Physics Informed Extreme Learning Machines With Residual Variation Diminishing Scheme for Nonlinear Problems With Discontinuous Surfaces

Abstract

This work extends Extreme Learning Machines (ELM) to obtain solutions for nonlinear higher order partial differential equations that govern the physics of different domains. The ELM operates in sync with a novel Residual Variation Diminishing Scheme (RVDS) developed as a part of this work and described here, and is able to generate solutions around surfaces of discontinuity in the field and boundaries, as typically appearing in Aerospace configurations. Proofs of Concept of RVDS and its synchronization with ELM are presented. The Physics Informed Learning Machine developed here is demonstrated to obviate the major limitations of Physics Informed Neural Networks (PINN), namely, inability to handle directed differencing and its corollary, surfaces of discontinuity, which translates into a major constraint for Aerospace applications, susceptibility to spectral bias which makes it difficult to learn regions with sharp gradients, and slow convergence that is avoided by a coupling of ELM with RVDS.

Keywords

• Aerodynamics
• artificial neural networks
• extreme learning machines
• finite difference schemes
• fluid mechanics
• nonlinear differential equations