Deep operator networks (DeepONets) have demonstrated their capability of approximating nonlinear operators for initial- and boundary-value problems. One attractive feature of DeepONets is their versatility since they do not rely on prior knowledge about the solution structure of a problem and can thus be directly applied to a large class of problems. However, convergence in identifying the parameters of the networks may sometimes be slow. In order to improve on DeepONets for approximating the wave equation, we introduce the Green operator networks (GreenONets), which use the representation of the exact solution to the homogeneous wave equation in term of the Green's function. The performance of GreenONets and DeepONets is compared on a series of numerical experiments for homogeneous and heterogeneous media in one and two dimensions.

We define a metric in the space of positive finite positive measures that extends the 2-Wasserstein metric, i.e. its restriction to the set of probability measures is the 2-Wasserstein metric. We prove a dual and a dynamic formulation and extend the gradient flow machinery of the Wasserstein space. In addition, we relate the barycenter in this space to the barycenter in the Wasserstein space of the normalized measures.

Methods based on class activation maps (CAM) provide a simple mechanism to interpret predictions of convolutional neural networks by using linear combinations of feature maps as saliency maps. By contrast, masking-based methods optimize a saliency map directly in the image space or learn it by training another network on additional data. In this work we introduce Opti-CAM, combining ideas from CAM-based and masking-based approaches. Our saliency map is a linear combination of feature maps, where weights are optimized per image such that the logit of the masked image for a given class is maximized. We also fix a fundamental flaw in two of the most common evaluation metrics of attribution methods. On several datasets, Opti-CAM largely outperforms other CAM-based approaches according to the most relevant classification metrics. We provide empirical evidence supporting that localization and classifier interpretability are not necessarily aligned.

This is the third in a series of three papers in which we study a lattice gas subject to Kawasaki dynamics at inverse temperature \beta&gt;0 in a large finite box $\Lambda_\beta \subset \mathbb{Z}^2$ whose volume depends on $\beta$. Each pair of neighbouring particles has a negative binding energy -U&lt;0, while each particle has a positive activation energy \Delta&gt;0. The initial configuration is drawn from the grand-canonical ensemble restricted to the set of configurations where all the droplets are subcritical. Our goal is to describe, in the metastable regime $\Delta \in (U,2U)$ and in the limit as $\beta\to\infty$, how and when the system nucleates, i.e., creates a critical droplet somewhere in $\Lambda_\beta$ that subsequently grows by absorbing particles from the surrounding gas. In the first paper we showed that subcritical droplets behave as quasi-random walks. In the second paper we used the results in the first paper to analyse how subcritical droplets form and dissolve on multiple space-time scales when the volume is moderately large, namely, $|\Lambda_\beta| = \mathrm{e}^{\theta\beta}$with$\Delta < \theta < 2\Delta-U$. In the present paper we consider the setting where the volume is very large, namely, $|\Lambda_\beta| = \mathrm{e}^{\Theta\beta}$ with $\Delta < \Theta < \Gamma-(2\Delta-U)$, where$\Gamma$ is the energy of the critical droplet in the local model with fixed volume, and use the results in the first two papers to identify the nucleation time and the tube of typical trajectories towards nucleation. We will see that in a very large volume critical droplets appear more or less independently in boxes of moderate volume, a phenomenon referred to as homogeneous nucleation. One of the key ingredients in the proof is an estimate showing that no information can travel between these boxes on relevant time scales.

Learning from an imbalanced distribution presents a major challenge in predictive modeling, as it generally leads to a reduction in the performance of standard algorithms. Various approaches exist to address this issue, but many of them concern classification problems, with a limited focus on regression. In this paper, we introduce a novel method aimed at enhancing learning on tabular data in the Imbalanced Regression (IR) framework, which remains a significant problem. We propose to use variational autoencoders (VAE) which are known as a powerful tool for synthetic data generation, offering an interesting approach to modeling and capturing latent representations of complex distributions. However, VAEs can be inefficient when dealing with IR. Therefore, we develop a novel approach for generating data, combining VAE with a smoothed bootstrap, specifically designed to address the challenges of IR. We numerically investigate the scope of this method by comparing it against its competitors on simulations and datasets known for IR.

