Certified Neural Ritz Solver

Integrate this as a differentiable spectral-eigensolver head for a neural PDE model, not as a generic classifier loss. Choose a self-adjoint operator such as -Delta+V on a bounded domain and a known basis {phi_j}_{j=1}^M satisfying the boundary conditions. Let the MLP receive the potential V, geometry parameters, or a problem instance and output M candidate coefficient vectors c_1,...,c_K. Form the candidate functions u_k=sum_j c_{k,j} phi_j. Compute the stiffness and mass matrices by quadrature, A_{ij}=a(phi_i,phi_j) and B_{ij}=b(phi_i,phi_j), or use the network outputs directly to assemble a compressed K-dimensional pair A_c=C^TAC and B_c=C^TBC, where C=[c_1,...,c_K]. Solve A_c q=lambda B_c q, orthogonalizing with respect to B_c; these are the neural Ritz values lambda_{k,h}. For each k compute the certified target lambda_cert[k]=lambda_h[k]/(1+C_h^2*lambda_h[k]) and optimize an ordering-aware loss such as sum_k(lambda_cert[k]-target_k)^2 plus eigenvector residuals ||A_c q_k-lambda_h[k]B_c q_k||^2 and B-orthogonality penalties. The paper's mathematics supplies the projection inequality and the lower-bound transform. For a Fourier or Laplacian eigenbasis, set C_h=lambda_{M+1}^{-1/2}; otherwise estimate a conservative C_h from a validated approximation-space bound or deliberately use an inflated value alpha*C_est with alpha>1. Do not backpropagate through an uncertified estimate while claiming a certificate. First test on 1D and 2D Schrödinger operators with analytic Laplacian spectra, comparing a neural Ritz solver trained with raw eigenvalue loss against the certified loss at identical basis size and FLOPs. Measure actual eigenvalue error, violation rate of lambda_cert<=lambda_exact, residual norms, and the number of basis modes needed to reach a fixed accuracy. Success means zero lower-bound violations and comparable or improved eigenvalue accuracy, with the certificate remaining useful under distribution shift in V.