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IB-UQ: Information bottleneck based uncertainty quantification for neural function regression and neural operator learning

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arXiv 2023年

作者： Guo, Ling Wu, Hao Zhou, Wenwen Wang, Yan Zhou, Tao Department of Mathematics Shanghai Normal University Shanghai China School of Mathematical Sciences Institute of Natural Sciences MOE-LSC Shanghai Jiaotong University Shanghai China School of Mathematical Sciences Tongji University Shanghai China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

We propose a novel framework for uncertainty quantification via information bottleneck (IB-UQ) for scientific machine learning tasks, including deep neural network (DNN) regression and neural operator learning (DeepONet). Specifically, we incorporate the bottleneck by a confidence-aware encoder, which encodes inputs into latent representations according to the confidence of the input data belonging to the region where training data is located, and utilize a Gaussian decoder to predict means and variances of outputs conditional on representation variables. Furthermore, we propose a data augmentation based information bottleneck objective which can enhance the quantification quality of the extrapolation uncertainty, and the encoder and decoder can be both trained by minimizing a tractable variational bound of the objective. In comparison to uncertainty quantification (UQ) methods for scientific learning tasks that rely on Bayesian neural networks with Hamiltonian Monte Carlo posterior estimators, the model we propose is computationally efficient, particularly when dealing with large-scale data sets. The effectiveness of the IB-UQ model has been demonstrated through several representative examples, such as regression for discontinuous functions, real-world data set regression, learning nonlinear operators for partial differential equations, and a large-scale climate model. The experimental results indicate that the IB-UQ model can handle noisy data, generate robust predictions, and provide confident uncertainty evaluation for out-of-distribution data. © 2023, CC BY-NC-ND.

关键词： Monte Carlo methods

A PRIMAL FINITE ELEMENT SCHEME OF THE HODGE LAPLACE PROBLEM

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arXiv 2022年

作者： Zhang, Shuo LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing100190 China University of Chinese Academy of Sciences Beijing100049 China

In this paper, a unified family, for any n ≽ 2 and 1 ≼ k ≼ n − 1, of nonconforming finite element schemes are presented for the primal weak formulation of the n-dimensional Hodge-Laplace equation on (Equation presented) and on the simplicial subdivisions of the domain. The finite element scheme possesses an O(h)-order convergence rate for sufficiently regular data, and an O(hs)order rate on any s-regular domain, 0 Copyright © 2022, The Authors. All rights reserved.

关键词： Finite element method

PARTIALLY ADJOINT DISCRETIZATIONS OF ADJOINT OPERATORS AND NONCONFORMING FINITE ELEMENT EXTERIOR CALCULUS

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arXiv 2022年

This paper concerns the discretizations in pair of adjoint operators between Hilbert spaces so that the adjoint properties can be preserved. Due to the finite-dimensional essence of discretized operators, a new framework, theory of partially adjoint operators, is motivated and presented in this paper, so that adjoint properties can be figured out for finite-dimensional operators which can not be non-trivially densely defined in other background spaces. A formal methodology is presented under the framework to construct partially adjoint discretizations by a conforming discretization (CD) and an accompanied-by-conforming discretization (ABCD) for each of the operators. Moreover, the methodology leads to an asymptotic uniformity of an infinite family of finite-dimensional operators. A theory of nonconforming finite element exterior calculus (NCFEEC) is developed by constructing a systematic family of nonconforming finite element differential forms with certain structures by the theoretical framework and the formal construction of discretizations;by the NCFEEC, together with the classical FEEC, adjoint properties are reconstructed between discretizations of adjoint operators and between discretizations of variational *** Codes 47A05, 47A65, 47N40, 65J10, 65N30 Copyright © 2022, The Authors. All rights reserved.

关键词： Calculations

Multiple-scattering frequency-time hybrid solver for the wave equation in interior domains

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arXiv 2022年

作者： Bruno, Oscar P. Yin, Tao Department of Computing & Mathematical Sciences California Institute of Technology 1200 East California Blvd. CA91125 United States LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

This paper proposes a frequency-time hybrid solver for the time-dependent wave equation in two-dimensional interior spatial domains. The approach relies on four main elements, namely, 1) A multiple scattering strategy that decomposes a given interior time-domain problem into a sequence of limited-duration time-domain problems of scattering by overlapping open arcs, each one of which is reduced (by means of the Fourier transform) to a sequence of Helmholtz frequency-domain problems;2) Boundary integral equations on overlapping boundary patches for the solution of the frequency-domain problems in point 1);3) A smooth "Time-windowing and recentering" methodology that enables both treatment of incident signals of long duration and long time simulation;and, 4) A Fourier transform algorithm that delivers numerically dispersionless, spectrally-accurate time evolution for given incident fields. By recasting the interior time-domain problem in terms of a sequence of open-arc multiple scattering events, the proposed approach regularizes the full interior frequency domain problem-which, if obtained by either Fourier or Laplace transformation of the corresponding interior time-domain problem, must encapsulate infinitely many scattering events, giving rise to non-uniqueness and eigenfunctions in the Fourier case, and ill conditioning in the Laplace case. Numerical examples are included which demonstrate the accuracy and efficiency of the proposed *** Codes 35L05, 65M80, 65T99, 65R20 Copyright © 2022, The Authors. All rights reserved.