We present 4D topological textures in (quasi)monochromatic nonparaxial optical lattices that contain all possible polarization ellipses with every combination of ellipticity and orientation in 3D space. These fields span the nonparaxial polarization space (a complex projective plane) and a 4-sphere within specific spatiotemporal regions, forming 4D skyrmionic structures. Constructed from five plane waves with adiabatically varying relative amplitudes, they are experimentally realizable in free space by focusing a temporally variant beam with a high numerical aperture lens.

We demonstrate pixelation-free real-time widefield endoscopic imaging through an aperiodic multicore fiber (MCF) without any distal opto-mechanical elements or proximal scanners. Exploiting the memory effect in MCFs the images in our system are directly obtained without any post-processing using a static wavefront correction obtained from a single calibration procedure. Our approach allows for video-rate 3D widefield imaging of incoherently illuminated objects with imaging speed not limited by the wavefront shaping device refresh rate.

Phase retrieval is an inverse problem that, on one hand, is crucial in many applications across imaging and physics, and, on the other hand, leads to deep research questions in theoretical signal processing and applied harmonic analysis. This survey paper is an outcome of the recent workshop Phase Retrieval in Mathematics and Applications (PRiMA) (held on August 5--9 2024 at the Lorentz Center in Leiden, The Netherlands) that brought together experts working on theoretical and practical aspects of the phase retrieval problem with the purpose to formulate and explore essential open problems in the field.

We describe a phase-retrieval-based imaging method to directly spatially resolve the vector lattice distortions in an extended crystalline sample by explicit coupling of independent Bragg ptychography data sets into the reconstruction process. Our method addresses this multi-peak Bragg ptychography (MPBP) inverse problem by explicit gradient descent optimization of an objective function based on modeling of the probe-lattice interaction, along with corrective steps to address spurious reconstruction artifacts. Robust convergence of the optimization process is ensured by computing exact gradients with the automatic differentiation capabilities of high-performance computing software packages. We demonstrate MPBP reconstruction with simulated ptychography data mimicking diffraction from a single crystal membrane containing heterogeneities that manifest as phase discontinuities in the diffracted wave. We show the superior ability of such an optimization-based approach in removing reconstruction artifacts compared to existing phase retrieval and lattice distortion reconstruction approaches.

The logarithmic derivative of a point process plays a key role in the general approach, due to the third author, to constructing diffusions preserving a given point process. In this paper we explicitly compute the logarithmic derivative for determinantal processes on $\mathbb{R}$ with integrable kernels, a large class that includes all the classical processes of random matrix theory as well as processes associated with de Branges spaces. The argument uses the quasi-invariance of our processes established by the first author.

We define a metric in the space of positive finite positive measures that extends the 2-Wasserstein metric, i.e. its restriction to the set of probability measures is the 2-Wasserstein metric. We prove a dual and a dynamic formulation and extend the gradient flow machinery of the Wasserstein space. In addition, we relate the barycenter in this space to the barycenter in the Wasserstein space of the normalized measures.

Objective. The main goal of this work is to develop a model for multi-sensor signals such as MEG or EEG signals, that accounts for the inter-trial variability, suitable for corresponding binary classification problems. An important constraint is that the model be simple enough to handle small size and unbalanced datasets, as often encountered in BCI type experiments. Approach. The method involves linear mixed effects statistical model, wavelet transform and spatial filtering, and aims at the characterization of localized discriminant features in multi-sensor signals. After discrete wavelet transform and spatial filtering, a projection onto the relevant wavelet and spatial channels subspaces is used for dimension reduction. The projected signals are then decomposed as the sum of a signal of interest (i.e. discriminant) and background noise, using a very simple Gaussian linear mixed model. Main results. Thanks to the simplicity of the model, the corresponding parameter estimation problem is simplified. Robust estimates of class-covariance matrices are obtained from small sample sizes and an effective Bayes plug-in classifier is derived. The approach is applied to the detection of error potentials in multichannel EEG data, in a very unbalanced situation (detection of rare events). Classification results prove the relevance of the proposed approach in such a context. Significance. The combination of linear mixed model, wavelet transform and spatial filtering for EEG classification is, to the best of our knowledge, an original approach, which is proven to be effective. This paper improves on earlier results on similar problems, and the three main ingredients all play an important role.

