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We solve by Chebyshev spectral collocation some genuinely nonlinear Liouville-Bratu-Gelfand type, 1D and a 2D boundary value problems. The problems are formulated on the square domain $[-1, 1]\times[-1, 1]$ and the boundary condition…

Simplicial complexes prove effective in modeling data with multiway dependencies, such as data defined along the edges of networks or within other higher-order structures. Their spectrum can be decomposed into three interpretable subspaces…

Chebyshev interpolation polynomials exhibit the exponential approximation property to analytic functions on a cube. Based on the Chebyshev interpolation polynomial approximation, we propose iterative polynomial approximation algorithms to…

Probabilistic representations, such as Bayesian and Markov networks, are fundamental to much of statistical machine learning. Thus, learning probabilistic representations directly from data is a deep challenge, the main computational…

Estimating treatment effects from observational data is challenging due to two main reasons: (a) hidden confounding, and (b) covariate mismatch (control and treatment groups not having identical distributions). Long lines of works exist…

Calculating the spectral function of two dimensional systems is arguably one of the most pressing challenges in modern computational condensed matter physics. While efficient techniques are available in lower dimensions, two dimensional…

Score-based explainable machine-learning techniques are often used to understand the logic behind black-box models. However, such explanation techniques are often computationally expensive, which limits their application in time-critical…

Harmonic Balance is one of the most popular methods for computing periodic solutions of nonlinear dynamical systems. In this work, we address two of its major shortcomings: First, we investigate to what extent the computational burden of…

A Chebyshev expansion is a series in the basis of Chebyshev polynomials of the first kind. When such a series solves a linear differential equation, its coefficients satisfy a linear recurrence equation. We interpret this equation as the…

Machine learning models that first learn a representation of a domain in terms of human-understandable concepts, then use it to make predictions, have been proposed to facilitate interpretation and interaction with models trained on…

Frequency-dependent correlations, such as the spectral function and the dynamical structure factor, help understand condensed matter experiments. Within the density matrix renormalization group (DMRG) framework, an accurate method for…

For smooth random dynamical systems we consider the quenched linear and higher-order response of equivariant physical measures to perturbations of the random dynamics. We show that the spectral perturbation theory of Gou\"ezel, Keller, and…

We propose an inverse-design approach for computational spectrometers in which the scattering media are topology-optimized to achieve better performance in inference of unknown spectra. Unlike traditional end-to-end approaches, our inverse…

We introduce the conformally-invariant scalar product, originally devised for radiation fields, to the study of the modes of optical resonators. This scalar product allows one to normalize and compare resonant modes using their…

Conformal Prediction (CP) is a principled framework for quantifying uncertainty in blackbox learning models, by constructing prediction sets with finite-sample coverage guarantees. Traditional approaches rely on scalar nonconformity scores,…

Resonance based numerical schemes are those in which cancellations in the oscillatory components of the equation are taken advantage of in order to reduce the regularity required of the initial data to achieve a particular order of error…

Statistical analysis on compositional data has gained a lot of attention due to their great potential of applications. A feature of these data is that they are multivariate vectors that lie in the simplex, that is, the components of each…

Using neural networks to solve variational problems, and other scientific machine learning tasks, has been limited by a lack of consistency and an inability to exactly integrate expressions involving neural network architectures. We address…

Functional quadratic regression models postulate a polynomial relationship between a scalar response rather than a linear one. As in functional linear regression, vertical and specially high-leverage outliers may affect the classical…

Pseudospectral analysis is fundamental for quantifying the sensitivity and transient behavior of nonnormal matrices, yet its computational cost scales cubically with dimension, rendering it prohibitive for large-scale systems. While…