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Formally investigating the sources of students' difficulties around specific subjects is crucial for developing appropriate strategies to help students. We have been studying difficulties in understanding magnetism encountered by students…
Separation of variables can be a powerful technique for solving many of the partial differential equations that arise in physics contexts. Upper-division physics students encounter this technique in multiple topical areas including…
Usage, manipulation, transport, delivery, and mixing of granular or particulate media, comprised of spherical or polyhedral particles, is commonly encountered in industrial sectors of construction (cement and rock fragments), pharmaceutics…
Stochastic mathematical models are essential tools for understanding and predicting complex phenomena. The purpose of this work is to study the exit times of a stochastic dynamical system-specifically, the mean exit time and the…
Mixed-dimensional partial differential equations arise in several physical applications, wherein parts of the domain have extreme aspect ratios. In this case, it is often appealing to model these features as lower-dimensional manifolds…
We provide a novel method for sensitivity analysis of parametric robust Markov chains. These models incorporate parameters and sets of probability distributions to alleviate the often unrealistic assumption that precise probabilities are…
In todays world the request for very complex models for huge data sets is rising steadily. The problem with these models is that by raising the complexity of the models, it gets much harder to interpret them. The growing field of…
We will discuss two approaches to estimating partial derivatives and the metric components; one utilizing past work describing a causal set $\Box$ operator, and one using a construction from linear algebra called the Moore-Penrose inverse.…
Physics-based and first-principles models pervade the engineering and physical sciences, allowing for the ability to model the dynamics of complex systems with a prescribed accuracy. The approximations used in deriving governing equations…
This paper presents with justifications a technique that is useful for the study of piecewise deterministic Markov decision processes (PDMDPs) with general policies and unbounded transition intensities. This technique produces an auxiliary…
Machine-learning models are ubiquitous. In some domains, for instance, in medicine, the models' predictions must be interpretable. Decision trees, classification rules, and subgroup discovery are three broad categories of supervised…
Physics-informed deep operator networks (DeepONets) have emerged as a promising approach toward numerically approximating the solution of partial differential equations (PDEs). In this work, we aim to develop further understanding of what…
Partial differential equations are a convenient way to describe reaction- advection-diffusion processes of signalling models. If only one cell type is present, and tissue dynamics can be neglected, the equations can be solved directly.…
Interpreting a nonparametric regression model with many predictors is known to be a challenging problem. There has been renewed interest in this topic due to the extensive use of machine learning algorithms and the difficulty in…
Partial-wave analysis is one step in a process connecting experimental measurements to the N* states we are studying. Progress has been made in the area of `model-independent' analysis. However, more model-dependent approaches are needed to…
A long-standing problem at the interface of artificial intelligence and applied mathematics is to devise an algorithm capable of achieving human level or even superhuman proficiency in transforming observed data into predictive mathematical…
Differentiation is a cornerstone of computing and data analysis in every discipline of science and engineering. Indeed, most fundamental physics laws are expressed as relationships between derivatives in space and time. However, derivatives…
The use in the action integral of a volume element of the form $\Phi d^{D}x$ where $\Phi$ is a metric independent measure can give new interesting results in all types of known generally coordinate invariant theories: (1) 4-D theories of…
In this article we examine recent developments in the research area concerning the creation of end-to-end models for the complete optimization of measuring instruments. The models we consider rely on differentiable programming methods and…
Scientists and practitioners increasingly rely on machine learning to model data and draw conclusions. Compared to statistical modeling approaches, machine learning makes fewer explicit assumptions about data structures, such as linearity.…