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Neural operators such as the Fourier Neural Operator (FNO) have been shown to provide resolution-independent deep learning models that can learn mappings between function spaces. For example, an initial condition can be mapped to the solution of a partial differential equation (PDE) at a future time-step using a neural operator. Despite the popularity of neural operators, their use to predict solution functions over a domain given only data over the boundary (such as a spatially varying Dirichlet boundary condition) remains unexplored. In this paper, we refer to such problems as boundary-to-domain problems; they have a wide range of applications in areas such as fluid mechanics, solid mechanics, heat transfer etc. We present a novel FNO-based architecture, named Lifting Product FNO (or LP-FNO) which can map arbitrary boundary functions defined on the lower-dimensional boundary to a solution in the entire domain. Specifically, two FNOs defined on the lower-dimensional boundary are lifted into the higher dimensional domain using our proposed lifting product layer. We demonstrate the efficacy and resolution independence of the proposed LP-FNO for the 2D Poisson equation., Comment: Accepted by ICML 2024 AI for Science Workshop

We present the highest resolution images to date of caustics formed by wave dark matter ($\psi$DM) fluctuations near the critical curves of cluster gravitational lenses. We describe the basic magnification features of $\psi$DM in the source plane at high macromodel magnification and discuss specific differences between the $\psi$DM and standard cold dark matter (CDM) models. The unique generation of demagnified counterimages formed outside the Einstein radius for $\psi$DM is highlighted. Substructure in CDM cannot generate such demagnified images of positive parity, thus providing a definitive way to distinguish $\psi$DM from CDM. Highly magnified background sources with sizes $r\approx 1pc$, or approximately a factor of ten smaller than the expected de Broglie wavelength of $\psi$DM, offer the best possibility of discriminating between $\psi$DM and CDM. These include objects such as very compact stellar clusters at high redshift that JWST is finding in abundance., Comment: 8 pages

We have characterized the ionized, neutral, and warm molecular gas kinematics in the Seyfert 1 galaxy NGC 3227 using observations from the Hubble Space Telescope Space Telescope Imaging Spectrograph, Apache Point Observatory's Kitt Peak Ohio State Multi-Object Spectrograph, Gemini-North's Near-Infrared Integral Field Spectrometer, and the Atacama Large Millimeter Array. We fit multiple Gaussians to several spatially-resolved emission lines observed with long-slit and integral-field spectroscopy and isolate the kinematics based on apparent rotational and outflowing motions. We use the kinematics to determine an orientation for the bicone along which the outflows travel, and find that the biconical structure has an inclination of $40 ^{+5}_{-4}${\deg} from our line of sight, and a half-opening angle with an inner and outer boundary of $47 ^{+6}_{-2}${\deg} and $68 ^{+1}_{-1}${\deg}, respectively. We observe ionized outflows traveling 500 km s$^{-1}$ at distances up to 7$''$ (800 pc) from the SMBH, and disturbed ionized gas up to a distance of 15$''$ (1.7 kpc). Our analysis reveals that the ionized outflows are launched from within 20 pc of the SMBH, at the same location as a bridge of cold gas across the nucleus detected in ALMA CO(2-1) observations. We measure a turnover radius where the gas starts decelerating at a distance of $26 \pm 6$ pc from the AGN. Compared to a turnover radius in the range of $31- 63$ pc from a radiative driving model, we confirm that radiative driving is the dominant acceleration mechanism for the narrow line region (NLR) outflows in NGC 3227., Comment: 26 pages, 17 figures

Microlensed stars recently discovered by JWST & HST follow closely the winding critical curve of A370 along all sections of the ``Dragon Arc" traversed by the critical curve. These transients are fainter than $m_{AB}>26.5$, corresponding to the Asymptotic Giant Branch (AGB) and microlensed by diffuse cluster stars observed with $\simeq 18M_\odot/pc^2$, or about $\simeq 1$\% of the projected dark matter density. Most microlensed stars appear along the inner edge of the critical curve, following an asymmetric band of width $\simeq 4$kpc that is skewed by $-0.7\pm0.2$kpc. Some skewness is expected as the most magnified images should form along the inner edge of the critical curve with negative parity, but the predicted shift is small $\simeq -0.04$kpc and the band of predicted detections is narrow, $\simeq 1.4$kpc. Adding CDM-like dark halos of $10^{6-8}M_\odot$ broadens the band as desired but favours detections along the outer edge of the critical curve, in the wrong direction, where sub-halos generate local Einstein rings. Instead, the interference inherent to ``Wave Dark Matter" as a Bose-Einstein condensate ($\psi$DM) forms a symmetric band of critical curves that favours negative parity detections. A de Broglie wavelength of $\simeq 10$pc matches well the observed $4$kpc band of microlenses and predicts negative skewness $\simeq -0.6$kpc, similar to the data. The implied corresponding boson mass is $\simeq 10^{-22}$eV, in good agreement with estimates from dwarf galaxy cores when scaled by momentum. Further JWST imaging may reveal the pattern of critical curves by simply ``joining the dots" between microlensed stars, allowing wave corrugations of $\psi$DM to be distinguished from CDM sub-halos, Comment: 12 pages, 5 figures

