Fenchel-Gap Certified Neural PDE Training

1. Integration point: replace the scalar strong-form PINN residual loss with a two-network variational module. The primal MLP outputs $u_\theta(x)$ and the dual MLP outputs a vector flux $q_\phi(x)\in\mathbb{R}^d$. Use automatic differentiation only for $\nabla u_\theta$ and, optionally, $\nabla\cdot q_\phi$; do not differentiate the rapidly varying coefficient inside a strong residual. 2. Pseudocode: sample energy points $x_n$ and equilibrium points $z_m$; compute $g_n=\nabla u_\theta(x_n)$; evaluate $W(x_n,g_n)$ and $W^*(x_n,q_\phi(x_n))$; form $G_n=W+W^*-q_\phi\cdot g_n$; compute $r_m=\nabla\cdot q_\phi(z_m)+f(z_m)$; set $L=N^{-1}\sum_nG_n+\lambda M^{-1}\sum_m r_m^2+L_{boundary}$; jointly update $\theta$ and $\phi$ with Adam or LBFGS. For quadratic $W(x,\xi)=\frac12\xi^TK(x)\xi$, compute $W^*(x,q)=\frac12q^TK(x)^{-1}q$; for a general monotone $W$, evaluate the conjugate by a small inner Newton solve or use a learned convex potential with analytic conjugate. 3. The paper-derived quantity is the nonnegative Fenchel gap and its monotonicity-based state-error certificate. Estimate the full certificate by fresh Monte Carlo points, while training uses minibatch estimates; report the gap, equilibrium residual, and confidence intervals separately. Increase $\lambda$ only until equilibrium violation is comparable to the gap, avoiding an excessively stiff penalty. 4. First cheap experiment: use a 2D periodic nonlinear coefficient with a known finite-element reference solution, and compare a strong-residual PINN, a primal-only Deep Ritz model, and the primal-dual model at $\varepsilon=1/8,1/16$. Plot true $H^1$ error against the reported gap and test whether the gap upper-bounds error throughout training. Success is a monotone reduction in the certificate, strong correlation between certificate and true error, and stable convergence at small $\varepsilon$ where the strong-residual baseline exhibits gradient variance or sample-size degradation.