【作者】 张惠；

【导师】 刘继军；

【作者基本信息】 东南大学 ， 数学， 2024， 博士

【摘要】 偏微分方程作为描述物理现象的重要工具之一,一直是数学研究的关键领域.偏微分方程反问题在生物医学成像、数值天气预报、材料无损检测等领域具有广泛应用.很多情形下,系统源项、初始状态等其他参数是未知的,或者不精确的.通过可测量的附加信息来确定这些未知信息属于偏微分方程反问题范畴.本文基于局部测量数据,研究了两类偏微分方程反问题,分别是椭圆方程内部源项识别问题和浅水波方程初始源反演问题,具体包含以下内容.对于椭圆偏微分方程,考虑利用内部小区域上测量数据识别系统内部源项问题.针对这个线性反问题,第二章基于内部控制的损失函数,提出了一种神经网络求解方法.该算法利用两个神经网络分别近似未知源项和系统解,通过同时优化两组网络参数的方式来重构源项.理论上基于解的解析延拓的稳定性结果,利用源项反演的条件稳定性和解的正则性严格推导了所提神经网络算法的泛化误差.数值实验表明所提算法适用于高维情况,且抗噪能力较强.为避免上述椭圆方程反源问题本身的不适定性,第三章提出了带有正则化项的神经网络求解方法.关于带噪测量数据下神经网络近似解的收敛性分析工作大致分为两大步.第一步,间接通过正则化经验损失函数建立一般的期望损失函数的一个上界;在此上界基础上,适当选择与训练集采样点数目相关的正则化参数和正则化项,定量地建立一般期望损失函数在正则化经验带噪损失函数极小值处的收敛性.第二步,若采样点数目与噪声水平相关,可在源项的先验约束下建立神经网络近似解的误差阶.数值验证了理论分析工作的合理性.不同于以上椭圆方程稳态内源的辨识工作,第四章考虑了一种特殊的发展方程-浅水波方程的瞬态初始源反演问题.基于一段时间内二维局部区域上有限观测数据,提出了一种高效数据同化算法.此算法充分考虑了科氏力与海底地形因素,以确保初始状态的精准恢复.鉴于高度非线性且耦合的流体力学方程特点,在求解反问题之前,证明了具有适当边界条件的模型的解的唯一性,并在特定能量表示下建立了守恒定律,推广了忽略科氏力和地形因素的著名能量守恒律.在初始状态同化过程中,利用变分伴随法导出目标泛函梯度,降低算法实现难度.此外,在Arakawa C网格框架下建立了控制方程的离散格式,数学上分析了离散格式在近似意义下的保能量性质.数值算例表明了所提出的数据同化算法的有效性.更多还原

【Abstract】 Partial differential equations,as one of the important tools for describing physical phenomena,have been a key area of mathematical research.Inverse problems of partial differential equations have a wide range of applications in biomedical imaging,numerical weather prediction,and nondestructive testing of materials.In many cases,the source term,initial state and other parameters of the system are unknown or imprecise.Determining these unknowns through measurable additional information b elongs to the category of inverse problems for partial differential equations.In this paper,we study two types of inverse problems of partial differential equations using local measurement data,namely,the recovery of internal source for an elliptic equation and the reconstruction of initial source for shallow water equations,which contain the following details.For an elliptic partial differential equation,we consider the problem of identifying the internal source of the system using measurements on the small internal region.For this linear inverse roblem,Chapter 2 proposes a neural network method based on the loss fun ction of the internal control.The algorithm uses two neural networks to approximate the unknown source term and the system solution,respectively,and reconstructs the source term by simultaneously optimizing b oth sets of network parameters.Based on the stability result for the analytic extension of the solution,we strictly estimate the generalization error caused by the proposed algorithm employing the property of conditional stability and the regularity of the solution.Numerical experiments show that the proposed algorithm is suitable for the high-dimensional case with high noise immunity.In order to avoid the ill-posedness of the inverse source problem for the elliptic equation mentioned above,a neural network solution with regularizing terms is proposed in Chapter 3.The analysis on the convergence of the neural network approximate solutions with noisy measurement data is roughly divided into two major steps.In the first step,an upper bound on the general expected loss function is established indirectly by regularizing the empirical loss function;based on this upper bound,the convergence of the general expected loss function at the minimizer of the regularized empirical noisy loss function is established quantitatively by choosing the regularizing parameter and the regularization term that are related to the number of sampling points in the training set.In the second step,if the number of sampling points is correlated with the noise level,we can establish the error order of the neural network solution with some a-priori restrictions on the source.The numerical validation justifies the theoretical analysis.Different from the above work on the identification of steady state internal source for an elliptic equ ation,Chapter 4 considers the problem of transient initial source inversion for a special evolution equation,called the shallow water equations.We develop an efficient data assimilation algorithm on the basis of limited observations over a period of time in a localized 2-dimensional Our algorithm takes both the comp lete Coriolis force and the seafloor topography into account to ensure the accurate recovery of the initial state.In view of the characteristics of the highly nonlinear and coupled properties of hydrodynamic equations,we prove the uniqu eness of the solution of the moedl with appropriate boundary conditions before solving the problem and establish the conservation laws for the suitably defined energy quantity,which generalises the well-known law of energy conservation that ignoros the Coriolis force and topographic factors,In the process of initial state assimilation,we apply the variational adjoint approach to derive the gradient of objective function,which reduces the difficulty of algorithm implementation.In addition,we establish a discrete scheme for the governing equations in the Arakawa C-grid framework,from which we analyze the ener gy conservation properties of the discrete format mathematically in an approximate sense.Numerical examples demonstrate the effectiven ess of the proposed data assimilation algorithm.更多还原