Data-Driven Identification of Parametric Partial Differential Equations

Rudy, Samuel; Alla, Alessandro; Brunton, Steven L.; Kutz, J. Nathan

doi:10.1137/18m1191944

Cited by 250 publications

(185 citation statements)

References 74 publications

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“…Figure 6 shows the predicted mean and standard deviation of the solution u versus the reference. Figure 7 shows our DNN prediction of three u modes where the reference modes are calculated by Equation 17. We can see that the NN-aPC method makes accurate predictions of the mean and standard deviation of the solution u(x; ω) and learns the arbitrary polynomial chaos modes.…”

Section: Forward Problem: Stochastic Poisson Equation (A Pedagogical mentioning

confidence: 96%

“…For example, for the forward problem, some of the popular choices of machine learning tools are Gaussian process [3,4,5,6,7] and deep neural networks (DNNs) [8,9,10,11,12]. For inverse problems, similar methods have been advanced, e.g., Bayesian estimation [13] and variational Bayes inference [14], and have been proposed for a wide variety of objectives, from parameter estimation [15] to discovering partial differential equations [16,17,18,19] to learning constitutive relationships [20].…”

Section: Introductionmentioning

confidence: 99%

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Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems

Zhang

Guo

et al. 2019

Journal of Computational Physics

343

172

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Physics-informed neural networks (PINNs) have recently emerged as an alternative way of solving partial differential equations (PDEs) without the need of building elaborate grids, instead, using a straightforward implementation. In particular, in addition to the deep neural network (DNN) for the solution, a second DNN is considered that represents the residual of the PDE. The residual is then combined with the mismatch in the given data of the solution in order to formulate the loss function. This framework is effective but is lacking uncertainty quantification of the solution due to the inherent randomness in the data or due to the approximation limitations of the DNN architecture. Here, we propose a new method with the objective of endowing the DNN with uncertainty quantification for both sources of uncertainty, i.e., the parametric uncertainty and the approximation uncertainty. We first account for the parametric uncertainty when the parameter in the differential equation is represented as a stochastic process. Multiple DNNs are designed to learn the modal functions of the arbitrary polynomial chaos (aPC) expansion of its solution by using stochastic data from sparse sensors. We can then make predictions from new sensor measurements very efficiently with the trained DNNs. Moreover, we employ dropout to correct the overfitting and also to quantify the uncertainty of DNNs in approximating the modal functions. We then design an active learning strategy based on the dropout uncertainty to place new sensors in the domain in order to improve * Corresponding Author the predictions of DNNs. Several numerical tests are conducted for both the forward and the inverse problems to quantify the effectiveness of PINNs combined with uncertainty quantification. This NN-aPC new paradigm of physics-informed deep learning with uncertainty quantification can be readily applied to other types of stochastic PDEs in multi-dimensions.

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Section: Forward Problem: Stochastic Poisson Equation (A Pedagogical mentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems

Zhang

Guo

et al. 2019

Journal of Computational Physics

343

172

View full text Add to dashboard Cite

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“…SINDy solves an overdetermined linear system of equations by sparsity-promoting regularization. The basic algorithmic structure of SINDy has been modified to discover parametrically-dependent systems [40], resolve multiscale physics [8], infer biological networks [34], discover spatiotemporal systems [41], and identify nonlinear systems with control [6,22]. Consider a dynamical system of the forṁ…”

Section: Sensorsmentioning

confidence: 99%

“…Emerging data-driven methods are allowing for the discovery of physical and engineering principles directly arXiv:1809.05707v1 [nlin.PS] 15 Sep 2018 from time-series recordings. Our focus is on the SINDy architecture [5], which has been demonstrated on a diverse set of problems, including spatio-temporal [41], parametric [40], networked [34], control [6], and multiscale [8] systems. Importantly, the SINDy architecture can be directly related to model selection theory [34] in order to assess the quality and robustness of the model discovered.…”

Section: Introductionmentioning

confidence: 99%

Discovering time-varying aerodynamics of a prototype bridge by sparse identification of nonlinear dynamical systems

Kaiser

Laima

et al. 2019

Phys. Rev. E

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We develop data-driven dynamical models of the nonlinear aeroelastic effects on a long-span suspension bridge from sparse, noisy sensor measurements which monitor the bridge. Using the sparse identification of nonlinear dynamics (SINDy) algorithm, we are able to identify parsimonious, time-varying dynamical systems that capture vortex-induced vibration (VIV) events in the bridge. Thus we are able to posit new, data-driven models highlighting the aeroelastic interaction of the bridge structure with VIV events. The bridge dynamics are shown to have distinct, time-dependent modes of behavior, thus requiring parametric models to account for the diversity of dynamics. Our method generates hitherto unknown bridge-wind interaction models that go beyond current theoretical and computational descriptions. Our proposed method for real-time monitoring and model discovery allow us to move our model predictions beyond lab theory to practical engineering design, which has the potential to assess bad engineering configurations that are susceptible to deleterious bridge-wind interactions. With the rise of real-time sensor networks on major bridges, our model discovery methods can enhance an engineers ability to assess the nonlinear aeroelastic interactions of the bridge with its wind environment.

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“…The proposed framework replaces the guessing work often involved in such model development with a data-driven approach that uncovers the closure from data in a systematic fashion. Our approach draws inspiration from the early and contemporary contributions in deep learning for partial differential equations [14][15][16][17][18][19][20] and data-driven modeling strategies [21][22][23], and in particular relies on recent developments in physics-informed deep learning [24] and deep hidden physics models [25].…”

Section: Introductionmentioning

confidence: 99%

Deep learning of turbulent scalar mixing

2019

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Based on recent developments in physics-informed deep learning and deep hidden physics models, we put forth a framework for discovering turbulence models from scattered and potentially noisy spatio-temporal measurements of the probability density function (PDF). The models are for the conditional expected diffusion and the conditional expected dissipation of a Fickian scalar described by its transported single-point PDF equation. The discovered model are appraised against exact solution derived by the amplitude mapping closure (AMC)/ Johnsohn-Edgeworth translation (JET) model of binary scalar mixing in homogeneous turbulence.

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Data-Driven Identification of Parametric Partial Differential Equations

Cited by 250 publications

References 74 publications

Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems

Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems

Discovering time-varying aerodynamics of a prototype bridge by sparse identification of nonlinear dynamical systems

Deep learning of turbulent scalar mixing

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