Corrector-Enriched Two-Scale Network

1. Integration point: modify the solution MLP used by a variational PINN or Deep Ritz solver. Keep a coarse MLP $u_{0,\theta_0}(x)$, add a corrector MLP $\chi_{\theta_1}(x,y)$, and evaluate it at $y=x/\varepsilon$; enforce homogeneous boundary conditions by multiplying the final output by a known boundary mask $b(x)$ or by adding a boundary extension $g(x)$. Use automatic differentiation to obtain $\nabla_x u_\theta^\varepsilon(x)$, including the chain-rule derivative through $y=x/\varepsilon$. 2. Pseudocode: sample collocation points $x_n$; set $y_n=x_n/\varepsilon$; compute $u_0=u_0(x_n)$ and $J_0=\nabla_xu_0(x_n)$; compute $\chi=\chi(x_n,y_n)$; form $u=u_0+\varepsilon\sum_j\chi_jJ_{0,j}$; compute $L=N^{-1}\sum_n[W(x_n,\nabla u_n)-f_nu_n]$; backpropagate and update both parameter sets. 3. The theoretical objects are the coarse and first-order feature errors $\Phi_{0,m_0}$ and $\Phi_{1,m_1}$; estimate them empirically using held-out energy error and gradient error. If coefficient correctors are known, initialize or freeze $\chi_j$ from cell-problem data; otherwise pretrain them on periodic cell residuals and enforce periodicity with Fourier features or modulo-one coordinates. 4. First experiment: solve a 2D nonlinear periodic elliptic equation at $\varepsilon\in\{1/8,1/16,1/32\}$ using the same MLP width and the same number of quadrature points. Compare a plain coordinate MLP against the two-scale ansatz at equal parameter count and equal FLOPs. Measure relative $H^1$ error, energy error, gradient variance, and updates to a fixed error. Success means error scaling remains roughly proportional to $\varepsilon$ plus feature error, while the plain MLP loses accuracy or requires substantially more samples as $\varepsilon$ shrinks.