Hyperbolic umbilic (HU) is a point singularity of the gravitational lens equation, giving rise to a ring-shaped image formation made of four highly magnified images, off-centred from the lens centre. Recent observations have revealed new strongly lensed image formations near HU singularities, and many more are expected in ongoing and future observations. Like fold/cusp, image formations near HU also satisfy magnification relation ($R_{\rm hu}$), i.e., the signed magnification sum of the four images equals zero. Here, we study how $R_{\rm hu}$ deviates from zero as a function of area ($A_{\rm hu}$) covered by the image formation near HU and the distance ($d$) of the central maxima image (which is part of the HU image formation) from the lens centre for ideal single- and double-component cluster-scale lenses. For lens ellipticity values $\geq0.3$, the central maxima image will form sufficiently far from the lens centre ($d\gtrsim5''$), similar to the observed HU image formations with $R_{\rm hu}\simeq0$. We also find that, in some cases, double-component and actual cluster-scale lenses can lead to large cross-sections for HU image formations for sources at $z\gtrsim5$, effectively increasing the chances to observe HU image formation at high redshifts. Finally, we study the time delay distribution in the observed HU image formations, finding that not only are these images highly magnified, but the relative time delay between various pairs of HU characteristic image formation has a typical value of $\sim100$ days, an order of magnitude smaller than generic five-image formations in cluster lenses, making such image formations optimal targets for time delay cosmography studies., Comment: 14 pages, 11 figures, 1 table. Accepted in OJAp

We investigate the dielectric and impedance spectroscopic investigation of Pr-doped NASICON type Na$_{3+x}$Zr$_{2-x}$Pr$_{x}$Si$_2$PO$_{\rm 12}$ ($x=$ 0.05--0.2) samples as a function of temperature and frequency. The Rietveld refinement of x-ray diffraction patterns confirms the monoclinic phase having C2/c space groups for all the samples. The scanning electron microscopy shows the granular-like structure and energy dispersive x-ray analysis confirms the desired compositions. The temperature (90--400~K) and frequency (20 Hz-2 MHz) dependence of electric permittivity are explained using Maxwell-Wagner-Sillars (MWS) polarization and space charge polarization mechanisms. The dielectric relaxation shows nearly equal activation energy for all the samples with a non-Debye type of relaxation in the measured temperature range. The complex impedance analysis shows the presence of broad grain and grain boundary relaxation peaks. The stretched exponent analysis of electric modulus using the Kohlrausch-Williams-Watts (KWW) function further confirms the non-Debye type of relaxation. Moreover, scaling analysis of the electric modulus shows a similar type of relaxation for all the samples. The {\it a.c.} conductivity data are fitted using modified power law, where the temperature dependence of exponent ($s$) confirms the correlated barrier hopping (CBH) type conduction for all the samples. Our results indicate that the Pr doped NASICON samples are potential candidates for charge storage devices due to their large electric permittivity., Comment: submitted

In 1990, Jakeman (see \cite{jakeman1990statistics}) defined the binomial process as a special case of the classical birth-death process, where the probability of birth is proportional to the difference between a fixed number and the number of individuals present. Later, a fractional generalization of the binomial process was studied by Cahoy and Polito (2012) (see \cite{cahoy2012fractional}) and called it as fractional binomial process (FBP). In this paper, we study second-order properties of the FBP and the long-range behavior of the FBP and its noise process. We also estimate the parameters of the FBP using the method of moments procedure. Finally, we present the simulated sample paths and its algorithm for the FBP., Comment: 18 pages, 4 figures

