基于对偶性求解偏微分方程的变分公式：使用B样条与机器学习近似函数
Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants
AI总结：
针对缺乏原变分结构的偏微分方程，本文提出基于对偶场的变分原理，利用B样条与浅层神经网络作为近似函数进行Galerkin离散化求解，并在对流扩散等问题上验证了其精度与收敛率。
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型和双曲型方程，并不具有精确的原变分结构。最近，一种基于对偶（Lagrange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作为约束，并引入一个任意选取的、具有强凸性质的辅助势能进行优化。在要求Lagrangian关于原变量的梯度消失后，便得到一个从对偶场到原场的映射。这导致需要最小化一个凸对偶泛函，并受限于对偶变量上的Dirichlet边界条件，从而保证即使是不具备原变分结构的PDEs也能通过变分原理求解。对偶泛函的一阶变分消失，在对偶场Dirichlet边界条件之外，等价于融入了对偶到原变量代换的原PDE问题的弱形式。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，其试探函数和测试函数选为带有RePU激活函数的浅层神经网络或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，采用时空Galerkin实现，以张量积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ODEs和PDEs具有良好的精度，并在$L^2$范数和$H^1$半范数下确立了稳态对流扩散问题的收敛速率。
英文摘要：
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.
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