A Generalized Probabilistic Learning Approach for Multi-Fidelity Uncertainty Propagation in Complex Physical Simulations

Two of the most significant challenges in uncertainty propagation pertain to the high computational cost for the simulation of complex physical models and the high dimension of the random inputs. In applications of practical interest both of these problems are encountered and standard methods for uncertainty quantification either fail or are not feasible. To overcome the current limitations, we propose a probabilistic multi-fidelity framework that can exploit lower-fidelity model versions of the original problem in a small data regime. The approach circumvents the curse of dimensionality by learning dependencies between the outputs of high-fidelity models and lower-fidelity models instead of explicitly accounting for the high-dimensional inputs. We complement the information provided by a low-fidelity model with a low-dimensional set of informative features of the stochastic input, which are discovered by employing a combination of supervised and unsupervised dimensionality reduction techniques. The goal of our analysis is an efficient and accurate estimation of the full probabilistic response for a high-fidelity model. Despite the incomplete and noisy information that low-fidelity predictors provide, we demonstrate that accurate and certifiable estimates for the quantities of interest can be obtained in the small data regime, i.e., with significantly fewer high-fidelity model runs than state-of-the-art methods for uncertainty propagation. We illustrate our approach by applying it to challenging numerical examples such as Navier-Stokes flow simulations and monolithic fluid-structure interaction problems.Comment: 31 pages, 14 figure

Tumour growth: An approach to calibrate parameters of a multiphase porous media model based on in vitro observations of Neuroblastoma spheroid growth in a hydrogel microenvironment

To unravel processes that lead to the growth of solid tumours, it is necessary to link knowledge of cancer biology with the physical properties of the tumour and its interaction with the surrounding microenvironment. Our understanding of the underlying mechanisms is however still imprecise. We therefore developed computational physics-based models, which incorporate the interaction of the tumour with its surroundings based on the theory of porous media. However, the experimental validation of such models represents a challenge to its clinical use as a prognostic tool. This study combines a physics-based model with in vitro experiments based on microfluidic devices used to mimic a three-dimensional tumour microenvironment. By conducting a global sensitivity analysis, we identify the most influential input parameters and infer their posterior distribution based on Bayesian calibration. The resulting probability density is in agreement with the scattering of the experimental data and thus validates the proposed workflow. This study demonstrates the huge challenges associated with determining precise parameters with usually only limited data for such complex processes and models, but also demonstrates in general how to indirectly characterise the mechanical properties of neuroblastoma spheroids that cannot feasibly be measured experimentally

Bayesian calibration of coupled computational mechanics models under uncertainty based on interface deformation

Calibration or parameter identification is used with computational mechanics models related to observed data of the modeled process to find model parameters such that good similarity between model prediction and observation is achieved. We present a Bayesian calibration approach for surface coupled problems in computational mechanics based on measured deformation of an interface when no displacement data of material points is available. The interpretation of such a calibration problem as a statistical inference problem, in contrast to deterministic model calibration, is computationally more robust and allows the analyst to find a posterior distribution over possible solutions rather than a single point estimate. The proposed framework also enables the consideration of unavoidable uncertainties that are present in every experiment and are expected to play an important role in the model calibration process. To mitigate the computational costs of expensive forward model evaluations, we propose to learn the log-likelihood function from a controllable amount of parallel simulation runs using Gaussian process regression. We introduce and specifically study the effect of three different discrepancy measures for deformed interfaces between reference data and simulation. We show that a statistically based discrepancy measure results in the most expressive posterior distribution. We further apply the approach to numerical examples in higher model parameter dimensions and interpret the resulting posterior under uncertainty. In the examples, we investigate coupled multi-physics models of fluid-structure interaction effects in biofilms and find that the model parameters affect the results in a coupled manner