Physics-informed learning of governing equations from scarce data

Chen, Zhao; Liu, Yang; Sun, Hao

doi:10.1038/s41467-021-26434-1

Cited by 209 publications

(110 citation statements)

References 55 publications

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“…To serve this purpose well, a random sampling method that provides good coverage of the parameter space and avoids an unconscious bias reflected in the collected data is required [39]. A superior input group is also helpful to facilitate the metamodels to capture data characteristics, optimize the training speed, and improve the applicability of the trained model [40,41]. Recognizing these, the Sobol sequence method [42] was adopted to sample different wave parameters to form the input group, which is a high-efficient quasi-Monte Carlo sequence and has been proven effective in reducing model learning costs [43,44].…”

Section: Tested Wave Casesmentioning

confidence: 99%

3D Numerical Modeling and Quantification of Oblique Wave Forces on Coastal Bridge Superstructures

Jia¹,

Zhu

Dong

2022

JMSE

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Simply supported bridges comprise the majority of bridge systems in coastal communities and are susceptible to severe damage from extreme waves induced by storms or tsunamis. However, the effects of oblique wave impacts have been less investigated due to the lack of appropriate numerical models. To address this issue, this study investigates the effects of wave incident angles on coastal bridge superstructures by developing an advanced computational fluid dynamics (CFD) model. Different wave scenarios, including wave height, relative clearance, incident angle, and wavelength are tested. It is found that the maximum wave forces in the horizontal and longitudinal directions could reach 1901 and 862 kN under extreme conditions, respectively, destroying bearing connections. Three surrogate models, i.e., the Gaussian Kriging surrogate model, the Artificial Neural Network (ANN), and the Polynomial Chaos Expansion (PCE), are established by correlating the wave parameters with the maximum wave forces. Through comparisons among the three surrogate models, it is found that the 3-order PCE model has better performance in predicting loads in vertical and horizontal directions, while the ANN model is more suitable for results in the longitudinal direction. This study contributes to the optimized design of coastal bridges and also offers an opportunity for future studies to investigate hazard damage-mitigation measures.

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Section: Tested Wave Casesmentioning

confidence: 99%

3D Numerical Modeling and Quantification of Oblique Wave Forces on Coastal Bridge Superstructures

Jia¹,

Zhu

Dong

2022

JMSE

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“…In recent years, utilizing physics information to encode in the losses of a deep neural network (NN), promising faster accurate learning of physics with NN that respect basic physics laws with less labeled data, commonly recognized as Physics-Informed Neural Networks (PINNs) [22,23]. One of the celebrated characteristics of PINNs is they can learn from sparse data [24] as physics doesn't generate humongous data as easily in other commercial fields. Upon the proposed PINN framework, various types of PINNs designed for different engineering applications emerge in fields, with their most renowned works in predicting fluid fields [25,26], but also include electronics [28,29].…”

Section: Inductormentioning

confidence: 99%

Physics-Informed Deep Operator Control: Controlling chaos in van der Pol oscillating circuits

Zhai,

Sands

2021

Preprint

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Controlling chaos is a long-standing problem in engineering. By linearizing nonlinear systems, traditional control methods cannot learn features from chaotic data for control. Here, we introduce Physics-Informed Deep Operator Control (PIDOC), by encoding control signal and initial position into the losses of a Physics-Informed Neural Network (PINN), the nonlinear system exhibits the desired route given the control signal. The framework has a clear application scenario: receiving signals as physics commands and learning chaotic data from the nonlinear van der Pol system, PIDOC is able to execute the controlled signal as the output of the PINN. PIDOC is first applied to a benchmark problem, unveils a fluctuating behavior of the acceleration that seemingly behave in the same frequency of the desired signal, indicating the learning of neural networks (NN) elicit stochasticity for higher-order terms, yet still successfully impose the control that enforces the system behave as in the same frequency of the control signal. PIDOC is also proved capable of converging to different desired trajectories based on case studies. Initial positions slightly affect the control accuracy at the beginning stage yet still converge to the control signal based on numerical experiments. To note, for highly nonlinear systems, PIDOC is not able to execute control as high accuracies compared with the benchmark problem, yet still, PIDOC converges to decent trajectories corresponding to given signals with fair predictable behavior. The depth and width of the NN structure did not heavily variate the convergence of PIDOC based on case studies on van der Pol systems with low and high nonlinearities. Surprisingly, by enlarging the control signal by multiplying a Lagrangian multiplier, PIDOC failed to converge to the desired trajectory, indicating enlarging the control signal does not increase control quality, which is not intuitive. The proposed framework can be potentially applied to many nonlinear systems for controlling chaos.

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“…Some of the simultaneous methods employ neural networks [15] and automatic differentiation [16] to decompose the noisy measurements into the deterministic and random noisy components to recover the governing equations more accurately. Others use smoothing splines [17] and deep neural networks [18] as a surrogate model to approximate the state variables for the data loss which then can be differentiated to obtain state time derivatives required for the state derivative error. However, simultaneous methods often require solving challenging nonlinear optimization problems and fine tuning of the hyperparameters of the algorithm.…”

Section: Introductionmentioning

confidence: 99%

A Priori Denoising Strategies for Sparse Identification of Nonlinear Dynamical Systems: A Comparative Study

Cortiella¹,

Park²,

Doostan³

2022

Preprint

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In recent years, identification of nonlinear dynamical systems from data has become increasingly popular. Sparse regression approaches, such as Sparse Identification of Nonlinear Dynamics (SINDy), fostered the development of novel governing equation identification algorithms assuming the state variables are known a priori and the governing equations lend themselves to sparse, linear expansions in a (nonlinear) basis of the state variables. In the context of the identification of governing equations of nonlinear dynamical systems, one faces the problem of identifiability of model parameters when state measurements are corrupted by noise. Measurement noise affects the stability of the recovery process yielding incorrect sparsity patterns and inaccurate estimation of coefficients of the governing equations. In this work, we investigate and compare the performance of several local and global smoothing techniques to a priori denoise the state measurements and numerically estimate the state time-derivatives to improve the accuracy and robustness of two sparse regression methods to recover governing equations: Sequentially Thresholded Least Squares (STLS) and Weighted Basis Pursuit Denoising (WBPDN) algorithms. We empirically show that, in general, global methods, which use the entire measurement data set, outperform local methods, which employ a neighboring data subset around a local point. We additionally compare Generalized Cross Validation (GCV) and Pareto curve criteria as model selection techniques to automatically estimate near optimal tuning parameters, and conclude that Pareto curves yield better results. The performance of the denoising strategies and sparse regression methods is empirically evaluated through well-known benchmark problems of nonlinear dynamical systems.

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Physics-informed learning of governing equations from scarce data

Cited by 209 publications

References 55 publications

3D Numerical Modeling and Quantification of Oblique Wave Forces on Coastal Bridge Superstructures

3D Numerical Modeling and Quantification of Oblique Wave Forces on Coastal Bridge Superstructures

Physics-Informed Deep Operator Control: Controlling chaos in van der Pol oscillating circuits

A Priori Denoising Strategies for Sparse Identification of Nonlinear Dynamical Systems: A Comparative Study

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