关键词： Wave equations

LOW-DEGREE ROBUST HELLINGER-REISSNER FINITE ELEMENT SCHEMES FOR PLANAR LINEAR ELASTICITY WITH SYMMETRIC STRESS TENSORS

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arXiv 2022年

作者： Zhang, Shuo Lsec Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing100190 China University of Chinese Academy of Sciences Beijing100049 China

In this paper, we study the construction of low-degree robust finite element schemes for planar linear elasticity on general triangulations. Firstly, we present a low-degree nonconformingHelling-Reissner finite element scheme. For the stress tensor space, the piecewise polynomial shape function space is (Formula presented) the dimension of the total space is asymptotically 8 times of the number of vertices, and the supports of the basis functions are each a patch of an edge. The piecewise rigid body space is used for the displacement. Robust error estimations in L2and broken H(div) norms are presented. Secondly, we investigate the theoretical construction of schemes with lowest-degree polynomial shape function spaces. Specifically, a Hellinger-Reissner finite element scheme is constructed, with the local shape function space for the stress tensor being (Formula presented) which is of the lowest degree for the local approximation of H(div;S), and the space for the displacement is piecewise constants. Robust error estimations in L2and broken H(div) norms are presented for regular solutions and data. Meanwhile, accompanied with this Hellinger-Reissner finite element scheme, a minimal Navier-Lamé finite element scheme is constructed, with the local shape function space for the displacement being (Formula presented) which is of lowest degree for the local approximation in the norm ||·||0+ ||ϵ(·)||0. Robust error estimations in broken energy norms are presented for regular solutions and data. Copyright © 2022, The Authors. All rights reserved.

关键词： Stress tensor

A PML method for signal-propagation problems in axon

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arXiv 2022年

作者： Jiang, Xue Lyu, Maohui Yin, Tao Zheng, Weiying Department of Mathematics Faculty of Science Beijing University of Technology Beijing100124 China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

This work is focused on the modelling of signal propagations in myelinated axons to characterize the functions of the myelin sheath in the neural structure. Based on reasonable assumptions on the medium properties, we derive a two-dimensional neural-signaling model in cylindrical coordinates from the time-harmonic Maxwell’s equations. The well-posedness of model is established upon Dirichlet boundary conditions at the two ends of the neural structure and the radiative condition in the radial direction of the structure. Using the perfectly matched layer (PML) method, we truncate the unbounded background medium and propose an approximate problem on the truncated domain. The well-posedness of the PML problem and the exponential convergence of the approximate solution to the exact solution are established. Numerical experiments based on finite element discretization are presented to demonstrate the theoretical results and the efficiency of our methods to simulate the signal propagation in axons. Copyright © 2022, The Authors. All rights reserved.

关键词： Maxwell equations

A PRIMAL FINITE ELEMENT SCHEME OF THE H(d) ∩ H(δ) ELLIPTIC PROBLEM

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arXiv 2022年

In this paper, a unified family, for n > 2 and 1 6 k 6 n − 1, of finite element schemes are presented for the primal weak formulation of the n-dimensional (Equation presented) elliptic problem. Copyright © 202... 详细信息

关键词： Finite element method

A RIEMANNIAN EXPONENTIAL AUGMENTED LAGRANGIAN METHOD FOR computing THE PROJECTION ROBUST WASSERSTEIN DISTANCE

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arXiv 2022年

作者： Jiang, Bo Liu, Ya-Feng Key Laboratory for NSLSCS of Jiangsu Province School of Mathematical Sciences Nanjing Normal University Nanjing China State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

Projecting the distance measures onto a low-dimensional space is an efficient way of mitigating the curse of dimensionality in the classical Wasserstein distance using optimal transport. The obtained maximized distance is referred to as projection robust Wasserstein (PRW) distance. In this paper, we equivalently reformulate the computation of the PRW distance as an optimization problem over the Cartesian product of the Stiefel manifold and the Euclidean space with additional nonlinear inequality constraints. We propose a Riemannian exponential augmented Lagrangian method (ReALM) with a global convergence guarantee to solve this problem. Compared with the existing approaches, ReALM can potentially avoid too small penalty parameters. Moreover, we propose a framework of inexact Riemannian gradient descent methods to solve the subproblems in ReALM efficiently. In particular, by using the special structure of the subproblem, we give a practical algorithm named as the inexact Riemannian Barzilai-Borwein method with Sinkhorn iteration (iRBBS). The remarkable features of iRBBS lie in that it performs a flexible number of Sinkhorn iterations to compute an inexact gradient with respect to the projection matrix of the problem and adopts the Barzilai-Borwein stepsize based on the inexact gradient information to improve the performance. We show that iRBBS can return an ǫ-stationary point of the original PRW distance problem within O(ǫ−3) iterations. Extensive numerical results on synthetic and real datasets demonstrate that our proposed ReALM as well as iRBBS outperform the state-of-the-art solvers for computing the PRW *** Codes 65K10, 90C26, 90C47 Copyright © 2022, The Authors. All rights reserved.

关键词： Constrained optimization

IMPROVED LAGUERRE SPECTRAL METHODS WITH LESS ROUND-OFF ERRORS AND BETTER STABILITY

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arXiv 2022年

作者： Huang, Shenghe Yu, Haijun School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China NCMIS & LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Beijing100190 China

Laguerre polynomials are orthogonal polynomials defined on positive half line with respect to weight e−x. They have wide applications in scientific and engineering computations. However, the exponential growth of Laguerre polynomials of high degree makes it hard to apply them to complicated systems that need to use large numbers of Laguerre bases. In this paper, we introduce modified three-term recurrence formula to reduce the round-off error and to avoid overflow and underflow issues in generating generalized Laguerre polynomials and Laguerre functions. We apply the improved Laguerre methods to solve an elliptic equation defined on the half line. More than one thousand Laguerre bases are used in this application and meanwhile accuracy close to machine precision is achieved. The optimal scaling factor of Laguerre methods are studied and found to be independent of number of quadrature points in two cases that Laguerre methods have better convergence speeds than mapped Jacobi *** Codes 65N35, 65D32, 65G50, 33F05 © 2022, CC BY-NC-ND.

关键词： Polynomials