相关论文: An Algebraic Programming Style for Numerical Softw…

This paper studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs), a.k.a. (Ordinary) Difference Equations. It presents a new framework using these equations as a central tool for computation and…

This book encompasses both traditional and modern methods treating partial differential equation (PDE) of first order and second order. There is a balance in making a selfcontained mathematical text and introducing new subjects. The Lie…

This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…

We introduce a general, analytical framework to express and to approximate partial differential equations (PDEs) numerically on graphs and networks of surfaces---generalized by the term hypergraphs. To this end, we consider PDEs on…

Complex systems of systems (SoS) are characterized by multiple interconnected subsystems. Typically, each subsystem is designed and analyzed using methodologies and formalisms that are specific to the particular subsystem model of…

Viewing optimization methods as numerical integrators for ordinary differential equations (ODEs) provides a thought-provoking modern framework for studying accelerated first-order optimizers. In this literature, acceleration is often…

Recently, various evolutionary partial differential equations (PDEs) with a mixed derivative have been emerged and drawn much attention. Nonetheless, their PDE-theoretical and numerical studies are still in their early stage. In this paper,…

A numerical framework is developed to solve various types of PDEs on complicated domains, including steady and time-dependent, non-linear and non-local PDEs, with different boundary conditions that can also include non-linear and non-local…

The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations…

The numerical solution of partial differential equations (PDEs) is difficult, having led to a century of research so far. Recently, there have been pushes to build neural--numerical hybrid solvers, which piggy-backs the modern trend towards…

Stochastic partial differential equations (SPDEs) represent a very active research field with numerous recent developments and breakthrough results. There are several well-established approaches and methods used to construct solutions for…

Algebraic data types (ADTs) are a construct classically found in functional programming languages that capture data structures like enumerated types, lists, and trees. In recent years, interest in ADTs has increased. For example, popular…

Designing software systems for Geometric Computing applications can be a challenging task. Software engineers typically use software abstractions to hide and manage the high complexity of such systems. Without the presence of a unifying…

Partial Differential Equations (PDEs) are central to science and engineering. Since solving them is computationally expensive, a lot of effort has been put into approximating their solution operator via both traditional and recently…

Abstraction is a well-known approach to simplify a complex problem by over-approximating it with a deliberate loss of information. It was not considered so far in Answer Set Programming (ASP), a convenient tool for problem solving. We…

In this note, we address the following question: Why certain nonassociative algebra structures emerge in the regularity theory of elliptic type PDEs and also in constructing nonclassical and singular solutions? The aim of the paper is…

Solving Partial Differential Equations (PDEs) is fundamental to numerous scientific and engineering disciplines. A common challenge arises from solving the PDE families, which are characterized by sharing an identical mathematical structure…

Despite the promise of scientific machine learning (SciML) in combining data-driven techniques with mechanistic modeling, existing approaches for incorporating hard constraints in neural differential equations (NDEs) face significant…

Epistemic logic programs (ELPs) are a popular generalization of standard Answer Set Programming (ASP) providing means for reasoning over answer sets within the language. This richer formalism comes at the price of higher computational…

Partial differential equations (PDEs) with spatially-varying coefficients arise throughout science and engineering, modeling rich heterogeneous material behavior. Yet conventional PDE solvers struggle with the immense complexity found in…