Reduced dimensional Gaussian process emulators of parametrized partial differential equations based on Isomap

Abstract

In this paper, Isomap and kernel Isomap are used to dramatically reduce the dimensionality of the output space to efficiently construct a Gaussian process emulator of parametrized partial differential equations. The output space consists of spatial or spatio-temporal fields that are functions of multiple input variables. For such problems, standard multioutput Gaussian process emulation strategies are computationally impractical and/or make restrictive assumptions regarding the correlation structure. The method we develop can be applied without modification to any problem involving vector-valued targets and vector-valued inputs. It also extends a method based on linear dimensionality reduction to response surfaces that cannot be described accurately by a linear subspace of the high dimensional output space. Comparisons to the linear method are made through examples that clearly demonstrate the advantages of nonlinear dimensionality reduction.

A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials

“…Although closely related, model reduction for single scale problems [264][265][266][267] differs to those in the multiscale setting [268][269][270][271]. In general, two nested BVPs are usually required in the multiscale setting.…”

“…Recently, several dimension reduction techniques have been proposed that seek meaningful low-dimensional structures hidden in high-dimensional data [264,266,268,[301][302][303][304][305][306][307]. In the context of mechanics of materials, this high-dimensional data may represent full-field experimental measurements and/or detailed RUC simulations.…”

“…Isomap has been used in several contexts. For example, a Gaussian process emulator of parametrized partial differential equations for single scale analysis has been developed by Xing et al [264] and Triantafyllidis et al [266]. Ganapathysubramanian and Zabaras employed Isomap for nonlinear dimension reduction of a data-driven stochastic input model [305], and Bhattacharjee and Matouš [268] used Isomap for manifold-based reduced order modeling of nonlinear heterogeneous hyperelastic materials as highlighted bellow.…”

Dimension Reduction for the Emulation of Cardiac Electrophysiology Models for Single Cells and Tissue

“…The features are the k most dominant eigenvectors of the kernel matrix, in the same fashion as kernel principal component analysis. Predictions for V (x, t) are formed by using GP emulation to predict the co-ordinates in feature space and then transforming back into the original space by undoing the isomap transform via local linear interpolation [7].…”

Manifold learning for the emulation of spatial fields from computational models

“…Subsurface flow in a porous medium can be modelled by Brinkman's equation (with a Boussinesq buoyancy term) and a thermal energy balance [62]: (28) in which v is the flow velocity, T is temperature, p is pressure, g is the gravita- [17] can be found in [19],…”