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.

翻译：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问题，以及瞬态抛物型和双曲型方程，并不具有精确的原始变分结构。最近，一种基于对偶（拉格朗日乘子）场的变分原理被提出。该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、具有强凸性质的辅助势函数进行优化。通过要求拉格朗日函数对原始变量的梯度为零，可得到从对偶场到原始场的映射。这导出了一个凸对偶泛函在满足对偶变量Dirichlet边界条件下的最小化问题，从而保证即使不具备原始变分结构的PDEs也能通过变分原理求解。该对偶泛函一阶变分为零的条件（除对偶场的Dirichlet边界条件外）即为原始PDE问题在结合对偶-原始变量变换后的弱形式。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Galerkin离散化方法，其试探函数和测试函数选取为浅层神经网络（使用RePU激活函数）或B样条的线性组合；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，并以张量积B样条作为逼近函数。数值实验展示了该方法在稳态/瞬态对流-扩散方程及瞬态热传导问题中的应用效果。所提方法在常微分方程和偏微分方程求解中表现出良好的精度，并在稳态对流-扩散问题的$L^2$范数和$H^1$半范数下建立了收敛率。