Discovering variable fractional orders of advection–dispersion equations from field data using multi-fidelity Bayesian optimization

Pang, Guofei; Perdikaris, Paris; Cai, Wei; Karniadakis, George Em

doi:10.1016/j.jcp.2017.07.052

Cited by 58 publications

(37 citation statements)

References 23 publications

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“…Multifidelity methods have much broader applications, not only Monte Carlo‐based methods, but also more general UQ aspects, for example, optimization with uncertainty (Bonfiglio, Perdikaris, Brizzolara, & Karniadakis, 2018; Heinkenschloss, Kramer, Takhtaganov, & Willcox, 2018; Pang, Perdikaris, Cai, & Karniadakis, 2017), multifidelity surrogate modeling (Chaudhuri, Lam, & Willcox, 2018; Giselle Ferńandez‐Godino, Park, Kim, & Haftka, 2019; Guo, Song, Park, Li, & Haftka, 2018; Parussini, Venturi, Perdikaris, & Karniadakis, 2017; Perdikaris, Venturi, Royset, & Karniadakis, 2015; Tian et al, 2020) and multifidelity information reuse, and fusion (Cook, Jarrett, & Willcox, 2018; Perdikaris, Venturi, & Karniadakis, 2016). We refer to (Park et al, 2017; Peherstorfer, Willcox, & Gunzburger, 2018) for a comprehensive introduction and in‐depth discussion of multifidelity methods for uncertainty propagation.…”

Section: Modern MC Methods For Uqmentioning

confidence: 99%

Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey

Zhang

2020

WIREs Computational Stats

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Uncertainty quantification (UQ) includes the characterization, integration, and propagation of uncertainties that result from stochastic variations and a lack of knowledge or data in the natural world. Monte Carlo (MC) method is a sampling‐based approach that has widely used for quantification and propagation of uncertainties. However, the standard MC method is often time‐consuming if the simulation‐based model is computationally intensive. This article gives an overview of modern MC methods to address the existing challenges of the standard MC in the context of UQ. Specifically, multilevel Monte Carlo (MLMC) extending the concept of control variates achieves a significant reduction of the computational cost by performing most evaluations with low accuracy and corresponding low cost, and relatively few evaluations at high accuracy and corresponding high cost. Multifidelity Monte Carlo (MFMC) accelerates the convergence of standard Monte Carlo by generalizing the control variates with different models having varying fidelities and varying computational costs. Multimodel Monte Carlo method (MMMC), having a different setting of MLMC and MFMC, aims to address the issue of UQ and propagation when data for characterizing probability distributions are limited. Multimodel inference combined with importance sampling is proposed for quantifying and efficiently propagating the uncertainties resulting from small data sets. All of these three modern MC methods achieve a significant improvement of computational efficiency for probabilistic UQ, particularly uncertainty propagation. An algorithm summary and the corresponding code implementation are provided for each of the modern MC methods. The extension and application of these methods are discussed in detail. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Monte Carlo Methods Statistical and Graphical Methods of Data Analysis > Sampling

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Section: Modern MC Methods For Uqmentioning

confidence: 99%

Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey

Zhang

2020

WIREs Computational Stats

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“…Poisson-Schrödinger simulations require tuning of a number of parameters, such as the damping step of the Newton solver that reaches equilibrium, and convergence is often an issue. Optimizing parameters is a general problem in computational sciences and physics 55,[67][68][69][70] and in machine learning. 71,72 Leduc et al 32 recently demonstrated how a surrogate model, using random forest (RF) to predict the convergence rates and E[I] to do the sampling, can be applied to speed up convergence of simulations, such as the P-S equations encoded in APSYS.…”

Section: Simulations For Targeted Designmentioning

confidence: 99%

Active learning in materials science with emphasis on adaptive sampling using uncertainties for targeted design

et al. 2019

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One of the main challenges in materials discovery is efficiently exploring the vast search space for targeted properties as approaches that rely on trial-and-error are impractical. We review how methods from the information sciences enable us to accelerate the search and discovery of new materials. In particular, active learning allows us to effectively navigate the search space iteratively to identify promising candidates for guiding experiments and computations. The approach relies on the use of uncertainties and making predictions from a surrogate model together with a utility function that prioritizes the decision making process on unexplored data. We discuss several utility functions and demonstrate their use in materials science applications, impacting both experimental and computational research. We summarize by indicating generalizations to multiple properties and multifidelity data, and identify challenges, future directions and opportunities in the emerging field of materials informatics.

