Isotonic surrogate modeling for computer experiments with many input variables

Virtual simulators are widely used for studying complex physical phenomena, from particle collisions to rocket propulsion. Such "computer experiments" can be highly time-intensive, and a Bayesian surrogate model can be used for efficient emulation with reliable uncertainty quantification. To train accurate surrogates with a limited sample size n, recent work has explored the incorporation of monotonicity (or isotonicity) information, which can often be elicited from physical systems. In practical applications with many input variables, however, existing Bayesian isotonic models can face statistical and computational limitations, which may result in worse performance compared to models that do not incorporate isotonicity. We propose a new transformed additive isotonic model (TAIM), which aims to tame this "curse-of-dimensionality". TAIM makes use of a flexible transformed additive isotonic modeling framework, which leverages a data-estimated link transformation and a monotone basis model with spike-and-slab priors on basis weights. Prediction-wise, TAIM achieves (up to log factors) a posterior contraction rate of O(n-1/3) when the true black-box function is in a transformed additive isotonic form with mild smoothness conditions. Such a rate does not depend on the input dimension d for terms involving n, which softens the effect of dimensionality on posterior predictions. Computation-wise, TAIM allows for efficient posterior inference via a carefully designed Gibbs sampler, where each sampling iteration requires only linear work in d. We further present an extension of TAIM that can model potential deviations from transformed additivity. Numerical experiments and two applications show the effectiveness of TAIM for isotonic surrogate modeling with many input variables.