8.3. Experiment 3: the Dirichlet operator layer🔗

Use Example 7.9 with Gaussian activation and observable \varphi=1. For the Dirichlet Green's function g, the output weight is w(y)=\int_0^1g(y,t)\,\mathrm dt=1-\cosh(y-1/2)/\cosh(1/2)>0 and V=\int_0^1w(y)\,\mathrm dy=1-2\tanh(1/2)\approx0.0758. Use d=64,512,4096 midpoint nodes and 100 test inputs with 30 sine modes. Appendix E gives the input and parameter distributions.

For sampling, compare f_N(x)=(V/N)\sum_j e^{-\langle g(y_j,\cdot),x\rangle^2/2}, with y_j\sim w/V, against the integral layer over 20 trials. At resolutions 512 and 4096 the error follows N^{-1/2}. At resolution 64 it levels off near 10^{-5}, the quadrature error of that reference discretization.

For an 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. A Monte Carlo estimate with 200000 Gaussian inputs and Karhunen–Loève truncations d'=8,64,512 agrees with the closed-form values within about three standard errors: