Shape-Constrained Bayesian Active Learning of Self-Limiting Saturation Curves

Saturation curves, monotone responses that rise from zero to a plateau, recur across gas adsorption, enzyme kinetics, and self-limiting growth. Learning one from a few costly, noisy measurements is a common bottleneck. We present a Bayesian active-learning platform built on a penalized monotone I-spline. Non-negative weights make every posterior draw rise from zero and never fall, a survey measurement anchors the plateau in place of a flatness prior, and the input reaching any saturation level is read from the posterior by level crossing, assuming no kinetic form. Each measurement is placed where the surrogate is least certain, which beats random designs and edges a log-spaced grid of the same budget. One fixed configuration recovers five kinetically distinct regimes from twenty measurements to a saturation error of 0.037 to 0.041, below the output noise, and averages 0.045 on 250 curves from an unseen distribution. Running unconstrained Gaussian processes (GPs) and two unpenalized monotone bases through the same loop separates each ingredient's contribution. Logarithmic resolution and the roughness penalty carry the accuracy, while monotonicity guarantees a level crossing unique on every draw. The penalty keeps the surrogate the most accurate model at every noise level, and its second-order form centers the recommended input for saturation where a first-order penalty reads late (a bias of +0.32 s with 94% coverage, against +0.44 s and 64% for the matched unconstrained GP). Because the recommended input lies where the curve is nearly flat, no method can pin it down precisely, and we report it as a cost-indexed quantile with the probability of falling short.