Isotonic surrogate modeling for computer experiments with many input variables

2026-09-28

AI Summary

This study addresses the statistical and computational bottlenecks of Bayesian order-preserving surrogate models in high-dimensional computer experiments by proposing the Transformed Additive Isotonic Model (TAIM). The framework estimates the link function in a data-driven manner and incorporates spike-and-slab priors to induce sparsity among base models. Theoretically, we establish that the posterior contraction rate is independent of input dimensionality, effectively overcoming the curse of dimensionality. Computationally, a Gibbs sampler with linear complexity is developed and further extended to non-additive modeling. Numerical experiments and real-world applications demonstrate that TAIM achieves superior predictive accuracy and computational efficiency in high-dimensional settings, significantly outperforming conventional order-preserving models.

Abstract

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.

Problem

• Isotonic surrogate modeling
• Computer experiments
• Curse of dimensionality
• Bayesian isotonic models
• High-dimensional inputs

Innovation

• Isotonic surrogate modeling
• Transformed additive model
• Curse-of-dimensionality
• Spike-and-slab priors
• Gibbs sampler