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

TL;DR

This work develops and demonstrates a dual variational principle for PDEs without a primal variational structure by introducing a convex auxiliary potential and optimizing over Lagrange multipliers. The resulting dual functional S(λ) or S(λ,μ) is maximized, while a dual-to-primal (DtP) mapping recovers the primal fields, enabling variational treatment of convection-diffusion, Laplace, and heat problems. The approach yields a symmetric, well-posed discrete system, and is implemented with space-time Galerkin methods using high-order B-splines and RePU neural networks to approximate dual fields, achieving accurate primal solutions and convergence in $L^2$ and $H^1$ norms. Terminal-time behavior and edge-corner effects in space-time discretizations are analyzed, with proposed remedies such as time slicing or buffering to improve robustness. The results establish a viable variational framework for non-variational PDEs and point to broad applications and future extensions to nonlinear and constrained systems.

Abstract

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.

Figures (19)

• Figure 1: Plot of the dual function $S({\bm{\lambda}})$ for the system of two quadratic equations.
• Figure 2: Plots of RePU functions that are used to form the dual fields. Filled circles in blue denote the location of the knots, $x_i$$(i = 0,1,\dots,5)$. Knots are uniformly spaced. Linear combination of the functions (a) $\sigma(x-x_i;p=2)$ and (b) $\sigma(x_i-x;p=2)$ are used to form $\mu_\theta(x)$. Linear combination of the functions (c) $\sigma(x-x_i;q=3)$ are used to form $\lambda_\theta^+(x)$ and linear combination of the functions (d) $\sigma(x_i-x;q=3)$ are used to form $\lambda_\theta^-(x)$. The dual field $\lambda_\theta(x) = (1-x) \lambda_\theta^+(x) + x \lambda_\theta^-(x)$.
• Figure 3: Neural network solution for the steady-state convection-diffusion problem ($\alpha = 1$) using $n = 10$, $p = 2$ and $q = 3$. (a) Exact solutions for $u$ and $u^\prime$; (b) Dual fields, $\mu_\theta(x)$ and $\lambda_\theta(x)$; (c) Primal fields, $u_\theta(x)$ and $q_\theta (x)$; (d) Error in $u$; and (e) Error in $u^\prime$.
• Figure 4: Neural network solution for the steady-state convection-diffusion problem ($\alpha = 1$) using $n = 10$, $p = 3$, and $q = 4$. (a) $u_\theta(x), \ q_\theta(x)$; (b) $u - u_\theta$; and (c) $u^\prime - q_\theta$.
• Figure 5: Neural network solution for the steady-state convection-diffusion problem ($\alpha = 10$) using $n = 30$, $p = 2$ and $q = 3$. (a) Exact solutions; (b) Neural network solutions; (c) $u - u_\theta$; and (d) $u^\prime - q_\theta$.