Related papers: Defining Foundation Models for Computational Scien…

We apply foundation models to data discovery and exploration tasks. Foundation models include large language models (LLMs) that show promising performance on a range of diverse tasks unrelated to their training. We show that these models…

Foundation models, large-scale, pre-trained deep-learning models adapted to a wide range of downstream tasks have gained significant interest lately in various deep-learning problems undergoing a paradigm shift with the rise of these…

Advances in machine learning over the past decade have resulted in a proliferation of algorithmic applications for encoding, characterizing, and acting on complex data that may contain many high dimensional features. Recently, the emergence…

AI is undergoing a paradigm shift with the rise of models (e.g., BERT, DALL-E, GPT-3) that are trained on broad data at scale and are adaptable to a wide range of downstream tasks. We call these models foundation models to underscore their…

We propose a finite-element local basis-based operator learning framework for solving partial differential equations (PDEs). Operator learning aims to approximate mappings from input functions to output functions, where the latter are…

This is a survey on the theory of adaptive finite element methods (AFEMs), which are fundamental in modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and…

Foundation models (FMs) have shown transformative potential in radiology by performing diverse, complex tasks across imaging modalities. Here, we developed CT-FM, a large-scale 3D image-based pre-trained model designed explicitly for…

Foundation models (FMs) are general-purpose artificial intelligence (AI) models that have recently enabled multiple brand-new generative AI applications. The rapid advances in FMs serve as an important contextual backdrop for the vision of…

Numerical approximations of partial differential equations (PDEs) are routinely employed to formulate the solution of physics, engineering, and mathematical problems involving functions of several variables, such as the propagation of heat…

We show that the DDR method can be interpreted as defining a computable consistent discrete $\mathrm{L}^2$ product on a conforming FES defined by PDEs. Without modifying the numerical method itself, this point of view provides an…

Recent advances in artificial intelligence have witnessed the emergence of large-scale deep learning models capable of interpreting and generating both textual and imaging data. Such models, typically referred to as foundation models, are…

As industrial products become abundant and sophisticated, visual industrial defect detection receives much attention, including two-dimensional and three-dimensional visual feature modeling. Traditional methods use statistical analysis,…

This book explores an alternative to the current dominant paradigm where a discrete computer model is constructed as an attempt to approximate some continuum theory. We focus on a class of discrete computer models that are based on simple…

The advent of foundation models (FMs), large-scale pre-trained models with strong generalization capabilities, has opened new frontiers for financial engineering. While general-purpose FMs such as GPT-4 and Gemini have demonstrated…

This paper introduces and explores a new programming paradigm, Model-based Programming, designed to address the challenges inherent in applying deep learning models to real-world applications. Despite recent significant successes of deep…

The finite element method (FEM) is a cornerstone numerical technique for solving partial differential equations (PDEs). Here, we present $\textbf{Qu-FEM}$, a fault-tolerant era quantum algorithm for the finite element method. In contrast to…

In this article we consider the widely used immersed finite element method (IFEM), in both explicit and implicit form, and its relationship to our more recent one-field fictitious domain method (FDM). We review and extend the formulation of…

Modeling of physical systems includes extensive use of software packages that implement the accurate finite element method for solving differential equations considered along with the appropriate initial and boundary conditions. When the…

The Finite Element Method (FEM) is the gold standard for spatial discretization in numerical simulations for a wide spectrum of real-world engineering problems. Prototypical areas of interest include linear heat transfer and linear…

The impressive performance of ChatGPT and other foundation-model-based products in human language understanding has prompted both academia and industry to explore how these models can be tailored for specific industries and application…