Related papers: New differential equations in the six-vertex model

Machine learning (ML) models have emerged as a promising approach for solving partial differential equations (PDEs) in science and engineering. Previous ML models typically cannot generalize outside the training data; for example, a trained…

A new method is introduced for studying boundary value problems for a class of linear PDEs with {\it variable} coefficients. This method is based on ideas recently introduced by the author for the study of boundary value problems for PDEs…

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…

We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show…

Partial differential equations (PDEs) that fit scientific data can represent physical laws with explainable mechanisms for various mathematically-oriented subjects, such as physics and finance. The data-driven discovery of PDEs from…

We introduce a practical method to enforce partial differential equation (PDE) constraints for functions defined by neural networks (NNs), with a high degree of accuracy and up to a desired tolerance. We develop a differentiable…

We study a model elliptic pseudo-differential equation and simplest boundary value problems for a half-space and a special cone in Sobolev--Slobodetskii spaces which have different smoothness with respect to separate variables. Sufficient…

Solving partial differential equations (PDEs) with machine learning has recently attracted great attention, as PDEs are fundamental tools for modeling real-world systems that range from fundamental physical science to advanced engineering…

Partial Differential Equations (PDEs) are the bedrock for modern computational sciences and engineering, and inherently computationally expensive. While PDE foundation models have shown much promise for simulating such complex…

In this study, we propose parameter-varying neural ordinary differential equations (NODEs) where the evolution of model parameters is represented by partition-of-unity networks (POUNets), a mixture of experts architecture. The proposed…

This is a continuation of the papers [4] of Bleher and Fokin and [5] of Bleher and Liechty, in which the large $n$ asymptotics is obtained for the partition function $Z_n$ of the six-vertex model with domain wall boundary conditions in the…

In this article we study the well-posedness (uniqueness and existence of solutions) of nonlinear elliptic Partial Differential Equations (PDEs) on a finite graph. These results are obtained using the discrete comparison principle and…

Electrical circuits are present in a variety of technologies, making their design an important part of computer aided engineering. The growing number of parameters that affect the final design leads to a need for new approaches to quantify…

Mixed dimensional partial differential equations (PDEs) are equations coupling unknown fields defined over domains of differing topological dimension. Such equations naturally arise in a wide range of scientific fields including geology,…

This paper develops the necessary ingredients for the variational approach of initial boundary-value problems of parabolic partial differential equations on a fixed spatial domain containing evolving subdomains. In particular, we introduce…

Whether integrable, partially integrable or nonintegrable, nonlinear partial differential equations (PDEs) can be handled from scratch with essentially the same toolbox, when one looks for analytic solutions in closed form. The basic tool…

We enumerate staircases with fixed left and right columns. These objects correspond to ice-configurations, or alternating sign matrices, with fixed top and bottom parts. The resulting partition functions are equal, up to a normalization…

Many scientific and industrial applications require solving Partial Differential Equations (PDEs) to describe the physical phenomena of interest. Some examples can be found in the fields of aerodynamics, astrodynamics, combustion and many…

This book is mainly an exposition of the author's works and his joint works with his former students on explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic…

Partial differential equations (PDEs) are widely used across the physical and computational sciences. Decades of research and engineering went into designing fast iterative solution methods. Existing solvers are general purpose, but may be…