Extended Galerkin neural network approximation of singular variational problems with error control

TL;DR

Extended Galerkin neural networks (xGNN) provide a rigorous variational framework for approximating general boundary value problems with error control, building on prior GNN work and enabling a posteriori error estimation. The method extends Galerkin neural networks to non-self-adjoint and indefinite problems via a weighted least-squares variational formulation, and enriches the approximation space with knowledge-based, singular solution components. The authors develop a theoretical convergence framework showing that the network width can depend primarily on the smooth part of the solution, and they demonstrate the approach on Poisson and Stokes problems in polygonal domains with corner singularities, including learning of singular exponents. Numerically, xGNN with learning of knowledge-based functions achieves improved convergence and resolves singularities and Moffatt eddies, highlighting practical impact for simulations with singularities and complex boundary behavior.

Abstract

We present extended Galerkin neural networks (xGNN), a variational framework for approximating general boundary value problems (BVPs) with error control. The main contributions of this work are (1) a rigorous theory guiding the construction of new weighted least squares variational formulations suitable for use in neural network approximation of general BVPs (2) an ``extended'' feedforward network architecture which incorporates and is even capable of learning singular solution structures, thus greatly improving approximability of singular solutions. Numerical results are presented for several problems including steady Stokes flow around re-entrant corners and in convex corners with Moffatt eddies in order to demonstrate efficacy of the method.

Figures (16)

• Figure 1: Errors with respect to the number of basis functions learning using the Galerkin neural network approach applied to the variational problem \ref{['eq:coercivity example']} for $\delta =1$, $\mathcal{W} = L^{2}(\partial\Omega)$, and the sequence of problems $m=2,4,8,16$.
• Figure 1: The true solution $u(r,\theta) = r^{\lambda}\sin{\theta}$ (left) and approximate solutions obtain using Galerkin neural networks applied to the variational problem \ref{['eq:poisson lsq']} with $\beta = 0$ (middle) and $\beta=1$ (right).
• Figure 1: Example \ref{['ex:poisson Lshaped training']}: learning of $\lambda_{1}$ and over several runs with initialization $\mu \sim \mathcal{U}(0,1)$.
• Figure 2: Errors with respect to the number of basis functions learned using the Galerkin neural network applied to the variational problem \ref{['eq:coercivity example']} for several choices of penalty parameter $\delta = m^{2s}$ with $m=16$ and $s = 0,1/2,1,3/2,2,5/2$.
• Figure 2: Example \ref{['ex:poisson rlambda']}: for $\beta=0$, the exact errors $u-u_{i-1}$ (top row) and approximate errors which correspond to basis functions $\varphi_{i}^{NN} \approx u-u_{i-1}$ learned using the Galerkin neural network approach (bottom row).

Theorems & Definitions (20)

• Theorem 2.1
• Proof 1
• Example 2.2
• Lemma 2.3
• Proposition 2.4: gnn1
• Proposition 2.5
• Proof 2
• Corollary 2.6
• Proof 3
• Remark 2.7