There has been rapid progress recently on the application of deep networks to solution of partial differential equations, collectively labelled as Physics Informed Neural Networks (PINNs). In this paper, we develop Physics Informed Extreme Learning Machine (PIELM), a rapid version of PINNs which can be applied to stationary and time dependent linear partial differential equations. We demonstrate that PIELM matches or exceeds the accuracy of PINNs on a range of problems. We also discuss the limitations of neural network based approaches, including our PIELM, in the solution of PDEs on large domains and suggest an extension, a distributed version of our algorithm --DPIELM. We show that DPIELM produces excellent results comparable to conventional numerical techniques in the solution of time-dependent problems. Collectively, this work contributes towards making the use of neural networks in the solution of partial differential equations in complex domains as a competitive alternative to conventional discretization techniques.Keywords Partial differential equations · Physics informed neural networks · Extreme learning machine · Advection-diffusion equation * Use footnote for providing further information about author (webpage, alternative address)-not for acknowledging funding agencies.
This paper develops an extreme learning machine for solving linear partial differential equations (PDE) by extending the normal equations approach for linear regression. The normal equations method is typically used when the amount of available data is small. In PDEs, the only available ground truths are the boundary and initial conditions (BC and IC). We use the physics-based cost function used in state-of-the-art deep neural network-based PDE solvers called physics informed neural network (PINN) to compensate for the small data. However, unlike PINN, we derive the normal equations for PDEs and directly solve them to compute the network parameters. We demonstrate our method's feasibility and efficiency by solving several problems like function approximation, solving ordinary differential equations (ODEs), steady and unsteady PDEs on regular and complicated geometries. We also highlight our method's limitation in capturing sharp gradients and propose its domain distributed version to overcome this issue. We show that this approach is much faster than traditional gradient descent-based approaches and offers an alternative to conventional numerical methods in solving PDEs in complicated geometries.
Recently, physics informed neural networks (PINNs) have produced excellent results in solving a series of linear and nonlinear partial differential equations (PDEs) without using any prior data. However, due to slow training speed, PINNs are not directly competitive with existing numerical methods. To overcome this issue, the authors developed Physics Informed Extreme Learning Machine (PIELM), a rapid version of PINN, and tested it on a range of linear PDEs of first and second order. In this paper, we evaluate the effectiveness of PIELM on higher-order PDEs with practical engineering applications. Specifically, we demonstrate the efficacy of PIELM to the biharmonic equation. Biharmonic equations have numerous applications in solid and fluid mechanics, but they are hard to solve due to the presence of fourth-order derivative terms, especially in complicated geometries. Our numerical experiments show that PIELM is much faster than the original PINN on both regular and irregular domains. On irregular domains, it also offers an excellent alternative to traditional methods due to its meshless nature.
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