Partial Differential Equations as Three-Dimensional Inverse Problem of Moments

We considerer partial differential equations of second order, for example the Klein-Gordon equation, the Poisson equation, on a region E = （a1, b1 ） × （a2, b2 ） x （a3, b3 ）. We will see that with a common procedure in all cases, we can write the equation in partial derivatives as an Fredholm integral equation of first kind and will solve this latter with the techniques of inverse problem moments. We will find an approximated solution and bounds for the error of the estimated solution using the techniques on problem of moments.

A fractal approximation algorithm for inverse initial-value problems of nonlinear differential equations

A fractal approximation algorithm is developed to obtain approximate solutions to an inverse initial-value problem IVP(inverse IVP) for the differential equation. Numerical computational results are presented to demonstrate the effectiveness of this algorithm for solving inverse IVP for a class of specific differential equations.

FORCED OSCILLATIONS OF BOUNDARY VALUE PROBLEMS OF HIGHER ORDER FUNCTIONAL PARTIAL DIFFERENTIAL EQUATIONS

In this paper we study the forced oscillations of boundary value problems of a class of higher order functional partial differential equations.The principal tool is an everaging techniqe which enables one to establish oscillation in terms of related functional differential inequallities.

A Differential Continuation－Regularization Method for Solving Inverse Problems of Acoustic Wave Equations

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INITIAL BOUNDARY VALUE PROBLEMS FOR A CLASS OF NONLINEAR INTEGRO－PARTIAL DIFFERENTIAL EQUATIONS

This paper studies the global existence of the classical solutions to the following problem:This problem describes the nonlinear vibrations of finite rods with nonlinear vis-coelasticity. Under certain conditions on σand f , we obtained the unique existence of the global classical solution of this problem.

Incorporating Lasso Regression to Physics-Informed Neural Network for Inverse PDE Problem

Partial Differential Equation(PDE)is among the most fundamental tools employed to model dynamic systems.Existing PDE modeling methods are typically derived from established knowledge and known phenomena,which are time-consuming and labor-intensive.Recently,discovering governing PDEs from collected actual data via Physics Informed Neural Networks(PINNs)provides a more efficient way to analyze fresh dynamic systems and establish PEDmodels.This study proposes Sequentially Threshold Least Squares-Lasso(STLasso),a module constructed by incorporating Lasso regression into the Sequentially Threshold Least Squares(STLS)algorithm,which can complete sparse regression of PDE coefficients with the constraints of l0 norm.It further introduces PINN-STLasso,a physics informed neural network combined with Lasso sparse regression,able to find underlying PDEs from data with reduced data requirements and better interpretability.In addition,this research conducts experiments on canonical inverse PDE problems and compares the results to several recent methods.The results demonstrated that the proposed PINN-STLasso outperforms other methods,achieving lower error rates even with less data.

A SINGULAR PERTURBATION PROBLEM FOR PERIODIC BOUNDARY PARTIAL DIFFERENTIAL EQUATION

In this paper, we consider a singular perturbation elliptic-parabolic partial differential equation for periodic boundary value problem, and construct a difference scheme. Using the method of decomposing the singular term from its solution and combining an asymptotic expansion of the equation, we prove that the scheme constructed by this paper converges uniformly to the solution of its original problem with O(r+h2).

NUMERICAL SOLUTION OF A SINGULARLY PERTURBED ELLIPTIC-HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION ON A NONUNIFORM DISCRETIZATION MESH

In this paper, we consider the upwind difference scheme for singular perturbation problem (1.1). On a special discretization mesh, it is proved that the solution of the upwind difference scheme is first order convergent, uniformly in the small parameter e , to the solution of problem (1.1). Numerical results are finally provided.

A Computational Quadruple Laplace Transform for the Solution of Partial Differential Equations

In this paper, we proposed new results in quadruple Laplace transform and proved some properties concerned with quadruple Laplace transform. We also developed some applications based on these results and solved homogeneous as well as non-homogeneous partial differential equations involving four variables. The performance of quadruple Laplace transform is shown to be very encouraging by concrete examples. An elementary table of quadruple Laplace transform is also provided.

