PDE Methods for Nonlocal Models

In scientific and engineering calculation, the effective solution of partial differential equations (PDEs) has great significance. This paper presents an innovative method based on the combination of a U-Net neural network with Fourier neural operators, aiming to improve the accuracy and efficiency of solving partial differential equations. U-Net neural networks with a unique encoding–decoder structure and hopping connections can efficiently extract and integrate spatial-domain features and accurately describe the spatial structure of PDEs. The Fourier neural operator combines the advantages of neural networks and Fourier transform to learn the mapping relationship of high-dimensional data by training the linear transformation parameters in the detailed frequency domain of the data. Fourier transform is used to explore and process the frequency-domain features and capture the frequency characteristics of the solution of partial differential equations. By combining the two, this method not only retains spatial details but also makes full use of frequency-domain information, which significantly reduces the dependence of the model on large-scale data and improves the generalization ability of the model. The experimental results show that compared with the traditional method, this method performs well in various complex partial differential equation (PDE) solving tasks and achieves high accuracy.

A new higher order finite element method for elliptic partial differential equations on a stationary smooth surface $\Gamma$ is introduced and analyzed. We assume that $\Gamma$ is characterized as the zero level of a level set function $\phi$ and only a finite element approximation $\phi_h$ (of degree $k \geq 1$) of $\phi$ is known. For the discretization of the partial differential equation, finite elements (of degree $m\geq 1$) on a piecewise linear approximation of $\Gamma$ are used. The discretization is lifted to $\Gamma_h$, which denotes the zero level of $\phi_h$, using a quasi-orthogonal coordinate system that is constructed by applying a gradient recovery technique to $\phi_h$. A complete discretization error analysis is presented in which the error is split into a geometric error, a quadrature error, and a finite element approximation error. The main result is a $H^1(\Gamma)$-error bound of the form $c(h^m + h^{k+1})$. Results of numerical experiments illustrate the higher order convergence of t...

Interface solutions of partial differential equations with point singularity

The companion volume of this book, ‘Analytic methods for partial differential equations’, is concerned with solution of partial differential equations using classical methods which result in analytic solutions. These equations result when almost any physical situation is modelled, ranging from fluid mechanics problems, through electromagnetic problems to models of the economy. Some specific and fundamental problems were highlighted in the earlier volume, namely the three major classes of linear second order partial differential equations. The heat equation, the wave equation and Laplace’s equation will form a basis for study from a numerical point of view for the same reason as they did in the analytic case. That is, the three equations are the canonical forms to which any quasi-linear second order equation may be reduced using the characteristic transformation.KeywordsPartial Differential EquationCanonical FormSpectral RadiusLinear Algebraic EquationBackground MathematicThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We apply the homotopy perturbation method to obtain the solution of partial differential equations of fractional order. This method is powerful tool to find exact and approximate solution of many linear and nonlinear partial differential equations of fractional order. Convergence of the method is proved and the convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution. The fractional derivatives are described in the Caputo sense. Some examples are presented to verify convergence hypothesis and simplicity of the method.

Deep neural networks based temporal-difference methods for high-dimensional parabolic partial differential equations

Aiming at the heat conduction equation, the heat conduction model is a very typical example of the partial differential equation. This paper focuses on the development and preliminary knowledge of the partial differential equation, the establishment of the heat conduction model and the solution of the heat conduction model. The partial differential equation with fractional Laplace operator is a kind of typical fractional partial differential equation, which has important applications in the fields of science and engineering. Fractional Laplace operator is a kind of nonlocal quasi-differential operator, which is the infinitesimal generator of L é vy steady-state process. It is essentially different from the classical Laplace operator, which leads to the disappearance of some classical properties, which generally brings difficulties to the study of this kind of problems. It is very difficult to solve the explicit solution of partial differential equation with fractional Laplace operator. In order to improve the accuracy and speed of solving partial differential equation, a high precision method for solving partial differential equation based on deep learning network is proposed. A multilayer radial basis function neural network is constructed and its structural layers are determined. By introducing subnetworks into each layer of the multilayer radial basis function neural network, a composite multilayer radial basis function neural network is constructed to improve the real function approximation performance and operation accuracy of the multilayer radial basis function neural network. The high-precision compound multi-layer radial basis function neural network is used to solve the partial differential equation. By giving a specific example of solving the partial differential equation, the solution effect of the method is tested. The results show that the method has very high solution accuracy under the four-layer network structure, which can improve about 1.5 orders of magnitude compared with the first layer. Under different number of training samples, it has higher solution accuracy and speed, and the comprehensive performance is superior. Therefore, it is a practical but challenging work to study the existence of solutions of partial differential equations with fractional Laplace operators. This paper mainly describes the research progress and trends of the existence of solutions of several classes of partial differential equations with fractional Laplace operators, including some of the work done by the author in this field in recent years.

