8 research outputs found
Efficient sampling and solver enhancement for uncertainty quantification
No full textThe focus of this thesis is on the development of efficient and robust uncertainty quantiÞcation algorithms that can be used for a wide range of applications. The goal is to determine how likely certain outcomes are if some aspects of the application are not known exactly, i.e., in case of limited prior knowledge or uncertainties in the parameter values. We focus in particular on quantifying uncertainties in the output of a mathematical model induced by uncertainties in the
input parameters, often referred to as forward propagation of parametric uncertainties. The first part of this thesis discusses how to signiÞcantly reduce the required number of samples needed in the forward propagation of uncertainties. In particular, if the quantity of interest is highly non-linear or discontinuous depending on the input parameters, Gibbs phenomena may occur, which deteriorate the accuracy globally. As we often do not know beforehand if a quantity of
interest is smooth or discontinuous, we propose a new algorithm that works in both cases. In the second part of this thesis, instead of placing samples optimally in order to reduce the overall computational cost of forward propagation, we opt to increase the accuracy of the quantity of interest that is outputted by the solver. To clarify, when using numerical discretisation to compute outcomes of the mathematical model, the resolution of the computational grid determines the accuracy of the quantity of interest, but it also determines the computational time it takes to compute this quantity of interest. In general, a quantity of interest that is computed using a coarse computational grid is fast to compute but also inaccurate. As a remedy, we opt to use machine learning to increase the accuracy of a quantity of interest that is computed on a coarse grid, as it is able to deal with highly non-linear behaviour and high-dimensional input/ output relations. The main topics of this thesis are uncertainty quantiÞcation and machine learning. We assume that the reader possesses basic prior knowledge on these two subjects. Furthermore, the test cases discussed in both Part I and II of this thesis assume prior knowledge on basic numerical discretisation techniques, in particular Þnite volume methods
Intrusive deconvolutional neural networks for enhancing PIC/FLIP solutions
No full textTraditional fluid flow predictions require large computational resources. Despite
recent progress in parallel and GPU computing, the ability to run fluid flow predictions in real-time is often infeasible. Recently developed machine learning
approaches, which are trained on high-fidelity data, perform unsatisfactorily outside the training set and remove the ability of utilising legacy codes after training.
We propose a novel methodology that uses a deep learning approach that can be
used within a low-fidelity fluid flow solver to significantly increase the accuracy of
the low-fidelity simulations. The resulting solver enables accurate while reducing
computational times up to 100 times. The deep neural network is trained on a
combination of low- and high-fidelity data, and the resulting solver is referred to as
a multi-fidelity solver. The proposed methodology is demonstrated by means of
enhancing a fluid flow simulator, known as PIC/FLIP, which is a popular fluid flow
simulator in the field of computer generated imagery
Multi-level neural networks for PDEs with uncertain parameters
No full textA novel multi-level method for partial differential equations with uncertain parameters is proposed. The principle behind the method is that the error between grid levels in multi-level methods has a spatial structure that is by good approximation independent of the actual grid level. Our method learns this structure by employing a sequence of convolutional neural networks, that are well-suited to automatically detect local error features as latent quantities of the solution. Furthermore, by using the concept of transfer learning, the information of coarse grid levels is reused on fine grid levels in order to minimize the required number of samples on fine levels. The method outperforms state-of-the-art multi-level methods, especially in the case when complex PDEs (such as single-phase and free-surface flow problems) are concerned, or when high accuracy is required
PDE/PDF-informed adaptive sampling for efficient non-intrusive surrogate modelling
No full textA novel refinement measure for non-intrusive surrogate modelling of partial differential equations (PDEs) with uncertain parameters is proposed. Our approach uses an empirical interpolation procedure, where the proposed refinement measure is based on a PDE residual and probability density function of the uncertain parameters, and excludes parts of the PDE solution that are not used to compute the quantity of interest. The PDE residual used in the refinement measure is computed by using all the partial derivatives that enter the PDE separately. The proposed refinement measure is suited for efficient parametric surrogate construction when the underlying PDE is known, even when the parameter space is non-hypercube, and has no restrictions on the type of the discretisation method. Therefore, we are not restricted to conventional discretisation techniques, e.g., finite elements and finite volumes, and the proposed method is shown to be effective when used in combination with recently introduced neural network PDE solvers. We present several numerical examples with increasing complexity that demonstrate accuracy, efficiency and generality of the method
An adaptive minimum spanning tree multi-element method for uncertainty quantification of smooth and discontinuous responses
No full text\u3cp\u3eA novel approach for nonintrusive uncertainty propagation is proposed. Our approach overcomes the limitation of many traditional methods, such as generalized polynomial chaos methods, which may lack sufficient accuracy when the quantity of interest depends discontinuously on the input parameters. As a remedy we propose an adaptive sampling algorithm based on minimum spanning trees combined with a domain decomposition method based on support vector machines. The minimum spanning tree determines new sample locations based on both the probability density of the input parameters and the gradient in the quantity of interest. The support vector machine efficiently decomposes the random space in multiple elements, avoiding the appearance of Gibbs phenomena near discontinuities. On each element, local approximations are constructed by means of least orthogonal interpolation, in order to produce stable interpolation on the unstructured sample set. The resulting minimum spanning tree multielement method does not require initial knowledge of the behavior of the quantity of interest and automatically detects whether discontinuities are present. We present several numerical examples that demonstrate accuracy, efficiency, and generality of the method.\u3c/p\u3
PDE/PDF-informed adaptive sampling for efficient nonintrusive surrogate modeling
No full textA novel refinement measure for nonintrusive surrogate modeling of partial differential equations (PDEs) with uncertain parameters is proposed. Our approach uses an empirical interpolation procedure, where the proposed refinement measure is based on both the PDE residual and the probability density function. An important strength is that it ex-cludes parts of the PDE solution that are not required to compute the quantity of interest. The PDE residual used in the refinement measure is computed by using all the partial derivatives that enter the PDE separately. The proposed refinement measure is suited for efficient parametric surrogate construction when the underlying PDE is known, even when the parameter space is nonhypercube, and has no restrictions on the type of the discretization method. Therefore, we are not restricted to conventional discretization techniques, e.g., finite elements or finite volumes, and the proposed method is shown to be effective when used in combination with recently introduced neural network PDE solvers. We present several numerical examples with increasing complexity that demonstrate accuracy, efficiency, and generality of the method
Machine learning for closure models in multiphase flow applications
No full textMultiphase flows are described by the multiphase Navier-Stokes equations. Numerically solving these equations is computationally expensive, and performing many simulations for the purpose of design, optimization and uncertainty quantification is often prohibitively expensive. A simplified model, the so-called two-fluid model, can be derived from a spatial averaging process. The averaging process introduces a closure problem, which is represented by unknown friction terms in the two-fluid model. Correctly modeling these friction terms is a long-standing problem in two-fluid model development. In this work we take a new approach, and learn the closure terms in the two-fluid model from a set of unsteady high-fidelity simulations conducted with the open source code Gerris. These form the training data for a neural network. The neural network provides a functional relation between the two-fluid model's resolved quantities and the closure terms, which are added as source terms to the two-fluid model. With the addition of the locally defined interfacial slope as an input to the closure terms, the trained two-fluid model reproduces the dynamic behavior of high fidelity simulations better than the two-fluid model using a conventional set of closure terms