Application of <i>q</i>-Calculus to the Solution of Partial <i>q</i>-Differential Equations

We introduce the concept of q-calculus in quantum geometry. This involves the q-differential and q-integral operators. With these, we study the basic rules governing q-calculus as compared with the classical Newton-Leibnitz calculus, and obtain some important results. We introduce the reduced q-differential transform method (RqDTM) for solving partial q-differential equations. The solution is computed in the form of a convergent power series with easily computable coefficients. With the help of some test examples, we discover the effectiveness and performance of the proposed method and employing MathCAD 14 software for computation. It turns out that when q = 1, the solution coincides with that for the classical version of the given initial value problem. The results demonstrate that the RqDTM approach is quite efficient and convenient.

A CONTINUATION HOMOTOPY METHOD FOR THE INVERSE PROBLEM OF OPERATOR IDENTIFICATION AND ITS APPLICATION

A new widly convergent method for solving the problem of operator identification is illustrated. Numerical simulations are carried out to test the feasibility and to study the general characteristics of the technique without the real measurement data. This technique is a direct application of the continuation homo-topy method for solving nonlinear systems of equations. It is found that this method does give excellent results in solving the inverse problem of the elliptic differential equations.

Optimization of Random Feature Method in the High-Precision Regime

Machine learning has been widely used for solving partial differential equations(PDEs)in recent years,among which the random feature method(RFM)exhibits spectral accuracy and can compete with traditional solvers in terms of both accuracy and efficiency.Potentially,the optimization problem in the RFM is more difficult to solve than those that arise in traditional methods.Unlike the broader machine-learning research,which frequently targets tasks within the low-precision regime,our study focuses on the high-precision regime crucial for solving PDEs.In this work,we study this problem from the following aspects:(i)we analyze the coeffcient matrix that arises in the RFM by studying the distribution of singular values;(ii)we investigate whether the continuous training causes the overfitting issue;(ii)we test direct and iterative methods as well as randomized methods for solving the optimization problem.Based on these results,we find that direct methods are superior to other methods if memory is not an issue,while iterative methods typically have low accuracy and can be improved by preconditioning to some extent.

A Radial Basis Function Method with Improved Accuracy for Fourth Order Boundary Value Problems

Accurately approximating higher order derivatives is an inherently difficult problem. It is shown that a random variable shape parameter strategy can improve the accuracy of approximating higher order derivatives with Radial Basis Function methods. The method is used to solve fourth order boundary value problems. The use and location of ghost points are examined in order to enforce the extra boundary conditions that are necessary to make a fourth-order problem well posed. The use of ghost points versus solving an overdetermined linear system via least squares is studied. For a general fourth-order boundary value problem, the recommended approach is to either use one of two novel sets of ghost centers introduced here or else to use a least squares approach. When using either ghost centers or least squares, the random variable shape parameter strategy results in significantly better accuracy than when a constant shape parameter is used.

THE SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR HIGHER ORDER SEMILINEAR ELLIPTIC EQUATIONS

In this paper a singular perturbation of boundary value problem for elliptic partial differential equations of higher order is considered by using the differential inequalities. The uniformly valid asymptotic expansion in entire region is obtained.

PARAMETER IDENTIFICATION IN FRACTIONAL DIFFERENTIAL EQUATIONS

This article investigates the fractional derivative order identification, the coefficient identification, and the source identification in the fractional diffusion problems. If 1 〈 α〈 2, we prove the unique determination of the fractional derivative order and the dif- fusion coefficient p（x） by fo u（0, s）ds, 0 〈 t 〈 T for one-dimensional fractional diffusion-wave equations. Besides, if 0 〈 α 〈 1, we show the unique determination of the source term f（x, y） by U（0, 0, t）, 0 〈 t 〈 T for two-dimensional fractional diffusion equations. Here, a denotes the fractional derivative order over t.

