Physics Informed Extreme Learning Machine (PIELM)–A rapid method for the numerical solution of partial differential equations

Dwivedi, Vikas; Srinivasan, Balaji Vasan

doi:10.1016/j.neucom.2019.12.099

Cited by 144 publications

(92 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We note that Equation 15 was solved in Dwivedi and Srinivasan (2020) with the so-called physics informed extreme learning machine method. In this method, only one hidden layer is used and the coefficients associated with the last (output) neural network layer are computed by pseudo-inverse operation.…”

Section: Two-dimensional Time-dependent Ade: Instantaneous Gaussian Sourcementioning

confidence: 99%

See 1 more Smart Citation

Physics‐Informed Neural Network Method for Forward and Backward Advection‐Dispersion Equations

Tartakovsky

2021

Water Resources Research

View full text Add to dashboard Cite

Advection-dispersion equations (ADEs) are an important class of partial differential equations (PDEs) that are used to describe transport phenomena in the fields of hydrology (Ingham & Ma, 2005) and hydrogeology (Patil & Chore, 2014).We consider both forward and backward ADEs. In the former case, the initial conditions at time t 0 as well as boundary conditions are specified, and the solutions of forward ADEs are found for later times. This is a well-posed problem with well-established numerical methods. Nevertheless, there are some challenges in numerically solving forward ADEs, mainly associated with the advection-dominated problems. In backward ADEs, the concentration is known at later (terminal) times and solutions are sought for earlier times. Backward ADEs arise in the source identification problems (Neupauer & Wilson, 1999;Wilson & Liu, 1994) and could lead to numerically unstable grid-based solutions that require some form of regularization (Xiong et al., 2006) or should be solved as an inverse problem that is computationally more expensive because it requires solving forward problems multiple times (Atmadja & Bagtzoglou, 2001).Numerical discretization-based methods, including the finite element (FE) and finite difference (FD) methods, are commonly used for solving the Darcy flow equation and ADE. Discretization-based methods approximate the PDE solution with its values at a set of grid points distributed over the spatiotemporal domain. The discrete solution is obtained by discretizing the time and spatial derivatives of state variables. It is worth noting that the space-dependent parameters such as hydraulic conductivity are usually given not as a continuous field but as a set of values at the grid points.The combination of an advection (first-order) term and a dispersion (second-order) term in ADEs present several challenges for numerical methods. For example, for advection-dominated transport (i.e., Péclet number Pe ≫ 1), the numerical solutions can develop oscillations (over-or undershoot) or numerical dispersion (Huyakorn, 2012;Pinder & Gray, 2013). These two numerical issues are closely related, and a numerical scheme developed to reduce numerical dispersion generally causes oscillation, whereas the suppression of oscillation comes at the cost of increased numerical dispersion (Wang & Lacroix, 1997).

show abstract

Section: Two-dimensional Time-dependent Ade: Instantaneous Gaussian Sourcementioning

confidence: 99%

“…condition penalty terms in the loss function. The larger error in Dwivedi and Srinivasan (2020) can also be due to the ill-conditioned pseudo-inverse problem.…”

Section: Two-dimensional Time-dependent Ade: Instantaneous Gaussian Sourcementioning

confidence: 99%

Physics‐Informed Neural Network Method for Forward and Backward Advection‐Dispersion Equations

Tartakovsky

2021

Water Resources Research

View full text Add to dashboard Cite

show abstract

“…On the other hand, the use of ELMs for "traditional" numerical analysis tasks and in particular for the numerical solution of PDEs is still widely unexplored. Very recently in [20], it has been proposed a physics-informed ELM to solve stationary and time dependent linear PDEs, where the authors however report a failure of ELMs to deal, for example, with PDEs which solutions exhibit steep gradients. In [46], the authors used ELMs to solve ordinary and linear PDEs.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution and bifurcation analysis of nonlinear partial differential equations with extreme learning machines

et al. 2021

View full text Add to dashboard Cite

We address a new numerical method based on a class of machine learning methods, the so-called Extreme Learning Machines (ELM) with both sigmoidal and radial-basis functions, for the computation of steady-state solutions and the construction of (one-dimensional) bifurcation diagrams of nonlinear partial differential equations (PDEs). For our illustrations, we considered two benchmark problems, namely (a) the one-dimensional viscous Burgers with both homogeneous (Dirichlet) and non-homogeneous boundary conditions, and, (b) the one- and two-dimensional Liouville–Bratu–Gelfand PDEs with homogeneous Dirichlet boundary conditions. For the one-dimensional Burgers and Bratu PDEs, exact analytical solutions are available and used for comparison purposes against the numerical derived solutions. Furthermore, the numerical efficiency (in terms of numerical accuracy, size of the grid and execution times) of the proposed numerical machine-learning method is compared against central finite differences (FD) and Galerkin weighted-residuals finite-element (FEM) methods. We show that the proposed numerical machine learning method outperforms in terms of numerical accuracy both FD and FEM methods for medium to large sized grids, while provides equivalent results with the FEM for low to medium sized grids; both methods (ELM and FEM) outperform the FD scheme. Furthermore, the computational times required with the proposed machine learning scheme were comparable and in particular slightly smaller than the ones required with FEM.

show abstract

“…However, traditional discrete methods often involve tedious meshing and iterative solving of large sparse nonlinear systems, which are computationally cumbersome on modern parallelized architectures 5 – 8 . Moreover, the current meshing process is still a highly specialized activity that remains in the empirical, descriptive realm of knowledge, especially for complex geometries and physical configurations 9 , 10 .…”

Section: Introductionmentioning

confidence: 99%

An improved data-free surrogate model for solving partial differential equations using deep neural networks

Chen

Wan

et al. 2021

Sci Rep

View full text Add to dashboard Cite

Partial differential equations (PDEs) are ubiquitous in natural science and engineering problems. Traditional discrete methods for solving PDEs are usually time-consuming and labor-intensive due to the need for tedious mesh generation and numerical iterations. Recently, deep neural networks have shown new promise in cost-effective surrogate modeling because of their universal function approximation abilities. In this paper, we borrow the idea from physics-informed neural networks (PINNs) and propose an improved data-free surrogate model, DFS-Net. Specifically, we devise an attention-based neural structure containing a weighting mechanism to alleviate the problem of unstable or inaccurate predictions by PINNs. The proposed DFS-Net takes expanded spatial and temporal coordinates as the input and directly outputs the observables (quantities of interest). It approximates the PDE solution by minimizing the weighted residuals of the governing equations and data-fit terms, where no simulation or measured data are needed. The experimental results demonstrate that DFS-Net offers a good trade-off between accuracy and efficiency. It outperforms the widely used surrogate models in terms of prediction performance on different numerical benchmarks, including the Helmholtz, Klein–Gordon, and Navier–Stokes equations.

show abstract

Physics Informed Extreme Learning Machine (PIELM)–A rapid method for the numerical solution of partial differential equations

Cited by 144 publications

References 26 publications

Physics‐Informed Neural Network Method for Forward and Backward Advection‐Dispersion Equations

Physics‐Informed Neural Network Method for Forward and Backward Advection‐Dispersion Equations

Numerical solution and bifurcation analysis of nonlinear partial differential equations with extreme learning machines

An improved data-free surrogate model for solving partial differential equations using deep neural networks

Contact Info

Product

Resources

About