The section opens with a review well written by a giant in the field of numerical solution of partial differential equations, Roland Glowinski. In this featured review, he analyzes the book Think Before You Compute: A Prelude to Computational Fluid Dynamics, by E. J. Hinch, in much detail and contributes his own thoughts on appraisal or possibilities for improvement to every chapter of the book. The reviewer's verdict on the book is: “The book is a gem.” The following three reviewed books, written in an earlier and different world, take on a whole new relevance against the background of the current pandemic situation. First, we have the book by Irina Kareva and Georgy Karev, Modeling Evolution of Heterogeneous Populations: Theory and Applications. This book is diligently reviewed by Karen Page, who rates the book as “enlightening" even if it still contains small annoying errors. The following book, Delay-Adaptive Linear Control, by Yang Zhu and Miroslav Krstic, also seems to anticipate some aspects of the current modeling and control of the COVID-19 pandemic. This book is reviewed in detail by Bozenna Pasik-Duncan and is classified as an important contribution to control in delay systems. For the classification and mining of numerous medical data, such as those being generated right now, Mehrshad Sadria and Anita Layton suggest reading Bioinformatics, edited by Andreas D. Baxevanis, Gary D. Bader, and David S. Wishart, and recommend it as an “easy to read, greatly informative, and highly timely” book. Finally, I would like to mention the book by Zhendong Luo and Goong Chen on Proper Orthogonal Decomposition Methods for Partial Differential Equations, which is reviewed by the expert Stefan Volkwein, who praises the book as providing “a really good recipe for the application of the POD method to many nonstationary PDEs,” and also the book Basic Calculus of Planetary Orbits and Interplanetary Flight: The Missions of the Voyagers, Cassini, and Juno, by Alexander J. Hahn, that allows the reader to escape the current narrow earthly confinement in mathematical thoughts and delve into interplanetary explorations.

This work considers the partial solutions of partial differential equations for initial/boundary conditions using the Adomian decomposition method. The study formally shows that the partial solutions are always identical for all styles of boundary conditions. We also prove that the partial solution in the t-direction requires less computational work if compared with other partial solutions developed in any space variable direction. In addition, several mathematical models that govern the heat distribution and the wave propagation phenomenas have been tested, and the results obtained have shown that the t-solution minimizes the size of calculations if compared with the traditional techniques.

Tremendous progress has been achieved in image analysis and processing, and in particular, the use of partial differential equation (PDE) methods in image analysis has proliferated in recent years. However, PDE methods have seen little application to optical image-based structural damage detection for civil systems. This paper applies the Chan-Vese active contour model and its level set representation to detect concrete surface damage, in this case concrete cracks, in optical images. The detected cracks can be further characterized by extracted geometric quantities including their width, length and area of coverage, and associated with engineering design. With regard to crack width extraction, an approximate method is proposed, which relies on solving for a signed distance function. We test these methods by using synthetic images as well as real multi-temporal images from a laboratory experiment. This paper illustrates that using the proposed methods, cracks with complex topographical patterns can be successfully detected with sufficient accuracy.

In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the homotopy analysis method for partial differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

We conducted a six-month investigation of the design, analysis, and software implementation of a class of singularity-insensitive, scalable, parallel nonlinear iterative methods for the numerical solution of nonlinear partial differential equations. The solutions of nonlinear PDEs are often nonsmooth and have local singularities, such as sharp fronts. Traditional nonlinear iterative methods, such as Newton-like methods, are capable of reducing the global smooth nonlinearities at a nearly quadratic convergence rate but may become very slow once the local singularities appear somewhere in the computational domain. Even with global strategies such as line search or trust region the methods often stagnate at local minima of {parallel}F{parallel}, especially for problems with unbalanced nonlinearities, because the methods do not have built-in machinery to deal with the unbalanced nonlinearities. To find the same solution u* of F(u) = 0, we solve, instead, an equivalent nonlinearly preconditioned system G(F(u*)) = 0 whose nonlinearities are more balanced. In this project, we proposed and studied a nonlinear additive Schwarz based parallel nonlinear preconditioner and showed numerically that the new method converges well even for some difficult problems, such as high Reynolds number flows, when a traditional inexact Newton method fails.

