Robin-Boundary Diffusion Sampler

Use a two-dimensional constrained latent space whose admissible region is a wedge, or locally approximate a more general constraint manifold by a wedge. Train a score network s_theta(x,t) with the standard denoising score-matching objective, then add boundary points and a Robin residual. If the network models a density p_theta, use R_b = normal_derivative p_theta + q p_theta. If it models a score, estimate the normal derivative by finite differences and penalize the equivalent logarithmic condition n dot gradient_x log p_theta + q = 0. In reverse Euler-Maruyama sampling, detect a proposed boundary crossing. Reflect the normal displacement with probability 1 minus min(1, q times sqrt(delta_t divided by D)), and absorb or terminate the trajectory otherwise; calibrate this discretization against direct local-time estimates. The paper contributes the Robin PDE, the local-time Feynman-Kac weight, and the absorbing persistence exponent; the network estimates the score and boundary density. First experiment: train a two-dimensional variance-preserving diffusion on Gaussian data restricted to wedges with alpha equal to pi/2, pi/3, and pi/4, comparing hard rejection, pure reflection, and Robin sampling for q equal to 0.1, 0.3, 1, and 3. Measure survival curves and their local log-slopes. The predicted signature is a crossover near t_c = 1/(D q^2), within a factor of two, followed by a late-time slope near -pi/(2 alpha), within 20 percent. If the crossover does not shift as q raised to the power -2, or the late slope is angle-independent, reject the implementation.