Physics-informed machine learning for reduced-order modeling of nonlinear problems

Chen, Wenqian; Wang, Qian; Hesthaven, Jan S.; Zhang, Chuhua

doi:10.1016/j.jcp.2021.110666

Cited by 158 publications

(63 citation statements)

References 63 publications

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“…[4,49,57]. Other nonintrusive alternatives for defining φ have also been suggested, such as: Gaussian process regression [30], polynomial chaos expansions [35] and neural networks [14,61].…”

Section: General Backgroundmentioning

confidence: 99%

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A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equations

Franco¹,

Manzoni²,

Zunino³

2021

Preprint

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Within the framework of parameter dependent PDEs, we develop a constructive approach based on Deep Neural Networks for the efficient approximation of the parameter-to-solution map. The research is motivated by the limitations and drawbacks of state-of-the-art algorithms, such as the Reduced Basis method, when addressing problems that show a slow decay in the Kolmogorov n-width. Our work is based on the use of deep autoencoders, which we employ for encoding and decoding a high fidelity approximation of the solution manifold. In order to fully exploit the approximation capabilities of neural networks, we consider a nonlinear version of the Kolmogorov n-width over which we base the concept of a minimal latent dimension. We show that this minimal dimension is intimately related to the topological properties of the solution manifold, and we provide some theoretical results with particular emphasis on second order elliptic PDEs. Finally, we report numerical experiments where we compare the proposed approach with classical POD-Galerkin reduced order models. In particular, we consider parametrized advection-diffusion PDEs, and we test the methodology in the presence of strong transport fields, singular terms and stochastic coefficients.

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Section: General Backgroundmentioning

confidence: 99%

“…Our construction is mostly inspired by the recent advancements in nonlinear approximation theory, e.g. [16,17,55], and the increasing use of deep-learning techniques for parametrized PDEs, as in [14,24,39,44].…”

Section: Introductionmentioning

confidence: 99%

A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equations

Franco¹,

Manzoni²,

Zunino³

2021

Preprint

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“…In [53,46], latent dynamics and nonlinear mappings are modeled as neural ODEs and autoencoders, respectively; in [49,43,65,47], autoencoders are used to learn approximate invariant subspaces of the Koopman operator. Relatedly, there have been studies on learning direct mappings via e.g., a neural network, from problem-specific parameters to either latent states or approximate solution states [64,19,58,10,35,69], where the latent states are computed by using autoencoder or linear POD.…”

Section: Related Workmentioning

confidence: 99%

“…That is, for the given network architecture (Table 1), increasing the amount of training/validating data does not have significant effect on the performance of the framework. On the other hand, increasing the number of training/validating parameter instances in a way that is shown in Figure 7 has significantly improved the accuracy of the approximations for the other testing parameter instances {µ (10) , µ (11) , µ (12) }. This set of experiments essentially illustrates that, for a given network architecture, more accurate approximation can be achieved for testing parameter instances that lie in between training/validating parameter instances (i.e., interpolation in the parameter space) than for those that lie outside of training/validating parameter instances (i.e., extrapolation in the parameter space).…”

Section: Node Pnodementioning

confidence: 99%

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Parameterized Neural Ordinary Differential Equations: Applications to Computational Physics Problems

Lee,

Parish

2020

Preprint

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This work proposes an extension of neural ordinary differential equations (NODEs) by introducing an additional set of ODE input parameters to NODEs. This extension allows NODEs to learn multiple dynamics specified by the input parameter instances. Our extension is inspired by the concept of parameterized ordinary differential equations, which are widely investigated in computational science and engineering contexts, where characteristics of the governing equations vary over the input parameters. We apply the proposed parameterized NODEs (PNODEs) for learning latent dynamics of complex dynamical processes that arise in computational physics, which is an essential component for enabling rapid numerical simulations for time-critical physics applications. For this, we propose an encoder-decoder-type framework, which models latent dynamics as PNODEs. We demonstrate the effectiveness of PNODEs with important benchmark problems from computational physics.

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Parameterized nonintrusive reduced‐order model for general unsteady flow problems using artificial neural networks

Gabor

2020

Numerical Methods in Fluids

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A nonintrusive reduced‐order model for nonlinear parametric flow problems is developed. It is based on extracting a reduced‐order basis from high‐order snapshots via proper orthogonal decomposition and using multilayered feedforward artificial neural networks to approximate the reduced‐order coefficients. The model is a generic and efficient approach for the reduction of time‐dependent parametric systems, including those described by partial differential equations. Since it is nonintrusive, it is independent of the high‐order computational method and can be used together with black‐box solvers. Numerical studies are presented for steady‐state isentropic nozzle flow with geometric parameterization and unsteady parameterized viscous Burgers equation. An adaptive sampling strategy is proposed to increase the quality of the neural network approximation while minimizing the required number of parameter samples and, as a direct consequence, the number of high‐order snapshots and the size of the network training set. Results confirm the accuracy of the nonintrusive approach as well as the speed‐up achieved compared with intrusive hyper‐reduced‐order approaches.

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Physics-informed machine learning for reduced-order modeling of nonlinear problems

Cited by 158 publications

References 63 publications

A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equations

A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equations

Parameterized Neural Ordinary Differential Equations: Applications to Computational Physics Problems

Parameterized nonintrusive reduced‐order model for general unsteady flow problems using artificial neural networks

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