solution complexity by carrying out strong and weak scaling tests. Our numerical results show that the proposed multilevel FBPINNs consistently and significantly outperform PINNs across a range of problems with high frequency and multi-scale solutions. Furthermore, as expected in classical DDMs, we show that multilevel FBPINNs improve the accuracy of FBPINNs when using large numbers of subdomains by aiding global communication between subdomains.

Cellular scale decision making is modulated by the dynamics of signalling molecules and their diffusive trajectories from a source to small absorbing sites on the cellular surface. Diffusive capture problems are computationally challenging due to the complex geometry and the applied boundary conditions together with intrinsically long transients that occur before a particle is captured. This paper reports on a particle-based Kinetic Monte Carlo (KMC) method that provides rapid accurate simulation of arrival statistics for (i) a half-space bounded by a surface with a finite collection of absorbing traps and (ii) the domain exterior to a convex cell again with absorbing traps. We validate our method by replicating classical results and in addition, newly developed boundary homogenization theories and matched asymptotic expansions on capture rates. In the case of non-spherical domains, we describe a new shielding effect in which geometry can play a role in sharpening cellular estimates on the directionality of diffusive sources.

Making inference with spatial extremal dependence models can be computationally burdensome since they involve intractable and/or censored likelihoods. Building on recent advances in likelihood-free inference with neural Bayes estimators, that is, neural networks that approximate Bayes estimators, we develop highly efficient estimators for censored peaks-over-threshold models that {use data augmentation techniques} to encode censoring information in the neural network {input}. Our new method provides a paradigm shift that challenges traditional censored likelihood-based inference methods for spatial extremal dependence models. Our simulation studies highlight significant gains in both computational and statistical efficiency, relative to competing likelihood-based approaches, when applying our novel estimators to make inference with popular extremal dependence models, such as max-stable, $r$-Pareto, and random scale mixture process models. We also illustrate that it is possible to train a single neural Bayes estimator for a general censoring level, precluding the need to retrain the network when the censoring level is changed. We illustrate the efficacy of our estimators by making fast inference on hundreds-of-thousands of high-dimensional spatial extremal dependence models to assess extreme particulate matter 2.5 microns or less in diameter (${\rm PM}_{2.5}$) concentration over the whole of Saudi Arabia.

The approximation properties of infinitely wide shallow neural networks heavily depend on the choice of the activation function. To understand this influence, we study embeddings between Barron spaces with different activation functions. These embeddings are proven by providing push-forward maps on the measures $\mu$ used to represent functions $f$. An activation function of particular interest is the rectified power unit ($\operatorname{RePU}$) given by $\operatorname{RePU}_s(x)=\max(0,x)^s$. For many commonly used activation functions, the well-known Taylor remainder theorem can be used to construct a push-forward map, which allows us to prove the embedding of the associated Barron space into a Barron space with a $\operatorname{RePU}$ as activation function. Moreover, the Barron spaces associated with the $\operatorname{RePU}_s$ have a hierarchical structure similar to the Sobolev spaces $H^m$.

Protecting privacy during inference with deep neural networks is possible by adding noise to the activations in the last layers prior to the final classifiers or other task-specific layers. The activations in such layers are known as "features" (or, less commonly, as "embeddings" or "feature embeddings"). The added noise helps prevent reconstruction of the inputs from the noisy features. Lower bounding the variance of every possible unbiased estimator of the inputs quantifies the confidentiality arising from such added noise. Convenient, computationally tractable bounds are available from classic inequalities of Hammersley and of Chapman and Robbins -- the HCR bounds. Numerical experiments indicate that the HCR bounds are on the precipice of being effectual for small neural nets with the data sets, "MNIST" and "CIFAR-10," which contain 10 classes each for image classification. The HCR bounds appear to be insufficient on their own to guarantee confidentiality of the inputs to inference with standard deep neural nets, "ResNet-18" and "Swin-T," pre-trained on the data set, "ImageNet-1000," which contains 1000 classes. Supplementing the addition of noise to features with other methods for providing confidentiality may be warranted in the case of ImageNet. In all cases, the results reported here limit consideration to amounts of added noise that incur little degradation in the accuracy of classification from the noisy features. Thus, the added noise enhances confidentiality without much reduction in the accuracy on the task of image classification.

We use fixed point theory to analyze nonnegative neural networks, which we define as neural networks that map nonnegative vectors to nonnegative vectors. We first show that nonnegative neural networks with nonnegative weights and biases can be recognized as monotonic and (weakly) scalable mappings within the framework of nonlinear Perron-Frobenius theory. This fact enables us to provide conditions for the existence of fixed points of nonnegative neural networks having inputs and outputs of the same dimension, and these conditions are weaker than those recently obtained using arguments in convex analysis. Furthermore, we prove that the shape of the fixed point set of nonnegative neural networks with nonnegative weights and biases is an interval, which under mild conditions degenerates to a point. These results are then used to obtain the existence of fixed points of more general nonnegative neural networks. From a practical perspective, our results contribute to the understanding of the behavior of autoencoders, and we also offer valuable mathematical machinery for future developments in deep equilibrium models.

Recently, deep neural networks have been shown to be vulnerable to backdoor attacks. A backdoor is inserted into neural networks via this attack paradigm, thus compromising the integrity of the network. As soon as an attacker presents a trigger during the testing phase, the backdoor in the model is activated, allowing the network to make specific wrong predictions. It is extremely important to defend against backdoor attacks since they are very stealthy and dangerous. In this paper, we propose a novel defense mechanism, Neural Behavioral Alignment (NBA), for backdoor removal. NBA optimizes the distillation process in terms of knowledge form and distillation samples to improve defense performance according to the characteristics of backdoor defense. NBA builds high-level representations of neural behavior within networks in order to facilitate the transfer of knowledge. Additionally, NBA crafts pseudo samples to induce student models exhibit backdoor neural behavior. By aligning the backdoor neural behavior from the student network with the benign neural behavior from the teacher network, NBA enables the proactive removal of backdoors. Extensive experiments show that NBA can effectively defend against six different backdoor attacks and outperform five state-of-the-art defenses.

Recent advances in 3D fully convolutional networks (FCN) have made it feasible to produce dense voxel-wise predictions of volumetric images. In this work, we show that a multi-class 3D FCN trained on manually labeled CT scans of several anatomical structures (ranging from the large organs to thin vessels) can achieve competitive segmentation results, while avoiding the need for handcrafting features or training class-specific models. To this end, we propose a two-stage, coarse-to-fine approach that will first use a 3D FCN to roughly define a candidate region, which will then be used as input to a second 3D FCN. This reduces the number of voxels the second FCN has to classify to ~10% and allows it to focus on more detailed segmentation of the organs and vessels. We utilize training and validation sets consisting of 331 clinical CT images and test our models on a completely unseen data collection acquired at a different hospital that includes 150 CT scans, targeting three anatomical organs (liver, spleen, and pancreas). In challenging organs such as the pancreas, our cascaded approach improves the mean Dice score from 68.5 to 82.2%, achieving the highest reported average score on this dataset. We compare with a 2D FCN method on a separate dataset of 240 CT scans with 18 classes and achieve a significantly higher performance in small organs and vessels. Furthermore, we explore fine-tuning our models to different datasets. Our experiments illustrate the promise and robustness of current 3D FCN based semantic segmentation of medical images, achieving state-of-the-art results. Our code and trained models are available for download: //github.com/holgerroth/3Dunet_abdomen_cascade.