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

Self-limiting saturation curves, monotone responses that rise from zero to a plateau, govern gas adsorption, enzyme kinetics, dose-response pharmacology, and the growth per cycle of atomic layer deposition (ALD), yet mapping each curve from a handful of costly measurements is a shared bottleneck.

The standard surrogate, a stationary-kernel Gaussian process, enforces no shape constraint: on sparse, noisy data it produces unphysical dips that corrupt both the inferred curve and the uncertainty used to choose the next experiment.

We present an active-learning platform built on Bayesian monotonic I-spline regression, in which every posterior curve rises from exactly zero and never decreases, the plateau is set by a measurement at maximum exposure rather than a prior, and the input at any saturation level is read from the posterior by level crossing with no kinetic model assumed.

Benchmarked isotherm by isotherm on five kinetically distinct families (Langmuir, dissociative Michaelis-Menten, sigmoidal Sips, logarithmic Elovich, and dispersive Kohlrausch-Williams-Watts), with ALD process development as the working example, the same fixed surrogate recovers every curve to within measurement noise without a single non-monotone posterior draw, and noise-free sweeps show the basis itself is near-exact across each family's regimes, locating its single capacity boundary at the sharpest sigmoidal onset.

Driven by ordinary uncertainty sampling, the platform reaches noise-floor accuracy within a 20-measurement budget in every regime, in as few as seven measurements, whereas random sampling succeeds in only two of the five; the predicted pulse times err only on the conservative side, toward longer pulses, near saturation.

Because the basis privileges no kinetic form, the platform applies wherever a self-limiting response must be learned from scarce data.