arXiv subjects

Yahya Saleh

Publications and source records attributed to Yahya Saleh.

11 recordsLinked to original sources

arXiv2026

Convergence theory for Hermite approximations under adaptive coordinate transformations

Recent work has shown that parameterizing and optimizing coordinate transformations using normalizing flows, i.e., invertible neural networks, can significantly accelerate the convergence of spectral approximations. We present the first error estimates for approximating functions using Hermite expansions composed with adaptive coordinate transformations. Our analysis establishes an equivalence principle: approximating a function $f$ in the span of the transformed basis is equivalent to approximating the pullback of $f$ in the span of Hermite functions. This allows us to leverage the classical approximation theory of Hermite expansions to derive error estimates in transformed coordinates in terms of the regularity of the pullback. We present an example demonstrating how a nonlinear coordinate transformation can enhance the convergence of Hermite expansions. Focusing on smooth functions decaying along the real axis, we construct a monotone transport map that aligns the decay of the target function with the Hermite basis. This guarantees spectral convergence rates for the corresponding Hermite expansion. Our analysis provides theoretical insight into the convergence behavior of adaptive Hermite approximations based on normalizing flows, as recently explored in the computational quantum physics literature.

math.NA↗

arXiv2025

Enhancing polynomial approximation of continuous functions by composition with homeomorphisms

We enhance the approximation capabilities of algebraic polynomials by composing them with homeomorphisms. This composition yields families of functions that remain dense in the space of continuous functions, while enabling more accurate approximations. For univariate continuous functions exhibiting a finite number of local extrema, we prove that there exist a polynomial of finite degree and a homeomorphism whose composition approximates the target function to arbitrary accuracy. The construction is especially relevant for multivariate approximation problems, where standard numerical methods often suffer from the curse of dimensionality. To support our theoretical results, we investigate both regression tasks and the construction of molecular potential-energy surfaces, parametrizing the underlying homeomorphism using invertible neural networks. The numerical experiments show strong agreement with our theoretical analysis.

math.NA↗

arXiv2025

Inducing Riesz and orthonormal bases in $L^2$ via composition operators

Let $C_h$ be a composition operator mapping $L^2(Ω_1)$ into $L^2(Ω_2)$ for some open sets $Ω_1, Ω_2 \subseteq \mathbb{R}^n$. We characterize the mappings $h$ that transform Riesz bases of $L^2(Ω_1)$ into Riesz bases of $L^2(Ω_2)$. Restricting our analysis to differentiable mappings, we demonstrate that mappings $h$ that preserve Riesz bases have Jacobian determinants that are bounded away from zero and infinity. We discuss implications of these results for approximation theory, highlighting the potential of using bijective neural networks to construct Riesz bases with favorable approximation properties.

math.FA↗

arXiv2025

Transferability and interpretability of vibrational normalizing-flow coordinates

The choice of vibrational coordinates is crucial for the accuracy, efficiency, and interpretability of molecular vibrational dynamics and spectra calculations. We explore the recently proposed normalizing-flow vibrational coordinates, which are learned molecule-specific coordinate transformations optimized for a given basis set. Much like how spherical coordinates naturally simplify the hydrogen atom by embedding physical insight into the coordinate system, normalizing-flow coordinates offload complexity from the basis functions into the coordinate transformation itself. This shift not only improves basis-set convergence, but also enhances interpretability of vibrational motions. We provide an analysis of the utility, interpretation and associated constraints of normalizing-flow vibrational coordinates. Moreover, we demonstrate that these coordinates can be generalized across different isotopologues, and even structurally related molecules, achieved with minimal fine-tuning of selected output parameters.

physics.chem-ph↗

arXiv2025

Taylor-mode automatic differentiation for constructing molecular rovibrational Hamiltonian operators

We present an automated framework for constructing Taylor series expansions of rovibrational kinetic and potential energy operators for arbitrary molecules, internal coordinate systems, and molecular frame embedding conditions. Expressing operators in a sum-of-products form allows for computationally efficient evaluations of matrix elements in product basis sets. Our approach uses automatic differentiation tools from the Python machine learning ecosystem, particularly the JAX library, to efficiently and accurately generate high-order Taylor expansions of rovibrational operators. The implementation is available at https://github.com/robochimps/vibrojet.

physics.atm-clus↗

arXiv2025

Laser micromachining of arbitrarily complex and overhang-free SiN nanomechanical resonators

