Algorithm

Mathematical analysis

Artificial intelligence

Composite material

Materials science

Range (aeronautics)

Applied mathematics

Mathematics

Computer science

Artificial neural network

Partial derivative

Discretization

Partial differential equation

Numerical partial differential equations

Neurocomputing

There has been rapid progress recently on the application of deep networks to the solution of partial differential equations, collectively labeled 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.

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