[Paper Review] Gaussian process modeling for stochastic multi-fidelity simulators, with application to fire safety

Rémi Stroh, Julien Bect+3 more|arXiv (Cornell University)|May 9, 2016

Probabilistic and Robust Engineering Design4 citations

TL;DR

This paper extends Bayesian multi-fidelity Gaussian process modeling to stochastic simulators, specifically for fire dynamics simulations with varying mesh fidelity. By modeling both mean and variance as Gaussian processes and incorporating a non-stationary correlation structure, the method enables accurate prediction of high-fidelity fire behavior using low-fidelity simulations, achieving predictive performance comparable to high-fidelity models with significantly reduced computational cost.

ABSTRACT

To assess the possibility of evacuating a building in case of a fire, a standard method consists in simulating the propagation of fire, using finite difference methods and takes into account the random behavior of the fire, so that the result of a simulation is non-deterministic. The mesh fineness tunes the quality of the numerical model, and its computational cost. Depending on the mesh fineness, one simulation can last anywhere from a few minutes to several weeks. In this article, we focus on predicting the behavior of the fire simulator at fine meshes, using cheaper results, at coarser meshes. In the literature of the design and analysis of computer experiments, such a problem is referred to as multi-fidelity prediction. Our contribution is to extend to the case of stochastic simulators the Bayesian multi-fidelity model proposed by Picheny and Ginsbourger (2013) and Tuo et al. (2014).

Motivation & Objective

• To address the high computational cost of high-fidelity fire simulations by leveraging cheaper, lower-fidelity simulations.
• To extend deterministic multi-fidelity Gaussian process models to stochastic simulators with inherent variability.
• To enable accurate prediction of fire behavior at fine mesh resolution using data from coarser meshes.
• To estimate the probability of exceeding critical temperature thresholds in fire safety assessments.
• To develop a computationally efficient surrogate model that maintains predictive accuracy while reducing simulation burden.

Proposed method

• Models the simulator output as a stochastic process with mean and variance functions, assuming Gaussian distributions for outputs.
• Uses a decomposition of the mean function into an ideal simulator and a deterministic error term that decreases as mesh size approaches zero.
• Implements a non-stationary Gaussian process prior for the mean function, with a separable covariance structure depending on input and mesh size.
• Models the variance as a function of mesh size only, assuming it decreases with finer mesh resolution.
• Applies a joint Gaussian process prior on the mean and variance, enabling full Bayesian inference with hierarchical priors.
• Employs a multi-fidelity framework where low-fidelity data inform the prior, and high-fidelity data refine the posterior predictive distribution.

Experimental results

Research questions

• RQ1 Can a Bayesian multi-fidelity Gaussian process model be extended to handle stochastic simulators with inherent variability?
• RQ2 How well can a low-fidelity model predict high-fidelity fire simulation outcomes in terms of mean and variance?
• RQ3 What is the predictive accuracy of the proposed model compared to a high-fidelity model in estimating critical temperature thresholds?
• RQ4 How does the model's uncertainty compare to that of a high-fidelity reference model in estimating failure probabilities?
• RQ5 Can the proposed model achieve reliable safety assessments with a limited computational budget?

Key findings

• The proposed multi-fidelity non-stationary model achieves predictive performance comparable to the high-fidelity model in predicting temperature at 20 cm mesh size.
• The multi-fidelity stationary model also shows good fit, though it slightly underestimates high-temperature outputs.
• Posterior standard deviations of normalized residuals are close to one, indicating well-calibrated predictive variances.
• The high-fidelity model yields the narrowest posterior distribution for the probability of exceeding 60°C, but the multi-fidelity models remain compatible with the reference.
• The multi-fidelity stationary model achieves a small posterior uncertainty, though its support slightly diverges from the high-fidelity model.
• The proposed model achieves a balance between accuracy and computational efficiency, enabling reliable safety assessment with reduced simulation cost.