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When solving the time-dependent radiative transport equation (RTE), implicit time discretization is often employed for its robustness and stability. This results in a sequence of steady-state RTEs with identical cross-sections but varying…

The overfitting is one of the cursing subjects in the deep learning field. To solve this challenge, many approaches were proposed to regularize the learning models. They add some hyper-parameters to the model to extend the generalization;…

This article is an extended version of previous work of the authors [40, 41] on low-rank matrix estimation in the presence of constraints on the factors into which the matrix is factorized. Low-rank matrix factorization is one of the basic…

Large Language Models (LLMs) have transformed both everyday life and scientific research. However, adapting LLMs from general-purpose models to specialized tasks remains challenging, particularly in resource-constrained environments.…

In this work, we design and analyze a Discrete de Rham (DDR) scheme for a contact mechanics problem involving fractures along which a model of Tresca friction is considered. Our approach is based on a mixed formulation involving a…

In this paper, we give a rigorous mathematical justification for the relaxation time approximation (RTA) model. We find that only the RTA with an energy-independent relaxation time can be justified in the case of hard interactions.…

In continual learning (CL), catastrophic forgetting often arises due to feature drift. This challenge is particularly prominent in the exemplar-free continual learning (EFCL) setting, where samples from previous tasks cannot be retained,…

Compression has emerged as one of the essential deep learning research topics, especially for the edge devices that have limited computation power and storage capacity. Among the main compression techniques, low-rank compression via matrix…

We propose and analyze a second-order Strang splitting method for a class of stiff matrix differential equations with Sylvester-type structure. The method splits the dynamics into a stiff linear part, treated exactly via matrix…

We propose a numerical integrator for determining low-rank approximations to solutions of large-scale matrix differential equations. The considered differential equations are semilinear and stiff. Our method consists of first splitting the…

The computation of the radiative transfer equation is expensive mainly due to two stiff terms: the transport term and the collision operator. The stiffness in the former comes from the fact that particles (such as photons) travels at the…

Low-Rank Adaptation (LoRA) has become the de facto method for parameter-efficient fine-tuning of large language models (LLMs), enabling rapid adaptation to diverse domains. In production, LoRA-based models are served at scale, creating…

Atomic and molecular breakup reactions, such as multiple-ionisation, are described by a driven Schr\"odinger equation. This equation is equivalent to a high-dimensional Helmholtz equation and it has solutions that are outgoing waves,…

Collective coherent scattering of laser light by an ensemble of polarizable point particles creates long range interactions, whose properties can be tailored by choice of injected laser powers, frequencies and polarizations. We use a…

Fine-tuning Large Language Models (LLMs) remains structurally uncertain despite parameter-efficient methods such as Low-Rank Adaptation (LoRA), as the layer-specific roles of internal representations are poorly understood, leading to…

In recent years, Large Language Models (LLMs) have demonstrated remarkable capabilities across a wide range of natural language processing (NLP) tasks, such as question-answering, sentiment analysis, text summarization, and machine…

Large language models (LLMs) have rapidly scaled in size, bringing severe memory and computational challenges that hinder their deployment. Singular Value Decomposition (SVD)-based compression has emerged as an appealing post-training…

Stochastic reduced-order models are widely used to represent the effective dynamics of complex systems, but estimating their drift and diffusion coefficients from data remains challenging. Standard approaches often rely on short-time…

Rigorous coupled-wave analysis (RCWA) is a very effective tool for the studying optical properties of multilayered vertically invariant periodic structures. However, it fails to deal with arrays of small particles because of high gradients…

The substantial computational demands of modern large-scale deep learning present significant challenges for efficient training and deployment. Recent research has revealed a widespread phenomenon wherein deep networks inherently learn…