Complex-stretched resonance layer

1. Integration point: use a neural operator or neural state-space model for a 1D or 2D wave equation with a compact inhomogeneity K. Apply the deformation only to the spatial derivative stencil, Fourier differentiation matrix, or boundary-layer coordinates outside K; leave network features and physical coefficients unchanged inside K. On a radial grid define F(x)=f(|x|)x, with f=0 for |x|≤r0 and f=1 for |x|≥r1. Use a smooth cubic or quintic interpolation and enforce sup_r |r f'(r)|≤ℓ−1. Construct Φ_μ(x)=x+μF(x), usually with μ=iη and η>0. In one dimension replace ∂x by (1+μF'(x))^{-1}∂x; in multiple dimensions use J_μ=I+μ∇F and transform gradients by J_μ^{-T}. 2. Pseudocode: inputs are grid x, interaction mask K, stretch η, operator coefficients, and network parameters θ. Build F and J_μ; assemble the stretched operator H_{μ,θ}; run Arnoldi or differentiable shifted linear solves to obtain candidate eigenpairs H_{μ,θ}u_j≈z_ju_j; minimize L=L_data+λ_resΣ_j||H_{μ,θ}u_j−z_ju_j||²+λ_int||χ(H_{μ,θ}−z_j)^{-1}χ−R_target(z_j)||². Set χ=1 on K and zero in the stretched layer. 3. Computed from the paper’s mathematics are the identity-near-K deformation, Jacobian-scaled derivative, and interior localization principle. Estimated empirically are η, resonant eigenvalues z_j, eigenvectors u_j, and target resolvent actions. 4. First experiment: train a small Fourier neural operator or MLP neural operator on 1D Helmholtz scattering from compact random potentials. Compare hard Dirichlet, standard absorbing, and complex-stretched boundaries at equal grid resolution and parameter count. Measure boundary reflection, interior prediction error, first-three-resonance error, gradient-norm variance, and loss versus steps. Success means lower reflection and resonance error, unchanged interior accuracy, or faster loss descent at equal FLOPs.