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 PDE as constraints, and to invoke an arbitrarily chosen auxiliary potential with strong convexity properties to be optimized. 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 to obtain the discrete equations, with the trial and test functions chosen as linear combination of either RePU activation functions (shallow neural network) or B-spline basis functions; 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.

翻译：许多偏微分方程（例如流体力学中的Navier-Stokes方程、固体中的非弹性变形问题，以及瞬态抛物型和双曲型方程）并不具备精确的原始变分结构。近期，一种基于对偶场（拉格朗日乘子）的变分原理被提出。该方法的核心思想是将给定的偏微分方程视为约束条件，并引入一个具有强凸性且可任意选择的辅助势函数进行优化。这导出了一个需在满足对偶变量Dirichlet边界条件下最小化的凸对偶泛函，从而保证即使不具备原始形式变分结构的偏微分方程也能通过变分原理求解。对偶泛函一阶变分的消失（在对偶场满足Dirichlet边界条件的前提下）即为原始偏微分方程问题的弱形式，其中已包含对偶变量到原始变量的转换。本文推导了线性一维瞬态对流扩散方程的对偶弱形式。采用Galerkin离散化获得离散方程，其中试探函数与检验函数选取为RePU激活函数（浅层神经网络）或B样条基函数的线性组合；相应的刚度矩阵具有对称性。对于瞬态问题，采用时空Galerkin实现方法，并以张量积B样条作为逼近函数。文中展示了稳态/瞬态对流扩散方程及瞬态热传导问题的数值算例。所提方法在求解常微分方程与偏微分方程时表现出良好的精度，并在$L^2$范数与$H^1$半范数下建立了稳态对流扩散问题的收敛速率。