scholarly journals Data-driven identification of 2D Partial Differential Equations using extracted physical features

2021 ◽  
Vol 381 ◽  
pp. 113831
Author(s):  
Kazem Meidani ◽  
Amir Barati Farimani
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Data Driven ◽  
Partial Differential ◽  
Physical Features

scholarly journals Applied Koopman Theory for Partial Differential Equations and Data-Driven Modeling of Spatio-Temporal Systems

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-16 ◽  
Author(s):  
J. Nathan Kutz ◽  
J. L. Proctor ◽  
S. L. Brunton
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Data Driven ◽  
Nonlinear Pdes ◽  
Dynamic Mode ◽  
Partial Differential ◽  
Koopman Operator ◽  
Dimensional Approximation ◽  
Spatio Temporal ◽  
The Impact

We consider the application of Koopman theory to nonlinear partial differential equations and data-driven spatio-temporal systems. We demonstrate that the observables chosen for constructing the Koopman operator are critical for enabling an accurate approximation to the nonlinear dynamics. If such observables can be found, then the dynamic mode decomposition (DMD) algorithm can be enacted to compute a finite-dimensional approximation of the Koopman operator, including its eigenfunctions, eigenvalues, and Koopman modes. We demonstrate simple rules of thumb for selecting a parsimonious set of observables that can greatly improve the approximation of the Koopman operator. Further, we show that the clear goal in selecting observables is to place the DMD eigenvalues on the imaginary axis, thus giving an objective function for observable selection. Judiciously chosen observables lead to physically interpretable spatio-temporal features of the complex system under consideration and provide a connection to manifold learning methods. Our method provides a valuable intermediate, yet interpretable, approximation to the Koopman operator that lies between the DMD method and the computationally intensive extended DMD (EDMD). We demonstrate the impact of observable selection, including kernel methods, and construction of the Koopman operator on several canonical nonlinear PDEs: Burgers’ equation, the nonlinear Schrödinger equation, the cubic-quintic Ginzburg-Landau equation, and a reaction-diffusion system. These examples serve to highlight the most pressing and critical challenge of Koopman theory: a principled way to select appropriate observables.

scholarly journals Learning data-driven discretizations for partial differential equations

2019 ◽  
Vol 116 (31) ◽  
pp. 15344-15349 ◽  
Author(s):  
Yohai Bar-Sinai ◽  
Stephan Hoyer ◽  
Jason Hickey ◽  
Michael P. Brenner
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ad Hoc ◽  
Finite Difference Methods ◽  
Spatial Dimension ◽  
Data Driven ◽  
Coarse Grained ◽  
Small Scale ◽  
Partial Differential ◽  
Long Wavelength

The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length- and timescales. Often, it is computationally intractable to resolve the finest features in the solution. The only recourse is to use approximate coarse-grained representations, which aim to accurately represent long-wavelength dynamics while properly accounting for unresolved small-scale physics. Deriving such coarse-grained equations is notoriously difficult and often ad hoc. Here we introduce data-driven discretization, a method for learning optimized approximations to PDEs based on actual solutions to the known underlying equations. Our approach uses neural networks to estimate spatial derivatives, which are optimized end to end to best satisfy the equations on a low-resolution grid. The resulting numerical methods are remarkably accurate, allowing us to integrate in time a collection of nonlinear equations in 1 spatial dimension at resolutions 4× to 8× coarser than is possible with standard finite-difference methods.

scholarly journals Computational fluid dynamics iteration driven by data

2021 ◽  
pp. 227-227
Author(s):  
Zhijun Zhou ◽  
Zhang Qi ◽  
Xichuan Cai ◽  
Kun Li ◽  
Jingwei Zhao
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Burgers Equation ◽  
Fluid Dynamic ◽  
Data Driven ◽  
Space Distribution ◽  
Dynamic Problems ◽  
Partial Differential ◽  
Detailed Explanation ◽  
Potential Fluid

Data-driven approaches have achieved remarkable success in different applications; however, their use in solving partial differential equations (PDEs) has only recently emerged. Herein, we present the potential fluid method (PFM), which uses existing data to nest physical meanings into mathematical iterative processes. The PFM is suitable for partial differential equations, such as computational fluid dynamic problems, including Burgers? equation. PFM can iteratively determine the steady-state space distribution of PDEs. For mathematical reasons, we compare the PFM with the finite difference method (FDM) and give a detailed explanation.

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