Nonlinear information fusion algorithms for data-efficient multi-fidelity modelling

Abstract

Multi-fidelity modelling enables accurate inference of quantities of interest by synergistically combining realizations of low-cost/low-fidelity models with a small set of high-fidelity observations. This is particularly effective when the low- and high-fidelity models exhibit strong correlations, and can lead to significant computational gains over approaches that solely rely on high-fidelity models. However, in many cases of practical interest, low-fidelity models can only be well correlated to their high-fidelity counterparts for a specific range of input parameters, and potentially return wrong trends and erroneous predictions if probed outside of their validity regime. Here we put forth a probabilistic framework based on Gaussian process regression and nonlinear autoregressive schemes that is capable of learning complex nonlinear and space-dependent cross-correlations between models of variable fidelity, and can effectively safeguard against low-fidelity models that provide wrong trends. This introduces a new class of multi-fidelity information fusion algorithms that provide a fundamental extension to the existing linear autoregressive methodologies, while still maintaining the same algorithmic complexity and overall computational cost. The performance of the proposed methods is tested in several benchmark problems involving both synthetic and real multi-fidelity datasets from computational fluid dynamics simulations.

Adaptive multi‐index collocation for uncertainty quantification and sensitivity analysis

“…Finally it is important to note that multi‐level surrogate methods have been developed which do not use sparse grids to build multivariate approximations over the stochastic space. Gaussian processes were used in References and low‐rank reduced order models were used in Reference . These methods are not considered further here since, similar to multi‐level sparse grid collocation, if a model does indeed have multiple discretization hyper‐parameters it can be unclear how to define a suitable 1D model hierarchy.…”

Multifidelity approaches for uncertainty quantification

“…Perdikaris et al chose the covariance function in their paper. However, different choices are also conceivable.…”

“…The scheme put forth by Kennedy and O'Hagan can also be recovered as a special case. It is also important to note, that the variant proposed by Perdikaris et al is essentially a special case of so‐called deep GP introduced in Damianou and Lawrence, the use of which would offer even more flexibility at increased computational cost and a far more complex implementation.…”

“…In conclusion, the approach put forth in Perdikaris et al is very flexible and can cope with complex nonlinear and space‐dependent cross‐correlation between levels. It can be interpreted as generalization of the original scheme proposed by Kennedy and O'Hagan and is also relatively easy to implement.…”

“…However, to highlight the similarity, we can write the relation between low‐fidelity output and high‐fidelity output in the BMFMC scheme in terms of a covariance function. This allows us to close the loop to the scheme put forth in Perdikaris et al Using the notation introduced in Section 4.1 the relation between low‐fidelity and high‐fidelity model in the BMFMC scheme can be written as where g (·) is a GP with the following covariance structure: …”