Tangential probability-flow sampler

(1) Integration point: replace the drift evaluation in the reverse or probability-flow ODE sampler. Given state x, time t, score network s_theta(x,t), known forward drift f(x,t), diffusion scale sigma(t), domain function g(x), and normal nu(x), compute v=f-0.5 sigma^2 s_theta. Apply the collar correction only when abs(g(x)) is less than epsilon. For a box, use the normal of the closest active face; for a sphere, use nu=x divided by its norm; for a general implicit domain use nu=grad g divided by the norm of grad g. (2) Pseudocode: v = f(x,t) - 0.5*sigma(t)**2*score_theta(x,t); a=clip(1-abs(g(x))/eps,0,1); v=v-a*dot(v,nu(x))*nu(x); x=x+dt*v. If a solver step exits the domain, optionally apply one differentiable projection back to g<=0, but count these events separately. (3) During score training, add lambda_bc * a * dot(v,nu(x))**2 at noisy samples in the boundary collar. Estimate normals analytically for simple domains or with nu=grad(g)/norm(grad(g)); estimate violation frequency, boundary current, and compressibility empirically from particle histograms or the Jacobian trace of the corrected velocity. The paper supplies the no-flux and tangency rationale; epsilon, lambda_bc, and correction strength are experimental hyperparameters. (4) First experiment: train a small VP or VE score model on a two-dimensional Gaussian mixture truncated to [-1,1]^2 and on a disk. Compare ordinary probability flow, coordinate clipping, reflected Euler-Maruyama, and this tangent sampler at equal ODE function evaluations. Measure constraint violations, sliced Wasserstein distance to the target, boundary histogram error, and the fraction of trajectories requiring projection. The expected signal is near-zero wall crossing, fewer clipping artifacts, and lower distributional error at the same solver budget. Ablate the boundary penalty and collar correction to separate training and inference effects.