This preprint studies physics-informed neural networks and variational physics-informed neural networks as approximation methods for elliptic boundary value problems. The analysis adopts a function-space viewpoint in which training is described through probability measures induced by neural network parameterizations. Variational training objectives are interpreted as controlling weak residuals in expectation, and convergence is formulated in terms of concentration of these measures near the weak-solution set. For linear, continuous, coercive elliptic operators, the paper establishes that decay of the full dual residual implies concentration in Sobolev norms. A residual observability condition is introduced to characterize when restricted variational residuals control the full residual, in analogy with stability conditions in Petrov–Galerkin methods. The results are independent of optimization dynamics and are intended as a numerical stability analysis of variational neural approximations rather than an algorithmic or training-time study.
Tamal D. Chowdhury (Fri,) studied this question.