标题： 基于对偶的变分公式求解偏微分方程：B样条和机器学习近似器的应用 显示英文标题

标题： Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants

摘要： 许多偏微分方程（PDEs），如流体力学中的Navier--Stokes方程、固体中的非弹性变形以及瞬态抛物线和双曲方程，没有精确的原始变分结构。最近，提出了一种基于对偶（拉格朗日乘子）场的变分原理。这种方法的核心思想是将给定的PDEs视为约束条件，并引入一个任意选择的具有强凸性性质的辅助势能以进行优化。在要求拉格朗日函数相对于原始变量的梯度消失的情况下，得到从对偶场到原始场的映射。这导致需要在对偶变量上满足狄利克雷边界条件的情况下最小化一个凸的对偶泛函，保证即使那些在原始形式中不具有变分结构的PDEs也可以通过变分原理求解。在对偶场的狄利克雷边界条件下，对偶泛函的一阶变分为原始PDE问题的弱形式，并包含了从对偶到原始变量的变换。我们推导了线性、一维、瞬态对流-扩散方程的对偶弱形式。采用伽辽金离散化方法，试验函数和测试函数选择为浅层神经网络与RePU激活函数或B样条的线性组合；相应的刚度矩阵是对称的。对于瞬态问题，使用时空伽辽金实现，采用张量积B样条作为近似函数。给出了稳态和瞬态对流-扩散方程以及瞬态热传导的数值结果。所提出的方法对于常微分方程（ODEs）和偏微分方程（PDEs）具有良好的准确性，并在稳态对流-扩散问题中建立了$L^2$范数和$H^1$半范数的收敛率。 显示英文摘要

摘要： Many partial differential equations (PDEs) such as Navier--Stokes equations in fluid mechanics, inelastic deformation in solids, and transient parabolic and hyperbolic equations do not have an exact, primal variational structure. Recently, a variational principle based on the dual (Lagrange multiplier) field was proposed. The essential idea in this approach is to treat the given PDEs as constraints, and to invoke an arbitrarily chosen auxiliary potential with strong convexity properties to be optimized. On requiring the vanishing of the gradient of the Lagrangian with respect to the primal variables, a mapping from the dual to the primal fields is obtained. This leads to requiring a convex dual functional to be minimized subject to Dirichlet boundary conditions on dual variables, with the guarantee that even PDEs that do not possess a variational structure in primal form can be solved via a variational principle. The vanishing of the first variation of the dual functional is, up to Dirichlet boundary conditions on dual fields, the weak form of the primal PDE problem with the dual-to-primal change of variables incorporated. We derive the dual weak form for the linear, one-dimensional, transient convection-diffusion equation. A Galerkin discretization is used, with the trial and test functions chosen as linear combination of either shallow neural networks with RePU activation functions or B-splines; the corresponding stiffness matrix is symmetric. For transient problems, a space-time Galerkin implementation is used with tensor-product B-splines as approximating functions. Numerical results are presented for the steady-state and transient convection-diffusion equation, and transient heat conduction. The proposed method delivers sound accuracy for ODEs and PDEs and rates of convergence are established in the $L^2$ norm and $H^1$ seminorm for the steady-state convection-diffusion problem.