Parabolic PINN minimizers converge to the initial-boundary solution

Assume the Axiom of Choice and Countable Choice. Let $U\subset {\mathbb{R}}^d$ be nonempty, bounded and open with an exterior ball at every spatial boundary point, $T>0$, and $Q=U\times (0,T)$. Use the full parabolic coefficient/data hypotheses of the variable-coefficient existence theorem: differentiability of $a_{ij},b_i$, symmetry and uniform ellipticity, the source weighted parabolic Hölder coefficient bounds, bounded locally parabolic Hölder expanded drift and potential, bounded locally parabolic Hölder forcing, compatible Hölder initial and lateral data, and one-sided local Hölder control at interior points of the final face. Use $L_t[u]={\sum }_{i,j}{D}_i(a_{ij}{D}_{j}u)+{\sum }_i{D}_i(b_{i}u)+{\sum }_i{c}_i{D}_{i}u+du,$ ${\tilde{b}}_i={\sum }_j{D}_j{a}_{ij}+b_i+c_i,\tilde{c}={\sum }_i{D}_i{b}_i+d\le 0.$ Retain the source pointwise bounds with constants ${\lambda }_0,\mathrm{\Lambda },v$ and its auxiliary $\theta >0$ condition ${\theta }^2{a}_{11}+\theta {\tilde{b}}_1\ge 1$; the direct parabolic time barrier also works without that auxiliary condition. Boundedness is part of the source's order-zero Hölder norm, not a consequence of its weighted ${\delta }^{2}f$ bound alone.

Use the corrected small-scale sampling, data/trial-class, exact-zero-witness and exact regularization assumptions of the regularized PINN theorem in the ambient dimension $D=d+1$, with residual $-h_t+L_t[h]$, target $f$, and restriction to the initial-lateral boundary ${\mathrm{\Gamma }}_T$ with target $\psi$ combining the compatible data. Require $H_m\subset C^{2,1}(Q)\cap C(\bar{U}\times [0,T])$. The Euclidean Hölder assumptions in this sampling interface are retained in addition to the parabolic PDE regularity assumptions. There is one classical solution $u^{\ast }\in C^{2,1}(Q)\cap C(\bar{U}\times [0,T])$, locally parabolic Hölder in its displayed derivatives, to $-u_t^{\ast }+L_t[u^{\ast }]=f$ and the prescribed data.

Every sequence of global empirical regularized minimizers converges to $u^{\ast }$ in $C^0(Q)$ almost surely as both counts tend to infinity. For $m_r=O(m_b^{(d+1)\mathrm{/}d})$, the occupancy-event squared uniform solution error is $O(m_r^{-\alpha \mathrm{/}(d+1)})$, with the probability and constants of the regularized-loss theorem at dimension $D=d+1$. If all minimizers satisfy both initial and lateral data exactly and the displayed divergence expansion agrees distributionally with the classical operator (in particular if $a,b$ are spatially $C^1$ locally), then their errors belong to $L^2(0,T;H_0^1(U))$ and tend to zero in $L^2(0,T;H^1(U))$ almost surely. The energy conclusion uses the actual bounded divergence coefficients and their source $v$ bound. No closed-cylinder $C^{2,1}$ regularity of $u^{\ast }$ or convergence of an optimizer is asserted.

Variable-coefficient parabolic initial-boundary existence and residual stability

Regularized PINN minimizers have vanishing residuals under sample refinement

Zero-data bounded classical parabolic errors are weak energy solutions

Shin et al., Theorems 3.6–3.7 and Appendices F–G

The source's pointwise word “differentiable” is retained for its classical problem. Its integration-by-parts energy argument additionally uses the weak divergence identity; this requirement is explicit here, with local spatial $C^1$ as a sufficient interpretation. Merely continuous zero boundary data do not prove energy membership. The new energy-identification helper supplies that separate step once the divergence equation has its intended weak meaning. The source's unusual weighted $C_{2+\alpha }$ convention is preserved in the definition; this theorem does not claim that the limiting continuous-data solution has the printed weighted global trial norm.