Elliptic PINN minimizers converge to the Dirichlet solution

Assume the Axiom of Choice and Countable Choice. Use the corrected small-scale sampling, trial-class, exact-zero-witness, and exact regularization hypotheses of the regularized PINN theorem, with residual and boundary operators $L$ and boundary restriction. Require every admitted trial function in $H_m$ to belong to $C^2(U)\cap C(\bar{U})$. Let $d\ge 2$ and let $U\subset {\mathbb{R}}^d$ be bounded Lipschitz with an exterior sphere at every boundary point. For $0<\alpha <1$, let $a_{ij}=a_{ji}\in C^{1,\alpha }(U)$ have minimum eigenvalue at least ${\lambda }_0>0$, let $b_i\in C^{1,\alpha }(U)$ and $c_i,d\in C^{0,\alpha }(U)$, and put $L[u]={\sum }_{i,j}{D}_i(a_{ij}{D}_{j}u)+{\sum }_i{D}_i(b_{i}u)+{\sum }_i{c}_i{D}_{i}u+du,$ ${\tilde{b}}_j={\sum }_i{D}_i{a}_{ij}+b_j+c_j,\tilde{c}={\sum }_i{D}_i{b}_i+d\le 0.$ All indicated Hölder norms are finite global norms. Retain the source coefficient bounds ${\sum }_{i,j}\mathrm{\mid }{a}_{ij}{\mathrm{\mid }}^2\le \mathrm{\Lambda }$ and ${\lambda }_0^{-2}(\mathrm{\mid }b{\mathrm{\mid }}^2+\mathrm{\mid }c{\mathrm{\mid }}^2)+{\lambda }_0^{-1}\mathrm{\mid }d\mathrm{\mid }\le v^2$. Let $f\in C^{0,\alpha }(U)$ and $g\in C^{0,\alpha }(\mathrm{\partial }U)$. There is exactly one $u^{\ast }\in C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2,\alpha }(U)\cap C^0(\bar{U})$ with $L[u^{\ast }]=f$ and $u^{\ast }=g$ on $\mathrm{\partial }U$.

Every sequence of global regularized empirical minimizers $h_m$ converges to $u^{\ast }$ in $C^0(U)$ almost surely as both sample counts tend to infinity. If $m_r=O(m_b^{d\mathrm{/}(d-1)})$, the occupancy-event squared uniform error is $O(m_r^{-\alpha \mathrm{/}d})$, with the probability and constants of the regularized-loss theorem. If each minimizer satisfies the boundary data exactly, the errors belong to $H_0^1(U)$ and converge to zero in $H^1(U)$ almost surely. If in addition $U$ is $C^{2,\alpha }$ and $g$ has a $C^{2,\alpha }$ lifting, the usual global $C^{2,\alpha }(\bar{U})$ branch applies with its stated coefficient/data hypotheses. No global Hölder regularity is claimed from the exterior-sphere condition alone, and no optimizer convergence is asserted.

Regularized PINN minimizers have vanishing residuals under sample refinement

Exterior-sphere elliptic Dirichlet existence, energy and residual stability

Elliptic H1 error estimate from the source coefficient bound

Shin et al., Theorems 3.3–3.4 and Appendices D–E

The source's stronger global $C^{2,\alpha }(U)$ existence assertion for exterior-sphere domains is false for its own globally bounded Hölder convention; the native $2\pi \mathrm{/}3$ cutoff-sector counterexample is retained. The exact original carrier is archived in the proposal. The active divergence formula counts $\mathrm{div}(bu)$ once, consistently with the expanded coefficients and energy calculation; the original repeated-index printed expression is recorded in the operator definition's source discrepancy. The minimizer conclusion uses the repaired populated-cell proof and the corrected small-scale sampling convention, with no hidden balance premise.