We establish sparsity and summability results for coefficient sequences of Wiener-Hermite polynomial chaos expansions of countably-parametric solutions of linear elliptic and parabolic divergence-form partial differential equations with Gaussian random field inputs.
The novel proof technique developed here is based on analytic continuation of parametric solutions into the complex domain. It differs from previous works that used bootstrap arguments and induction on the differentiation order of solution derivatives with respect to the parameters. The present holomorphy-based argument allows a unified, ``differentiation-free'' proof of sparsity (expressed in terms of $\ell^p$-summability or weighted $\ell^2$-summability) of sequences of Wiener-Hermite coefficients in polynomial chaos expansions in various scales of function spaces. The analysis also implies corresponding analyticity and sparsity results for posterior densities in Bayesian inverse problems subject to Gaussian priors on uncertain inputs from function spaces.
Our results furthermore yield dimension-independent convergence rates of various \emph{constructive} high-dimensional deterministic numerical approximation schemes such as single-level and multi-level versions of Hermite-Smolyak anisotropic sparse-grid interpolation and quadrature in both forward and inverse computational uncertainty quantification.
The quantification of multivariate uncertainties in partial differential equations can easily exceed any computing capacity unless proper measures are taken to reduce the complexity of the model. In this work, we propose a multidimensional Galerkin Proper Orthogonal Decomposition that optimally reduces each dimension of a tensorized product space. We provide the analytical framework and results that define and quantify the low-dimensional approximation. We illustrate its application for uncertainty modeling with Polynomial Chaos Expansions and show its efficiency in a numerical example.
University of Tübingen（图宾根大学）；University of Genova（热那亚大学）；Istituto Italiano di Tecnologia（意大利技术研究院）；Massachusetts Institute of Technology（麻省理工学院）；Max-Planck-Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
Neural operators are a type of deep architecture that learns to solve (i.e. learns the nonlinear solution operator of) partial differential equations (PDEs). The current state of the art for these models does not provide explicit uncertainty quantification. This is arguably even more of a problem for this kind of tasks than elsewhere in machine learning, because the dynamical systems typically described by PDEs often exhibit subtle, multiscale structure that makes errors hard to spot by humans. In this work, we first provide a mathematically detailed Bayesian formulation of the ''shallow'' (li…
The probability density function (PDF) associated with a given set of samples is approximated by a piecewise-linear polynomial constructed with respect to a binning of the sample space. The kernel functions are a compactly supported basis for the space of such polynomials, i.e. finite element hat functions, that are centered at the bin nodes rather than at the samples, as is the case for the standard kernel density estimation approach. This feature naturally provides an approximation that is scalable with respect to the sample size. On the other hand, unlike other strategies that use a finite …
Sampling recovery in Bochner spaces and applications to parametric PDEs
Mathematisch-Geographische Fakultät, KU Eichstätt-Ingolstadt（凯泽大学艾希施泰特-因戈尔施塔特数学地理学院）；Information Technology Institute, Vietnam National University, Hanoi（越南国家大学河内信息技术研究所）
IGA-based Multi-Index Stochastic Collocation for random PDEs on arbitrary domains
King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）；Consiglio Nazionale delle Ricerche - Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes” (CNR-IMATI)（意大利国家研究委员会应用数学与信息技术研究所）；RWTH Aachen University（亚琛工业大学）
Multi-Fidelity Physics-Informed Neural Networks with Bayesian Uncertainty Quantification and Adaptive Residual Learning for Efficient Solution of Parametric Partial Differential Equations
Department of Applied Mathematics and Computer Science (DTU Compute)（应用数学与计算机科学系（DTU Compute））；Technical University of Denmark（丹麦技术大学）
Neural Measures for learning distributions of Random PDEs
DMAM, University of Crete, Greece（DMAM，希腊克里特大学）；IACM, Foundation for Research and Technology - Hellas, Greece（IACM，希腊基础研究与技术研究所）；MPS, University of Sussex, United Kingdom（MPS，英国 Sussex 大学）
How to solve inverse problems is the challenge of many engineering and industrial applications. Recently, physics-informed neural networks (PINNs) have emerged as a powerful approach to solve inverse problems efficiently. However, it is difficult for PINNs to quantify the uncertainty of results. Therefore, this study proposed ensemble PINNs (E-PINNs) to handle this issue. The E-PINN uses ensemble statistics of several basic models to provide uncertainty quantifications for the inverse solution based on the PINN framework, and it is employed to solve the inverse problems in which the unknown qu…
Deep set based operator learning with uncertainty quantification
Department of Mathematics, Shanghai Normal University（上海师范大学数学系）；School of Mathematical Sciences, Institute of Natural Sciences（自然科学院数学系）；MOE-LSC, Shanghai Jiao Tong University（教育部重点实验室，上海交通大学）；Institute of Computational Mathematics（计算数学研究所）；Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences（科学与工程计算，中国科学院数学研究所）
Partial differential equations (PDEs) are fundamental for theoretically describing numerous physical processes that are based on some input fields in spatial configurations. Understanding the physical process, in general, requires computational modeling of the PDE. Uncertainty in the computational model manifests through lack of precise knowledge of the input field or configuration. Uncertainty quantification (UQ) in the output physical process is typically carried out by modeling the uncertainty using a random field, governed by an appropriate covariance function. This leads to solving high-d…
IV-Net: A neural network for elliptic PDEs with random and highly varying coefficients
Oden Institute for Computational Science and Engineering, The University of Texas at Austin（计算科学与工程院，德克萨斯大学奥斯汀分校）；Walker Department of Mechanical Engineering, The University of Texas at Austin（机械工程系，德克萨斯大学奥斯汀分校）
Uncertainty Quantification of geochemical and mechanical compaction in layered sedimentary basins
Politecnico di Milano（米兰理工大学）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes” - CNR（应用数学与信息技术研究所（CNR））
A crucial role in the theory of uncertainty quantification (UQ) of PDEs is played by the regularity of the solution with respect to the stochastic parameters; indeed, a key property one seeks to establish is that the solution is holomorphic with respect to (the complex extensions of) the parameters. In the context of UQ for the high-frequency Helmholtz equation, a natural question is therefore: how does this parametric holomorphy depend on the wavenumber $k$? The recent paper [Ganesh, Kuo, Sloan 2021] showed for a particular nontrapping variable-coefficient Helmholtz problem with affine depend…
Probabilistic operator learning: generative modeling and uncertainty quantification for foundation models of differential equations
Division of Applied Mathematics, Brown University（布朗大学应用数学系）；Department of Mathematics, University of California, Riverside（加州大学河滨分校数学系）；Department of Mathematics, University of California, Los Angeles（加州大学洛杉矶分校数学系）；Department of Mathematics and Statistics, University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校数学与统计学系）
On the optimality of dimension truncation error rates for a class of parametric partial differential equations
Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（奥地利科学院约翰·拉东计算与应用数学研究所）；School of Engineering Sciences, LUT University（拉普兰理工大学工程学院）
Scalable hierarchical PDE sampler for generating spatially correlated random fields using non-matching meshes
Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；Università della Svizzera italiana（瑞士意大利语大学）；University of Texas, Austin（德克萨斯大学奥斯汀分校）；Portland State University（波特兰州立大学）