Research on silicon nitride (SiN) nanomechanical resonators produces an exceptionally rich variety of resonator geometries, for which there is currently no available rapid prototyping solution. Experimental advances in nanobeam, trampoline, phononic bandgap, and soft-clamping structures all rely on conventional nanofabrication involving e-beam or photolithography, followed by various etching steps. These techniques are typically time-consuming, relatively inflexible, and often result in spurious residual SiN overhang that can degrade mechanical quality factors. In contrast, recent work has shown that simple resonant structures, such as nanobeams, can be prototyped by direct laser ablation of free-standing SiN membranes using a spatially distributed sequence of microholes that limits stress concentration. However, these early demonstrations were restricted to basic shapes, created by manually combining ablation routines for circles and straight lines. Here, we demonstrate the fabrication of arbitrarily complex geometries using an open-source software toolset--released with this publication--that automatically generates laser-ablated hole sequences directly from standard semiconductor layout files (i.e., GDSII). The software includes a layout alignment tool that compensates for the membrane orientation and dimensional variations, limiting material overhang to ~2 um. Using this toolset, we fabricate several resonator geometries, each in under 1 hour, two of which are exhaustively characterized as candidate structures for high-performance radiation sensing. The measured quality factors of these structures closely match finite element simulations and reach values up to 3.7 x 10^6. From these measurements, we extract material quality factors above 3700, which is on par with low-stress SiN unablated plain membranes and with comparable structures produced using conventional fabrication methods.

physics.optics↗

arXiv2025

Computing excited states of molecules using normalizing flows

Calculations of highly excited and delocalized molecular vibrational states are computationally challenging tasks, which strongly depends on the choice of coordinates for describing vibrational motions. We introduce a new method that leverages normalizing flows -- parametrized invertible functions -- to learn optimal vibrational coordinates that satisfy the variational principle. This approach produces coordinates tailored to the vibrational problem at hand, significantly increasing the accuracy and enhancing basis-set convergence of the calculated energy spectrum. The efficiency of the method is demonstrated in calculations of the 100 lowest excited vibrational states of H$_2$S, H$_2$CO, and HCN/HNC. The method effectively captures the essential vibrational behavior of molecules by enhancing the separability of the Hamiltonian and hence allows for an effective assignment of approximate quantum numbers. We demonstrate that the optimized coordinates are transferable across different levels of basis-set truncation, enabling a cost-efficient protocol for computing vibrational spectra of high-dimensional systems.

physics.chem-ph↗

arXiv2024

Bounds on the Generalization Error in Active Learning

We establish empirical risk minimization principles for active learning by deriving a family of upper bounds on the generalization error. Aligning with empirical observations, the bounds suggest that superior query algorithms can be obtained by combining both informativeness and representativeness query strategies, where the latter is assessed using integral probability metrics. To facilitate the use of these bounds in application, we systematically link diverse active learning scenarios, characterized by their loss functions and hypothesis classes to their corresponding upper bounds. Our results show that regularization techniques used to constraint the complexity of various hypothesis classes are sufficient conditions to ensure the validity of the bounds. The present work enables principled construction and empirical quality-evaluation of query algorithms in active learning.

stat.ML↗

arXiv2023

Synthesizing Cough Audio with GAN for COVID-19 Detection

For this final year project, the goal is to add to the published works within data synthesis for health care. The end product of this project is a trained model that generates synthesized images that can be used to expand a medical dataset (Pierre, 2021). The chosen domain for this project is the Covid-19 cough recording which is have been proven to be a viable data source for detecting Covid. This is an under-explored domain despite its huge importance because of the limited dataset available for the task. Once this model is developed its impact will be illustrated by training state-of-the-art models with and without the expanded dataset and measuring the difference in performance. Lastly, everything will be put together by embedding the model within a web application to illustrate its power. To achieve the said goals, an extensive literature review will be conducted into the recent innovations for image synthesis using generative models.

cs.SD↗

arXiv2022

Augmenting Basis Sets by Normalizing Flows

Approximating functions by a linear span of truncated basis sets is a standard procedure for the numerical solution of differential and integral equations. Commonly used concepts of approximation methods are well-posed and convergent, by provable approximation orders. On the down side, however, these methods often suffer from the curse of dimensionality, which limits their approximation behavior, especially in situations of highly oscillatory target functions. Nonlinear approximation methods, such as neural networks, were shown to be very efficient in approximating high-dimensional functions. We investigate nonlinear approximation methods that are constructed by composing standard basis sets with normalizing flows. Such models yield richer approximation spaces while maintaining the density properties of the initial basis set, as we show. Simulations to approximate eigenfunctions of a perturbed quantum harmonic oscillator indicate convergence with respect to the size of the basis set.

math.NA↗

arXiv2021

Active learning of potential-energy surfaces of weakly-bound complexes with regression-tree ensembles

Several pool-based active learning algorithms (AL) were employed to model potential energy surfaces (PESs) with a minimum number of electronic structure calculations. Theoretical and empirical results suggest that superior strategies can be obtained by sampling molecular structures corresponding to large uncertainties in their predictions while at the same time not deviating much from the true distribution of the data. To model PESs in an AL framework we propose to use a regression version of stochastic query by forest, a hybrid method that samples points corresponding to large uncertainties while avoiding collecting too many points from sparse regions of space. The algorithm is implemented with decision trees that come with relatively small computational costs. We empirically show that this algorithm requires around half the data to converge to the same accuracy in comparison to the uncertainty-based query-by-committee algorithm. Moreover, the algorithm is fully automatic and does not require any prior knowledge of the PES. Simulations on a 6D PES of \pyrrolew show that $\mathord{<}15\,000$ configurations are enough to build a PES with a generalization error of 16~\invcm, whereas the final model with around 50\,000 configurations has a generalization error of 11~\invcm.

physics.chem-ph↗