Related papers: Experts' understanding of partial derivatives usin…
Fractional differential operators provide an attractive mathematical tool to model effects with limited regularity properties. Particular examples are image processing and phase field models in which jumps across lower dimensional subsets…
Partial dependence curves (FPD) introduced by Friedman, are an important model interpretation tool, but are often not accessible to business analysts and scientists who typically lack the skills to choose, tune, and assess machine learning…
Equations governing physico-chemical processes are usually known at microscopic spatial scales, yet one suspects that there exist equations, e.g. in the form of Partial Differential Equations (PDEs), that can explain the system evolution at…
We investigate how students solve partial differential equations and partial derivatives in the context of quantum mechanics. We use the resources framework to investigate students' discussion in a group problem-solving environment to…
Differential equations (DEs) are commonly used to describe dynamic systems evolving in one (ordinary differential equations or ODEs) or in more than one dimensions (partial differential equations or PDEs). In real data applications the…
The approximation of solutions of partial differential equations (PDEs) with numerical algorithms is a central topic in applied mathematics. For many decades, various types of methods for this purpose have been developed and extensively…
Finding accurate solutions to partial differential equations (PDEs) is a crucial task in all scientific and engineering disciplines. It has recently been shown that machine learning methods can improve the solution accuracy by correcting…
A state-of-the-art deep domain decomposition method (D3M) based on the variational principle is proposed for partial differential equations (PDEs). The solution of PDEs can be formulated as the solution of a constrained optimization…
INTRODUCTION This papers deals with partial differential equations of second order, linear, with constant and not constant coefficients, in two variables, which admit real characteristics. I face the study of PDEs with the mentality of the…
Partial differential equations (PDEs) govern physical phenomena across the full range of scientific scales, yet their computational solution remains one of the defining challenges of modern science. This critical review examines two mature…
Surface metrology is the area of engineering concerned with the study of geometric variation in surfaces. This paper explores the potential for modern techniques from spatial statistics to act as generative models for geometric variation in…
Planning under uncertainty is critical to robotics. The Partially Observable Markov Decision Process (POMDP) is a mathematical framework for such planning problems. It is powerful due to its careful quantification of the non-deterministic…
The article proposes formulating and codifying a set of applied numerical methods, coined as Deep Learning Discrete Calculus (DLDC), that uses the knowledge from discrete numerical methods to interpret the deep learning algorithms through…
Proficiency with calculating, reporting, and understanding measurement uncertainty is a nationally recognized learning outcome for undergraduate physics lab courses. The Physics Measurement Questionnaire (PMQ) is a research-based assessment…
We present a contribution to the field of system identification of partial differential equations (PDEs), with emphasis on discerning between competing mathematical models of pattern-forming physics. The motivation comes from developmental…
Piecewise deterministic Markov processes (PDMPs) are a class of stochastic processes with applications in several fields of applied mathematics spanning from mathematical modeling of physical phenomena to computational methods. A PDMP is…
Modelling complex physical systems through partial differential equations (PDEs) is central to many disciplines in science and engineering. Yet in most real applications, unknown or incomplete relationships such as constitutive or thermal…
In this paper we provide a definition of fractional gradient operators, related to directional derivatives. We develop a fractional vector calculus, providing a probabilistic interpretation and mathematical tools to treat multidimensional…
Relating two quantities to describe a physical system or process is at the heart of "doing physics" for novices and experts alike. In this paper, we explore the ways in which experts use covariational reasoning when solving introductory…
We utilize extreme-learning machines for the prediction of partial differential equations (PDEs). Our method splits the state space into multiple windows that are predicted individually using a single model. Despite requiring only few data…