Exact-boundary geometric PINN ansatz

Integrate this at the output head of a PINN whose inputs are disc coordinates $z=(z_1,z_2)$ and whose raw outputs are one scalar $a_\theta(z)$ and one vector $b_\theta(z)\in\mathbb R^n$. Precompute a differentiable boundary extension $E[\gamma]$; for a closed planar curve, a simple choice is the harmonic or Fourier extension obtained by evaluating the truncated Fourier series of $\gamma(e^{i\phi})$. At every collocation point compute $\rho=1-z_1^2-z_2^2$, then set $X=\rho\exp(a)$ and $Y=E[\gamma](z)+\rho b$. Compute first and second derivatives of $(X,Y)$ with respect to $z$ and form the pullback metric $g=X^{-2}J^\top J$; invert the $2\times2$ matrix $g$ with a stable determinant clamp. Assemble the hyperbolic tension residual using $\tau^\alpha=g^{ab}(\partial_{ab}u^\alpha+\Gamma^\alpha_{\beta\gamma}(u)\partial_a u^\beta\partial_b u^\gamma-\Gamma^c_{ab}(g)\partial_cu^\alpha)$, or use the coordinate expression already implemented for the target PDE. Optimize only the mean squared residual, with no boundary loss and no separate positivity penalty. Compare against an unconstrained network trained with boundary and positivity penalties on a small hyperbolic minimal-disc problem. Check boundary error, minimum predicted $X$, residual loss, and convergence variance across random seeds; success means boundary error remains at machine precision while residual reaches the same or lower value with fewer optimization steps.