Full-Field Phase-Boundary PINN

Use a complex-valued MLP E_theta(x), implemented as an MLP with two real outputs for real and imaginary parts, as a physics-informed eigenmode solver. The exact integration point is the training loss. Sample interior points x_j in [0,a], evaluate E_theta and its first and second derivatives by automatic differentiation, and optimize both network weights and a scalar eigenvalue parameter k or k^2. Compute z=cos(qa)+i sin(qa), r_j=E_theta''(x_j)+k^2E_theta(x_j), r_a=E_theta(a)-zE_theta(0), and d_a=E_theta'(a)-zE_theta'(0). Minimize L=mean_j|r_j|^2+lambda_bc(|r_a|^2+|d_a|^2)+lambda_norm(|E_theta(x_ref)|^2-1)^2. To remove arbitrary global phase, either penalize Im(E_theta(x_ref))^2 or align the prediction to a reference field during evaluation. Do not include the periodic-factor residual with 2iq and q^2 terms in the first experiment. The residual and phase conditions come directly from the mathematics; collocation density, loss weights, and initialization are empirical. Begin with the homogeneous cell, where exact solutions are E=exp(+-ikx), then test a piecewise-constant periodic coefficient and compare against a finite-element generalized eigenproblem. Sweep q and q+m\kappa and verify that recovered eigenvalues and reconstructed fields coincide. Success is fewer low-residual nonphysical modes, stable convergence across aliases, and lower eigenvalue error than a periodic-factor PINN at equal parameter count.