Most instructors manage two opposing currents in classroom dynamics. The first is a desire to describe the subject in its broadest sense including all the subtleties and nuances. It gives the students a road map to the field, and if they can appreciate it at the time, it provides structure to a course that might otherwise appear to be a loosely connected series of topics. The second is the desire to ground the subject with concrete examples that students can grasp at the finest level. The examples link prerequisite knowledge with one of the new subjects and provide students with a foundation upon which to construct a deeper understanding. Few courses draw together so many disparate topics as numerical methods for partial differential equations (PDEs). When first exposed to finite differences for numerical PDEs, students will readily notice the regular structures in matrices, but few texts address these structures systematically and connect them directly to the analytical properties of the solution. There are many good examples, but it's hard to see the broader landscape. In this Education section article, “Functions of Difference Matrices Are Toeplitz Plus Hankel,” authors Gil Strang and Shev MacNamara fill in some of the big picture by presenting some new results and observations about the structure of difference matrices for approximating solutions to PDEs. A common approach in the analysis of numerical methods for PDEs is to discretize the spatial operator and leave time continuous so that one studies the semidiscrete system. The exact solution to these systems will be functions of the underlying difference matrices, hence the structure of functions of difference matrices is a natural point of interest for students, instructors, and researchers. Toeplitz matrices, having entries that are constant along diagonals, arise naturally in difference matrices. As the authors point out, this is to be expected because of the translation-invariance of the differentiation operator. In an unbounded domain, if initial condition $u_0$ leads to a solution $u$, a translation of $u_0$ will lead to a solution that is a translation of $u$. The authors discover that solutions to the semidiscrete systems are Toeplitz plus Hankel. The role of Hankel matrices, matrices that are constant along antidiagonals, is a little more subtle to explain. If one considers the role of boundary conditions for canonical problems like the heat or wave equations, one can see how this might arise. In the case of a single Neumann boundary condition for the heat equation, the classic method of images provides some insight. Shifting the initial condition toward the boundary shifts the image solution toward the boundary outside the domain, which shifts the influence of the image away from the boundary within the domain. Thus, shifting the initial condition has both a Toeplitz part from transition invariance and a Hankel part from the influence of the boundaries. The Toeplitz plus Hankel decomposition paves the way for exact solutions to semidiscrete approximations, allowing better error estimates and a deeper general understanding of spatial discretization. While it makes sense intuitively that functions of difference matrices should be Toeplitz plus Hankel, the authors enthusiastically explore the topic using a variety of methods and techniques from advanced undergraduate or introductory graduate mathematics including spectral decompositions, Riemann sums, and Fourier analysis. In other words, the destination is interesting, but the voyage is even more fun! This paper would serve as an excellent module for a numerical analysis course or an integrative seminar for undergraduates or graduate students interested in numerical methods.

Recent years have witnessed a growth in mathematics for deep learning—which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust—and deep learning for mathematics, where deep learning algorithms are used to solve problems in mathematics. The latter has popularised the field of scientific machine learning where deep learning is applied to problems in scientific computing. Specifically, more and more neural network (NN) architectures have been developed to solve specific classes of partial differential equations (PDEs). Such methods exploit properties that are inherent to PDEs and thus solve the PDEs better than standard feed-forward NNs, recurrent NNs, or convolutional neural networks. This has had a great impact in the area of mathematical modelling where parametric PDEs are widely used to model most natural and physical processes arising in science and engineering. In this work, we review such methods as well as their extensions for parametric studies and for solving the related inverse problems. We also show their relevance in various industrial applications.

This paper explores the solutions of certain fractional partial differential equations using two methods; the first method involves separation of variables, which is a common technique for solving partial differential equations. However, since many equations cannot be separated in this way, the tensor product of Banach spaces method is applied to find the atomic solutions. To solve the resulting ordinary differential equations, the reproducing Kernel Hilbert space method is used to find numerical solutions, which are then used to find the numerical solution of the partial differential equation. The residual errors indicate that this method is effective and powerful. In summary, this paper presents a study on the solutions of certain fractional partial differential equations using two methods and demonstrates the effectiveness of these methods in finding numerical solutions.