B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data

Yang, Liu; Meng, Xuhui; Karniadakis, George Em

doi:10.48550/arxiv.2003.06097

Cited by 12 publications

(21 citation statements)

References 23 publications

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“…The incorporation of data from multiple sensor types, such as active piezo-electric sensors can also be naturally implemented in the GNN framework. In addition, GNN-based computation may be a viable alternative to Bayesian PDE inversion problems [13], where observations are sparse, in place of the grid-based CNN computational substrate. Finally, GNNs offer great potential for inference of data-driven Reduced Order Models (ROMs), directly from physical observations.…”

Section: Discussionmentioning

confidence: 99%

Bayesian graph neural networks for strain-based crack localization

Mylonas¹,

Tsialiamanis²,

Worden³

et al. 2020

Preprint

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A common shortcoming of vibration-based damage localization techniques is that localized damages, i.e. small cracks, have a limited influence on the spectral characteristics of a structure. In contrast, even the smallest of defects, under particular loading conditions, cause localized strain concentrations with predictable spatial configuration. However, the effect of a small defect on strain decays quickly with distance from the defect, making strain-based localization rather challenging. In this work, an attempt is made to approximate, in a fully data-driven manner, the posterior distribution of a crack location, given arbitrary dynamic strain measurements at arbitrary discrete locations on a structure. The proposed technique leverages Graph Neural Networks (GNNs) and recent developments in scalable learning for Bayesian neural networks. The technique is demonstrated on the problem of inferring the position of an unknown crack via patterns of dynamic strain field measurements at discrete locations. The dataset consists of simulations of a hollow tube under random time-dependent excitations with randomly sampled crack geometry and orientation.

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Section: Discussionmentioning

confidence: 99%

Bayesian graph neural networks for strain-based crack localization

Mylonas¹,

Tsialiamanis²,

Worden³

et al. 2020

Preprint

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“…In particular, the Hamiltonian Monte Carlo (HMC) method [16], which is frequently used as ground truth for posterior estimation, is adopted here to estimate the posterior distributions in BNN [17]. In general, the estimation of posterior distributions for hyperparameters in the BNN is computationally prohibitive for problems with big data using HMC [14]. However, the high-fidelity data are generally scarce due to high acquisition cost associated with high-fidelity data.…”

Section: Multi-fidelity Bayesian Neural Networkmentioning

confidence: 99%

“…Generally, P (D) is analytically intractable. Here, we employ HMC to sample from the unnormalized P (θ|D) [14,16]. Then, we can obtain predictions on u at any x (i.e., {ũ (i) (x)} M i=1 ) based on the posterior samples (i.e., {θ (i) } M i=1 ) from the HMC.…”

Section: Multi-fidelity Bayesian Neural Networkmentioning

confidence: 99%

“…1, the multi-fidelity Bayesian neural network is composed of three different neural networks: the first is a deep neural network (DNN) to approximate the low-fidelity data, while the second one is a Bayesian neural network (BNN) for learning the correlation (with uncertainty quantification) between the lowand high-fidelity data. The last one is the physics-informed neural network (PINN), which is employed to encode the physical laws represented by partial differential equations (PDEs) [13,14].…”

Section: Introductionmentioning

confidence: 99%

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Multi-fidelity Bayesian Neural Networks: Algorithms and Applications

Meng,

Babaee,

Karniadakis

2020

Preprint

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We propose a new class of Bayesian neural networks (BNNs) that can be trained using noisy data of variable fidelity, and we apply them to learn function approximations as well as to solve inverse problems based on partial differential equations (PDEs). These multi-fidelity BNNs consist of three neural networks: The first is a fully connected neural network, which is trained following the maximum a posteriori probability (MAP) method to fit the low-fidelity data; the second is a Bayesian neural network employed to capture the cross-correlation with uncertainty quantification between the low-and high-fidelity data; and the last one is the physics-informed neural network, which encodes the physical laws described by PDEs. For the training of the last two neural networks, we first employ the mean-field variational inference (VI) to maximize the evidence lower bound (ELBO) to obtain informative prior distributions for the hyperparameters in the BNNs, and subsequently we use the Hamiltonian Monte Carlo method to estimate accurately the posterior distributions for the corresponding hyperparameters. We demonstrate the accuracy of the present method using synthetic data as well as real measurements. Specifically, we first approximate a one-and four-dimensional function, and then infer the reaction rates in one-and two-dimensional diffusion-reaction systems. Moreover, we infer the sea surface temperature (SST) in the Massachusetts and Cape Cod Bays using satellite images and in-situ measurements. Taken together, our results demonstrate that the present method can capture both linear and nonlinear correlation between the low-and high-fideilty data adaptively, identify unknown parameters in PDEs, and quantify uncertainties in predictions, given a few scattered noisy high-fidelity data.

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“…It is particularly useful for systems where there is a high cost of data acquisition or lack of high resolution data [16]. The authors in [17] proposed a Bayesian approach for physics informed neural network to solve forward and inverse problems.…”

Section: Introductionmentioning

confidence: 99%

A Physics Informed Neural Network for Time-Dependent Nonlinear and Higher Order Partial Differential Equations

Mattey,

Ghosh

2021

Preprint

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A physics informed neural network (PINN) incorporates the physics of a system by satisfying its boundary value problem through a neural network's loss function. The PINN approach has shown great success in approximating the map between the solution of a partial differential equation (PDE) and its spatio-temporal input. However, for strongly non-linear and higher order partial differential equations PINN's accuracy reduces significantly. To resolve this problem, we propose a novel PINN scheme that solves the PDE sequentially over successive time segments using a single neural network. The key idea is to re-train the same neural network for solving the PDE over successive time segments while satisfying the already obtained solution for all previous time segments. Thus it is named as backward compatible PINN (bc-PINN). To illustrate the advantages of bc-PINN, we have used the Cahn Hilliard and Allen Cahn equations, which are widely used to describe phase separation and reaction diffusion systems. Our results show significant improvement in accuracy over the PINN method while using a smaller number of collocation points. Additionally, we have shown that using the phase space technique for a higher order PDE could further improve the accuracy and efficiency of the bc-PINN scheme.

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B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data

Cited by 12 publications

References 23 publications

Bayesian graph neural networks for strain-based crack localization

Bayesian graph neural networks for strain-based crack localization

Multi-fidelity Bayesian Neural Networks: Algorithms and Applications

A Physics Informed Neural Network for Time-Dependent Nonlinear and Higher Order Partial Differential Equations

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