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“…The works on applying machine learning to parameter identification problems can be mainly divided into two categories. The first category exploits only the information of observed data in the spatio-temporal domain, and employs surrogate models such as Gaussian process regression (GP regression) [16], stochastic collocation methods [17], and feedforward neural networks (NNs) [18,19] to approximate the mapping from the parameters to be identified to the numerical solutions of PDEs or their mismatch with observed data. The PDEs are numerically solved to obtain the training points, and hence the information from PDEs is used implicitly.…”

Section: Introductionmentioning

confidence: 99%

fPINNs: Fractional Physics-Informed Neural Networks

Pang¹,

Lu²,

Karniadakis³

2019

SIAM J. Sci. Comput.

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543

286

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Physics-informed neural networks (PINNs), introduced in [1], are effective in solving integerorder partial differential equations (PDEs) based on scattered and noisy data. PINNs employ standard feedforward neural networks (NNs) with the PDEs explicitly encoded into the NN using automatic differentiation, while the sum of the mean-squared PDE-residuals and the mean-squared error in initial/boundary conditions is minimized with respect to the NN parameters. Here we extend PINNs to fractional PINNs (fPINNs) to solve space-time fractional advection-diffusion equations (fractional ADEs), and we study systematically their convergence, hence explaining both of fPINNs and PINNs for first time. Specifically, we demonstrate their accuracy and effectiveness in solving multi-dimensional forward and inverse problems with forcing terms whose values are only known at randomly scattered spatio-temporal coordinates (black-box forcing terms). A novel element of the fPINNs is the hybrid approach that we introduce for constructing the residual in the loss function using both automatic differentiation for the integer-order operators and numerical discretization for the fractional operators. This approach bypasses the difficulties stemming from the fact that automatic differentiation is not applicable to fractional operators because the standard chain rule in integer calculus is not valid in fractional calculus. To discretize the fractional operators, we employ the Grünwald-Letnikov (GL) formula in one-dimensional fractional ADEs and the vector GL formula in conjunction with the directional fractional Laplacian in two-and three-dimensional fractional ADEs. We first consider the one-dimensional fractional Poisson equation and compare the convergence of the fPINNs against the finite difference method (FDM). We present the solution convergence using both the mean L 2 error as well as the standard deviation due to sensitivity to NN parameter initializations. Using different GL formulas we observe first-, second-, and third-order convergence rates for small size of training sets but the error saturates for larger training sets. We explain these results by analyzing the four sources of numerical errors due to discretization, sampling, NN approximation, and optimization. The total error decays monotonically (below 10 −5 for third order GL formula) but it saturates beyond that point due to the optimization error. We also analyze the relative balance between discretization and sampling errors and observe that the sampling size and the number of discretization points (auxiliary points) should be comparable to achieve the highest accuracy. As we increase the depth of the NN up to certain value, the mean error decreases and the standard deviation increases whereas the width has essentially no effect unless its value is either too small or too large. We next consider time-dependent fractional ADEs and compare white-box (WB) and black-box (BB) forcing. We observe that for the WB forcing, our results are similar to the aforementioned cases, however,...

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Discovering variable fractional orders of advection–dispersion equations from field data using multi-fidelity Bayesian optimization

Cited by 58 publications

References 23 publications

Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey

Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey

Active learning in materials science with emphasis on adaptive sampling using uncertainties for targeted design

fPINNs: Fractional Physics-Informed Neural Networks

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