{
 "artifacts": null,
 "category": "sampling",
 "description": "Train a score-based diffusion model on a domain with partially reactive constraints, replacing hard rejection or large boundary penalties by a Robin boundary condition. Samples approaching the constraint boundary acquire a hazard proportional to accumulated boundary local time, giving a continuous interpolation between reflection and absorption.",
 "formulas_latex": [
  "partial_t S_q(x,t) = D Delta S_q(x,t), and normal_derivative S_q(x,t) + q S_q(x,t) = 0 for x on the boundary of Omega.",
  "S_q(t given x_0) = E_x0[ exp(-q L_t) ].",
  "S_infinity(t given x_0) is proportional to t^(-pi/(2 alpha)) as t tends to infinity.",
  "t_c is proportional to 1/(D q^2), and d log S_q(t) divided by d log t tends to -pi/(2 alpha) when t is much larger than t_c."
 ],
 "id": 3075,
 "implementation": "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.",
 "math_summary": "Let X_t be Brownian motion with diffusion coefficient D in the wedge Omega = {(r, theta): r greater than 0, 0 less than theta less than alpha}. Let L_t denote boundary local time and q greater than or equal to 0 the boundary reactivity. The Robin survival weight is W_t = exp(-q L_t), so S_q(t given x_0) = E_x0[W_t]. The backward survival function obeys partial_t S_q = D Delta S_q in Omega, with normal_derivative S_q + q S_q = 0 on the wedge sides. For perfectly absorbing sides, the long-time survival law is S_infinity(t given x_0) proportional to t raised to the power -pi divided by (2 alpha). Dimensional analysis gives the Robin crossover time t_c proportional to 1 divided by (D q squared).",
 "math_tags": [
  "stochastic-processes",
  "pde",
  "dynamical-systems",
  "geometry"
 ],
 "ml_areas": [
  "diffusion",
  "diffusion-sampling",
  "architecture"
 ],
 "paper": {
  "arxiv_id": "2609.02665",
  "arxiv_url": "https://arxiv.org/abs/2609.02665",
  "summary_what_math_gives_to_ml": "The paper gives a constructive asymptotic mechanism for Brownian survival in a wedge with partially reactive boundaries. Robin reactivity introduces a boundary length scale and produces a crossover between nearly reflecting behavior at short times and the absorbing-wedge persistence law at long times. The transferable asset is the Feynman-Kac representation of Robin killing through accumulated boundary local time, which can be used to build diffusion samplers with soft, calibrated constraint handling instead of hard rejection or arbitrary penalties. The predicted crossover scale and late-time power-law exponent provide falsifiable diagnostics for the neural implementation.",
  "title": "Survival in a partially reactive wedge",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 8,
  "usefulness": 5
 },
 "solves": [
  "accuracy",
  "stability",
  "sample-efficiency"
 ],
 "title": "Robin-Boundary Diffusion Sampler",
 "url": "https://synthcore.org/idea/3075/robin-boundary-diffusion-sampler",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}