Wavelet Neural Operator for solving parametric partial differential equations in computational mechanics problems

Tripura, Tapas; Chakraborty, Souvik

doi:10.1016/j.cma.2022.115783

Cited by 47 publications

(24 citation statements)

References 24 publications

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“…Once the neural operators are trained, they can generalize for unseen cases, which means the same network parameters are shared across different input functions. We have implemented three operator networks that have shown promising results so far, the DeepONet [17], the Fourier neural operator (FNO) [18], and the Wavelet neural operator (WNO) [20]. Although the original DeepONet architecture proposed in [17] has shown remarkable success, several extensions have been proposed in [24][25][26] to modify its implementation and produce efficient and robust architectures.…”

Section: Neural Operatorsmentioning

confidence: 99%

“…The wavelet neural operator (WNO) proposed in [20] learns the network parameters in the wavelet space that are both frequency and spatial localized, hence can learn the patterns in the images and/or signals more effectively. Specifically, the Fourier integral of FNO is replaced by wavelet integrals for capturing the spatial behavior of a signal or for studying the system under complex boundary conditions.…”

Section: Wavelet Neural Operatormentioning

confidence: 99%

“…In 2019, Lu et al [17] extended the universal operator approximation theorem [19] to propose the idea of deep neural operators (DeepONet), which can learn nonlinear continuous operators accurately and efficiently from a relatively small dataset due to its small generalization error. Later, the idea of neural operators was extended in the frequency domain by modifying the architecture of DeepONet, and Li et al [18] proposed the Fourier Neural Operator (FNO) and Tripura and Chakraborty [20] developed the wavelet neural operator (WNO).…”

Section: Introductionmentioning

confidence: 99%

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Deep neural operators can predict the real-time response of floating offshore structures under irregular waves

Cao¹,

Goswami²,

Karniadakis³

et al. 2023

Preprint

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The use of neural operators in a digital twin model of an offshore floating structure can provide a paradigm shift in structural response prediction and health monitoring, providing valuable information for real-time control. In this work, the performance of three neural operators is evaluated, namely, deep operator network (DeepONet), Fourier neural operator (FNO), and Wavelet neural operator (WNO). We investigate the effectiveness of the operators to accurately capture the responses of a floating structure under six different sea state codes (3 − 8) based on the wave characteristics described by the World Meteorological Organization (WMO). The results demonstrate that these high-precision neural operators can deliver structural responses more efficiently, up to two orders of magnitude faster than a dynamic analysis using conventional numerical solvers. Additionally, compared to gated recurrent units (GRUs), a commonly used recurrent neural network for time-series estimation, neural operators are both more accurate and efficient, especially in situations with limited data availability. To further enhance the accuracy, novel extensions, such as wavelet-DeepONet and self-adaptive WNO, are proposed. Taken together, our study shows that FNO outperforms all other operators for approximating the mapping of one input functional space to the output space as well as for responses that have small bandwidth of the frequency spectrum, whereas for learning the mapping of multiple functions in the input space to the output space as well as for capturing responses within a large frequency spectrum, DeepONet with historical states provides the highest accuracy.

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Section: Neural Operatorsmentioning

confidence: 99%

Section: Wavelet Neural Operatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Deep neural operators can predict the real-time response of floating offshore structures under irregular waves

Cao¹,

Goswami²,

Karniadakis³

et al. 2023

Preprint

View full text Add to dashboard Cite

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“…With this setup, given that we can obtain the measurements for system states X = [X, Ẋ], we aim (i) to automate the discovery of the exact analytical form of the Lagrangian L(X, Ẋ) by constraining the Lagrangian to satisfy Eqs. (7) and ( 8), (ii) to automate the discovery of the governing equations of motion using the discovered Lagrangian L(X, Ẋ), which will have perpetual prediction capability, and (iii) to automate the discovery of conservation laws using the principle of energy conservation on discovered Lagrangian L(X, Ẋ). To achieve tasks (i)-(iii), we restrict our access to a single trajectory of the system states X ∈ R m and Ẋ ∈ R m only.…”

Section: Problem Formulationmentioning

confidence: 99%

“…Learning the Lagrangian of systems from data has gained some popularity in recent times. Due to the significant developments in data-driven [5,6,7], and physics-informed [8,9,10,11] neural network algorithms, researchers have suggested using neural network to extract Lagrangian from data. Initial works related to the discovery of Lagrangian can be linked to Hamiltonian Neural Networks (HNN) [12,13].…”

Section: Introductionmentioning

confidence: 99%

Discovering interpretable Lagrangian of dynamical systems from data

Tripura¹,

Chakraborty²

2023

Preprint

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A complete understanding of physical systems requires models that are accurate and obeys natural conservation laws. Recent trends in representation learning involve learning Lagrangian from data rather than the direct discovery of governing equations of motion. The generalization of equation discovery techniques has huge potential; however, existing Lagrangian discovery frameworks are black-box in nature. This raises a concern about the reusability of the discovered Lagrangian. In this article, we propose a novel data-driven machine-learning algorithm to automate the discovery of interpretable Lagrangian from data. The Lagrangian are derived in interpretable forms, which also allows the automated discovery of conservation laws and governing equations of motion. The architecture of the proposed framework is designed in such a way that it allows learning the Lagrangian from a subset of the underlying domain and then generalizing for an infinite-dimensional system. The fidelity of the proposed framework is exemplified using examples described by systems of ordinary differential equations and partial differential equations where the Lagrangian and conserved quantities are known.

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