Brain-computer interface (BCI) systems facilitate unique communication between humans and computers, benefiting severely disabled individuals. Despite decades of research, BCIs are not fully integrated into clinical and commercial settings. It's crucial to assess and explain BCI performance, offering clear explanations for potential users to avoid frustration when it doesn't work as expected. This work investigates the efficacy of different deep learning and Riemannian geometry-based classification models in the context of motor imagery (MI) based BCI using electroencephalography (EEG). We then propose an optimal transport theory-based approach using earth mover's distance (EMD) to quantify the comparison of the feature relevance map with the domain knowledge of neuroscience. For this, we utilized explainable AI (XAI) techniques for generating feature relevance in the spatial domain to identify important channels for model outcomes. Three state-of-the-art models are implemented - 1) Riemannian geometry-based classifier, 2) EEGNet, and 3) EEG Conformer, and the observed trend in the model's accuracy across different architectures on the dataset correlates with the proposed feature relevance metrics. The models with diverse architectures perform significantly better when trained on channels relevant to motor imagery than data-driven channel selection. This work focuses attention on the necessity for interpretability and incorporating metrics beyond accuracy, underscores the value of combining domain knowledge and quantifying model interpretations with data-driven approaches in creating reliable and robust Brain-Computer Interfaces (BCIs)., Comment: 8 pages, 3 figures, 3 tables, draft of the accepted work at IJCNN, WCCI 2024

Decoding EEG during motor imagery is pivotal for the Brain-Computer Interface (BCI) system, influencing its overall performance significantly. As end-to-end data-driven learning methods advance, the challenge lies in balancing model complexity with the need for human interpretability and trust. Despite strides in EEG-based BCIs, challenges like artefacts and low signal-to-noise ratio emphasise the ongoing importance of model transparency. This work proposes using post-hoc explanations to interpret model outcomes and validate them against domain knowledge. Leveraging the GradCAM post-hoc explanation technique on the motor imagery dataset, this work demonstrates that relying solely on accuracy metrics may be inadequate to ensure BCI performance and acceptability. A model trained using all EEG channels of the dataset achieves 72.60% accuracy, while a model trained with motor-imagery/movement-relevant channel data has a statistically insignificant decrease of 1.75%. However, the relevant features for both are very different based on neurophysiological facts. This work demonstrates that integrating domain-specific knowledge with XAI techniques emerges as a promising paradigm for validating the neurophysiological basis of model outcomes in BCIs. Our results reveal the significance of neurophysiological validation in evaluating BCI performance, highlighting the potential risks of exclusively relying on performance metrics when selecting models for dependable and transparent BCIs., Comment: 5 pages, 4 figures, 1 table, accepted draft for IEEE EMBC 2024

High-entropy alloy (HEA) superconductors have attracted significant attention due to their exceptional low-temperature mechanical and superconducting properties. We report the synthesis and thorough characterization of an equiatomic HEA superconductor with the composition Sc$_{0.20}$V$_{0.20}$Ti$_{0.20}$Hf$_{0.20}$Nb$_{0.20}$, crystallizing in a body-centered cubic crystal structure (Im3$\bar{m}$). Our investigation, using magnetization, transport, and heat capacity measurements, reveals the presence of weakly coupled, fully gapped superconductivity with a transition temperature of 4.17(3) K and the upper critical field exceeding the Pauli paramagnetic limit. The metallic nature, combined with a high upper critical field, positions it as a promising candidate for applications in superconducting devices., Comment: 7 pages, 5 figures

We develop time-series machine learning (ML) methods for closure modeling of the Unsteady Reynolds Averaged Navier Stokes (URANS) equations applied to stably stratified turbulence (SST). SST is strongly affected by fine balances between forces and becomes more anisotropic in time for decaying cases. Moreover, there is a limited understanding of the physical phenomena described by some of the terms in the URANS equations. Rather than attempting to model each term separately, it is attractive to explore the capability of machine learning to model groups of terms, i.e., to directly model the force balances. We consider decaying SST which are homogeneous and stably stratified by a uniform density gradient, enabling dimensionality reduction. We consider two time-series ML models: Long Short-Term Memory (LSTM) and Neural Ordinary Differential Equation (NODE). Both models perform accurately and are numerically stable in a posteriori tests. Furthermore, we explore the data requirements of the ML models by extracting physically relevant timescales of the complex system. We find that the ratio of the timescales of the minimum information required by the ML models to accurately capture the dynamics of the SST corresponds to the Reynolds number of the flow. The current framework provides the backbone to explore the capability of such models to capture the dynamics of higher-dimensional complex SST flows.