Stabilization and control of subcritical semilinear wave equation in bounded domain with Cauchy-Ventcel boundary conditions

We analyze the exponential decay property of solutions of the semilinear wave equation in bounded domain Ω of R^N with a damping term which is effective on the exterior of a ball and boundary conditions of the Cauchy-Ventcel type. Under suitable and natural assumptions on the nonlinearity, we prove that the exponential decay holds locally uniformly for finite energy solutions provided the nonlinearity is subcritical at infinity. Subcriticality means, roughly speaking, that the nonlinearity grows at infinity at most as a power p 〈 5. The results obtained in R^3 and RN by B. Dehman, G. Lebeau and E. Zuazua on the inequalities of the classical energy （which estimate the total energy of solutions in terms of the energy localized in the exterior of a ball） and on Strichartz＇s estimates, allow us to give an application to the stabilization controllability of the semilinear wave equation in a bounded domain of R^N with a subcritical nonlinearity on the domain and its boundary, and conditions on the boundary of Cauchy-Ventcel type.

Numerical Treatment of Initial-Boundary Value Problems with Mixed Boundary Conditions

In this paper, we extend the reliable modification of the Adomian Decom-position Method coupled to the Lesnic’s approach to solve boundary value problems and initial boundary value problems with mixed boundary conditions for linear and nonlinear partial differential equations. The method is applied to different forms of heat and wave equations as illustrative examples to exhibit the effectiveness of the method. The method provides the solution in a rapidly convergent series with components that can be computed iteratively. The numerical results for the illustrative examples obtained show remarkable agreement with the exact solutions. We also provide some graphical representations for clear-cut comparisons between the solutions using Maple software.

Accurate reconstruction of the optical parameter distribution in participating medium based on the frequency-domain radiative transfer equation

Reconstructing the distribution of optical parameters in the participating medium based on the frequency-domain radiative transfer equation （FD-RTE） to probe the internal structure of the medium is investigated in the present work. The forward model of FD-RTE is solved via the finite volume method （FVM）. The regularization term formatted by the generalized Gaussian Markov random field model is used in the objective function to overcome the ill-posed nature of the inverse problem. The multi-start conjugate gradient （MCG） method is employed to search the minimum of the objective function and increase the efficiency of convergence. A modified adjoint differentiation technique using the collimated radiative intensity is developed to calculate the gradient of the objective function with respect to the optical parameters. All simulation results show that the proposed reconstruction algorithm based on FD-RTE can obtain the accurate distributions of absorption and scattering coefficients. The reconstructed images of the scattering coefficient have less errors than those of the absorption coefficient, which indicates the former are more suitable to probing the inner structure.

Dirichlet-to-Neumann Map for a Hyperbolic Equation

In this paper, we provide an explicit expression for the full Dirichlet-to-Neumann map corresponding to a radial potential for a hyperbolic differential equation in 3-dimensional. We show that the Dirichlet-Neumann operators corresponding to a potential radial have the same properties for hyperbolic differential equations as for elliptic differential equations. We numerically implement the coefficients of the explicit formulas. Moreover, a Lipschitz type stability is established near the edge of the domain by an estimation constant. That is necessary for the reconstruction of the potential from Dirichlet-to-Neumann map in the inverse problem for a hyperbolic differential equation.

面向流体力学的物理神经网络综述

针对融合了物理控制方程,尤为适用于物理场预测的新兴神经网络方法——物理神经网络(PINN),开展深入的文献调研,形成对面向流体力学的物理神经网络方法发展趋势的研判。首先,对神经网络融合物理信息的思路进行溯源;其次,介绍当前物理神经网络基本架构,针对全连接型物理神经网络,从间断问题的高精度预测研究、偏微分方程(PDE)植入形式、流场重建问题、损失函数形式、多精度数据及多尺度问题以及训练控制等方面进行文献综述;再次,对于基于卷积神经网络(CNN)和其他新兴网络架构的物理神经网络进行文献梳理;最后,形成面向流体力学的物理神经网络发展趋势与思考。通过对2017年至2023年间近百篇文献的研究及相关数值实验可知,针对强间断的高分辨率预测是面向高速流动问题的物理神经网络研究中需要解决的重要问题;基于全连接网络的物理神经网络拥有无网格化的优势,可用于各类流动问题的求解;基于卷积网络的物理神经网络具备与已有传统数值方法深度融合的优势,可有效利用已有的流场图像、物理量云图等结构化数据,进行复杂流动问题的求解。