Explanations obtained from transformer-based architectures in the form of raw attention, can be seen as a class-agnostic saliency map. Additionally, attention-based pooling serves as a form of masking the in feature space. Motivated by this observation, we design an attention-based pooling mechanism intended to replace Global Average Pooling (GAP) at inference. This mechanism, called Cross-Attention Stream (CA-Stream), comprises a stream of cross attention blocks interacting with features at different network depths. CA-Stream enhances interpretability in models, while preserving recognition performance.

The classical Transfer-Matrix Method (TMM) is often used to calculate the input impedance of woodwind instruments. However, the TMM ignores the possible influence of the radiated sound from toneholes on other open holes. In this paper a method is proposed to account for external tonehole interactions. We describe the Transfer-Matrix Method with external Interaction (TMMI) and then compare results using this approach with the Finite Element Method (FEM) and TMM, as well as with experimental data. It is found that the external tonehole interactions increase the amount of radiated energy, reduce slightly the lower resonance frequencies, and modify significantly the response near and above the tonehole lattice cutoff frequency. In an appendix, a simple perturbation of the TMM to account for external interactions is investigated, though it is found to be inadequate at low frequencies and for holes spaced far apart.

Infotaxis is a popular search algorithm designed to track a source of odor in a turbulent environment using information provided by odor detections. To exemplify its capabilities, the source-tracking task was framed as a partially observable Markov decision process consisting in finding, as fast as possible, a stationary target hidden in a 2D grid using stochastic partial observations of the target location. Here we provide an extended review of infotaxis, together with a toolkit for devising better strategies. We first characterize the performance of infotaxis in domains from 1D to 4D. Our results show that, while being suboptimal, infotaxis is reliable (the probability of not reaching the source approaches zero), efficient (the mean search time scales as expected for the optimal strategy), and safe (the tail of the distribution of search times decays faster than any power law, though subexponentially). We then present three possible ways of beating infotaxis, all inspired by methods used in artificial intelligence: tree search, heuristic approximation of the value function, and deep reinforcement learning. The latter is able to find, without any prior human knowledge, the (near) optimal strategy. Altogether, our results provide evidence that the margin of improvement of infotaxis toward the optimal strategy gets smaller as the dimensionality increases.

Given its unrivaled potential of integration and scalability, silicon is likely to become a key platform for large-scale quantum technologies. Individual electron-encoded artificial atoms either formed by impurities or quantum dots have emerged as a promising solution for silicon-based integrated quantum circuits. However, single qubits featuring an optical interface needed for large-distance exchange of information have not yet been isolated in such a prevailing semiconductor. Here we show the isolation of single optically-active point defects in a commercial silicon-on-insulator wafer implanted with carbon atoms. These artificial atoms exhibit a bright, linearly polarized single-photon emission at telecom wavelengths suitable for long-distance propagation in optical fibers. Our results demonstrate that despite its small bandgap (~ 1.1 eV) a priori unfavorable towards such observation, silicon can accommodate point defects optically isolable at single scale, like in wide-bandgap semiconductors. This work opens numerous perspectives for silicon-based quantum technologies, from integrated quantum photonics to quantum communications and metrology.

We present here a general method based on the investigation of the relative energy of the system, that provides an unconditional error estimate for the approximate solution of the barotropic Navier Stokes equations obtained by time and space discretization. We use this methodology to derive an error estimate for a specific DG/finite element scheme for which the convergence was proved in [27]. This is an extended version of the paper submitted to IMAJNA.

The study of quantum quasi-particles at low temperatures including their statistics, is a frontier area in modern physics. In a seminal paper F.D. Haldane proposed a definition based on a generalization of the Pauli exclusion principle for fractional quantum statistics. The present paper is a study of quantum quasi-particles obeying Haldane statistics in a fully non-linear kinetic Boltzmann equation model with large initial data on a torus. Strong L1 solutions are obtained for the Cauchy problem. The main results concern existence, uniqueness and stability. Depending on the space dimension and the collision kernel, the results obtained are local or global in time.

We report the fabrication of G centers in silicon with an areal density compatible with single photon emission at optical telecommunication wavelengths. Our sample is made from a silicon-on-insulator wafer which is locally implanted with carbon ions and protons at various fluences. Decreasing the implantation fluences enables to gradually switch from large ensembles to isolated single defects, reaching areal densities of G centers down to $\sim$0.2 $\mu$m$^{-2}$. Single defect creation is demonstrated by photon antibunching in intensity-correlation experiments, thus establishing our approach as a reproducible procedure for generating single artificial atoms in silicon for quantum technologies.