B-DeepONet: An enhanced Bayesian DeepONet for solving noisy parametric PDEs using accelerated replica exchange SGLD

Lin, Guang; Moya, Christian; Zhang, Zecheng

doi:10.1016/j.jcp.2022.111713

Cited by 17 publications

(6 citation statements)

References 17 publications

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“…A highly effective approach for tackling the inverse problem involves adopting the Bayesian inference framework, as referenced in prior works [6,10,23,33,35]. In this context, we define the measured flux data as d ∈ R n d and represent the prior distribution of ξ as p 0 (ξ), where ξ is the unknowns used to parameterize the shape of the source and the inverse quantities of interests (QoIs).…”

Section: Bayesian Frameworkmentioning

confidence: 99%

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Restoring the discontinuous heat equation source using sparse boundary data and dynamic sensors

Lin,

Ou,

Zhang

et al. 2024

Inverse Problems

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This study focuses on addressing the inverse source problem associated with the parabolic equation. We rely on sparse boundary flux data as our measurements, which are acquired from a restricted section of the boundary. While it has been established that utilizing sparse boundary flux data can enable source recovery, the presence of a limited number of observation sensors poses a challenge for accurately tracing the inverse quantity of interest. To overcome this limitation, we introduce a sampling algorithm grounded in Langevin dynamics that incorporates dynamic sensors to capture the flux information. Furthermore, we propose and discuss two distinct sensor migration strategies. Remarkably, our findings demonstrate that even with only two observation sensors at our disposal, it remains feasible to successfully reconstruct the high-dimensional unknown parameters.

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Section: Bayesian Frameworkmentioning

confidence: 99%

“…The choice of the temperature parameter τ has crucial influence on the sampling. Motivated by the replicaexchange methods [23,24,27] extends the preconditioned LD-based methods to multiple chains with different temperature parameters to accelerate distribution simulation.…”

Section: Langevin Diffusion and Bayesian Samplingmentioning

confidence: 99%

Restoring the discontinuous heat equation source using sparse boundary data and dynamic sensors

Lin,

Ou,

Zhang

et al. 2024

Inverse Problems

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“…Furthermore, we aim to expand our research from bi-fidelity to multi-fidelity and Bayesian operator learning [28,[42][43][44]. This expansion aims to optimize the use of data while mitigating the computational costs associated with predicting computationally expensive data.…”

Section: Future Workmentioning

confidence: 99%

A Physics-Guided Bi-Fidelity Fourier-Featured Operator Learning Framework for Predicting Time Evolution of Drag and Lift Coefficients

Mollaali,

Sahin,

Raza

et al. 2023

Fluids

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In the pursuit of accurate experimental and computational data while minimizing effort, there is a constant need for high-fidelity results. However, achieving such results often requires significant computational resources. To address this challenge, this paper proposes a deep operator learning-based framework that requires a limited high-fidelity dataset for training. We introduce a novel physics-guided, bi-fidelity, Fourier-featured deep operator network (DeepONet) framework that effectively combines low- and high-fidelity datasets, leveraging the strengths of each. In our methodology, we begin by designing a physics-guided Fourier-featured DeepONet, drawing inspiration from the intrinsic physical behavior of the target solution. Subsequently, we train this network to primarily learn the low-fidelity solution, utilizing an extensive dataset. This process ensures a comprehensive grasp of the foundational solution patterns. Following this foundational learning, the low-fidelity deep operator network’s output is enhanced using a physics-guided Fourier-featured residual deep operator network. This network refines the initial low-fidelity output, achieving the high-fidelity solution by employing a small high-fidelity dataset for training. Notably, in our framework, we employ the Fourier feature network as the trunk network for the DeepONets, given its proficiency in capturing and learning the oscillatory nature of the target solution with high precision. We validate our approach using a well-known 2D benchmark cylinder problem, which aims to predict the time trajectories of lift and drag coefficients. The results highlight that the physics-guided Fourier-featured deep operator network, serving as a foundational building block of our framework, possesses superior predictive capability for the lift and drag coefficients compared to its data-driven counterparts. The bi-fidelity learning framework, built upon the physics-guided Fourier-featured deep operator, accurately forecasts the time trajectories of lift and drag coefficients. A thorough evaluation of the proposed bi-fidelity framework confirms that our approach closely matches the high-fidelity solution, with an error rate under 2%. This confirms the effectiveness and reliability of our framework, particularly given the limited high-fidelity dataset used during training.

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“…In that setting, the user can use fewer samples since the equations provide additional information to the training. Optimization algorithms which exploit DON structure have also been proposed to handle noisy data and train DON [19,20], which may be more practical for real data. DON has also been generalized to a wider class of nonlinear approximation problems using shifts [13].…”

Section: Introductionmentioning

confidence: 99%

BelNet: basis enhanced learning, a mesh-free neural operator

Zhang,

Wing Tat,

Schaeffer

2023

Proc. R. Soc. A.

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Operator learning trains a neural network to map functions to functions. An ideal operator learning framework should be mesh-free in the sense that the training does not require a particular choice of discretization for the input functions, allows for the input and output functions to be on different domains, and is able to have different grids between samples. We propose a mesh-free neural operator for solving parametric partial differential equations. The basis enhanced learning network (BelNet) projects the input function into a latent space and reconstructs the output functions. In particular, we construct part of the network to learn the ‘basis’ functions in the training process. This generalized the networks proposed in Chen & Chen (Chen and Chen 1995 IEEE Trans. Neural Netw. 49 , 911–917. ( doi:10.1109/72.392253 ) and 6 , 904–910. ( doi:10.1109/IJCNN.1993.716815 )) to account for differences in input and output meshes. Through several challenging high-contrast and multiscale problems, we show that our approach outperforms other operator learning methods for these tasks and allows for more freedom in the sampling and/or discretization process.

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B-DeepONet: An enhanced Bayesian DeepONet for solving noisy parametric PDEs using accelerated replica exchange SGLD

Cited by 17 publications

References 17 publications

Restoring the discontinuous heat equation source using sparse boundary data and dynamic sensors

Restoring the discontinuous heat equation source using sparse boundary data and dynamic sensors

A Physics-Guided Bi-Fidelity Fourier-Featured Operator Learning Framework for Predicting Time Evolution of Drag and Lift Coefficients

BelNet: basis enhanced learning, a mesh-free neural operator

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