Distributed learning machines for solving forward and inverse problems in partial differential equations

Dwivedi, Vikas; Parashar, Nishant; Srinivasan, Balaji Vasan

doi:10.1016/j.neucom.2020.09.006

Cited by 62 publications

(21 citation statements)

References 20 publications

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“…Raissi et al (2019) and Raissi & Karniadakis (2018) introduced the framework of physics-informed neural network (PINN) to constrain neural networks with PDE derivatives computed using Automatic Differentiation (AD) Baydin et al (2018). In the past couple of years, the PINN framework has been extended to solve complicated PDEs representing complex physics (Jin et al, 2021;Mao et al, 2020;Rao et al, 2020;Wu et al, 2018;Qian et al, 2020;Dwivedi et al, 2021;Nabian et al, 2021;Kharazmi et al, 2021;Cai et al, 2021a;Bode et al, 2021;Taghizadeh et al, 2021;Lu et al, 2021c;Shukla et al, 2021;Hennigh et al, 2020;Li et al, 2021). More recently, alternate approaches that use discretization techniques using higher order derivatives and specialize numerical schemes to compute derivatives have shown to provide better regularization for faster convergence (Ranade et al, 2021b;Gao et al, 2021;Wandel et al, 2020;He & Pathak, 2020).…”

Section: Related Workmentioning

confidence: 99%

A composable autoencoder-based iterative algorithm for accelerating numerical simulations

Ranade,

Hill,

et al. 2021

Preprint

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Numerical simulations for engineering applications solve partial differential equations (PDE) to model various physical processes. Traditional PDE solvers are very accurate but computationally costly. On the other hand, Machine Learning (ML) methods offer a significant computational speedup but face challenges with accuracy and generalization to different PDE conditions, such as geometry, boundary conditions, initial conditions and PDE source terms. In this work, we propose a novel ML-based approach, CoAE-MLSim (Composable AutoEncoder Machine Learning Simulation), which is an unsupervised, lower-dimensional, local method, that is motivated from key ideas used in commercial PDE solvers. This allows our approach to learn better with relatively fewer samples of PDE solutions. The proposed ML-approach is compared against commercial solvers for better benchmarks as well as latest ML-approaches for solving PDEs. It is tested for a variety of complex engineering cases to demonstrate its computational speed, accuracy, scalability, and generalization across different PDE conditions. The results show that our approach captures physics accurately across all metrics of comparison (including measures such as results on section cuts and lines).

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Section: Related Workmentioning

confidence: 99%

A composable autoencoder-based iterative algorithm for accelerating numerical simulations

Ranade,

Hill,

et al. 2021

Preprint

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“…The most peculiar property of the ELM is that its input layer weights are pre-set random values and fixed throughout the training process, and output layer weights are training parameters. Spirited by ELM and PINN, Dwivedi et al [26,27] develop an ELM-based PDE solver called physic-informed extreme learning machine (PIELM) as a rapid version of PINN to solve the linear PDE problems efficiently. The PDE problem is transformed into a linear least squares problem by incorporating physics laws into the ELM as the cost function.…”

Section: Introductionmentioning

confidence: 99%

Bayesian Physics-Informed Extreme Learning Machine for Forward and Inverse PDE Problems with Noisy Data

Liu¹,

Wen²,

Wei³

et al. 2022

Preprint

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Physics-informed extreme learning machine (PIELM) has recently received significant attention as a rapid version of physics-informed neural network (PINN) for solving partial differential equations (PDEs). The key characteristic is to fix the input layer weights with random values and use Moore-Penrose generalized inverse for the output layer weights. The framework is effective, but it easily suffers from overfitting noisy data and lacks uncertainty quantification for the solution under noise scenarios. To this end, we develop Bayesian physics-informed extreme learning machine (BPIELM) to solve both forward and inverse linear PDE problems with noisy data in a unified framework. In our framework, a prior probability distribution is introduced in the output layer for extreme learning machine with physic laws and the Bayesian method is used to estimate the posterior of parameters. Besides, for inverse PDE problems, problem parameters considered as new output layer weights are unified in a framework with forward PDE problems. Finally, we demonstrate BPIELM considering both forward problems, including Poisson, advection, and diffusion equations, as well as inverse problems, where unknown problem parameters are estimated.The results show that, compared with PIELM, BPIELM quantifies uncertainty arising from noisy data and provides more accurate predictions. In addition, BPIELM is considerably cheaper than PINN in terms of the computational cost.

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“…(Deng et al, 2019) proposed a network-based generative adversarial network-based artificial intelligence framework to enhance the spatial resolution of complicated wake flow behind two side-by-side cylinders; however, (Liu et al, 2020) proposed a multitime path CNN. To reduce the increase in the visual complexity to an LR input, which is caused by data-driven upsampling approaches, various efforts (Raissi et al, 2019;Gao et al, 2020;Dwivedi et al, 2021) have been proposed. Despite the advantages, the success of these DL models mainly relies on a large quantity of offline HR data as labels.…”

Section: Introductionmentioning

confidence: 99%

Super-resolution PIV using multi-frame displacement information

Yang¹,

Ouyang²,

Cao³

et al. 2021

ISPIV21

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High-resolution (HR) fluid-flow velocity information is important to reliably analyze fluid measurements in particle image velocimetry (PIV), such as the boundary layer and turbulent flow. Efforts in PIV to enhance the resolution of flow fields are mainly based on single-frame information, which follows the velocity field estimation and may influence the final reconstruction accuracy. In this study, we propose a novel super-resolution (SR) reconstruction technology from another perspective, which consists of two parts: a multi-frame imaging system and a Bayesian-based multi-frame SR reconstruction algorithm. First, a splitbased imaging system is developed to obtain particle image pairs with fixed displacements. Subsequently, we present a Bayesian-based multi-frame SR (BMFSR) reconstruction algorithm to obtain an SR particle image. Multi-frame particle images collected by the developed system are used as the input low-resolution images for the following novel SR reconstruction algorithm. Synthetic and experimental particle images have been tested to verify the performance of the proposed technology, and the results are compared with the traditional and advanced reconstruction methods in PIV. The results and comparisons show that the proposed technology successfully achieves good performance in obtaining finer particle images and a more accurate velocity field.

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Distributed learning machines for solving forward and inverse problems in partial differential equations

Cited by 62 publications

References 20 publications

A composable autoencoder-based iterative algorithm for accelerating numerical simulations

A composable autoencoder-based iterative algorithm for accelerating numerical simulations

Bayesian Physics-Informed Extreme Learning Machine for Forward and Inverse PDE Problems with Noisy Data

Super-resolution PIV using multi-frame displacement information

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