Arcosh-convex PINN output

Integrate this into a coordinate-based PINN for a scalar elliptic field. The MLP receives spatial coordinates x in R^d and outputs one scalar latent value w_theta(x). Reconstruct u_theta(x) = -2 log cosh(w_theta(x)); use a numerically stable implementation such as u = -2 times (abs(w) + softplus(-2 abs(w)) - log(2)). At interior collocation points, use automatic differentiation to compute D^2u_theta. For the Liouville equation, form the residual r = trace(D^2u_theta) - exp(u_theta). For the real two-Hessian equation, form r = sigma_2(D^2u_theta) - exp(2u_theta). Sample boundary points and penalize w_theta squared, which enforces the required boundary value u = 0. On every interior minibatch, compute the symmetric Hessian H_w = (D^2w_theta + D^2w_theta transpose) divided by 2. For dimensions at most three, calculate its smallest eigenvalue using a differentiable symmetric eigensolver. Add the convexity loss L_conv = mean(softplus(epsilon - lambda_min(H_w))), with epsilon between 0.001 and 0.01. The complete loss is mean(r squared) + lambda_bc mean(w_boundary squared) + lambda_conv L_conv. The paper supplies the transformation and desired convexity property; all Hessians, eigenvalues, residuals, and gradients are computed numerically. Start with a unit disk and an ellipse in two dimensions, using the same four-layer tanh MLP, collocation points, optimizer, and training budget for both the proposed and unconstrained baselines. Compare PDE residual, boundary error, relative error against a high-resolution finite-element solution, minimum transformed Hessian eigenvalue, and variance over random seeds. The method is successful if it reduces error or residual at equal training cost, eliminates spurious nonconvex regions, and improves convergence reliability rather than merely lowering the auxiliary convexity loss.