E.3 Dirichlet layer and transform verification🔗

Experiment 3 uses d=64,512,4096 midpoint nodes and 100 test inputs x=\sum_{n\le30}(u_n/n)e_n with independent uniform u_n\in[-1,1]. With the Dirichlet Green's function g and observable 1, the parameter law is w(y)\,\mathrm dy/V, where w(y)=1-\cosh(y-1/2)/\cosh(1/2) and V=1-2\tanh(1/2). The networks approximate the integral layer over 20 trials. Midpoint quadrature uses weight 1/d. Draws from w/V invert a cumulative distribution computed by the trapezoidal rule on 20001 uniform grid points with linear interpolation; each Green direction is then evaluated on the input grid. The coarsest quadrature produces the error floor reported in Section 8.

For the independent transform check, use the filter from Experiment 2, a(t)=\sqrt2\sin(\pi t)+(\sqrt2/2)\sin(3\pi t) and biases b=0,-1/2,-1. Monte Carlo integration over 200000 Gaussian inputs uses Karhunen–Loève truncations d'=8,64,512. The comparisons and standard errors are reported in Section 8. The closed-form transform uses 2048 midpoints for inner products and 400 for the outer integral. Gaussian draws have covariance eigenvalues \lambda_n=(1+\pi^2n^2)^{-1} and are evaluated on the same 2048-point grid. The standard error is the empirical standard deviation of the integrand divided by the square root of the number of draws. This check at the specified direction and biases does not eliminate quadrature error or Gaussian-input truncation error.