A Normal Equation-Based Extreme Learning Machine for Solving Linear Partial Differential Equations

Vikas Dwivedi, Balaji Srinivasan

2021Journal of Computing and Information Science in Engineering21 citationsDOI

Abstract

Abstract This paper develops an extreme learning machine for solving linear partial differential equations (PDEs) 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 use 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), and 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.

Topics & Concepts

Partial differential equationArtificial neural networkOrdinary differential equationExtreme learning machineOdeApplied mathematicsComputer scienceMathematicsNumerical partial differential equationsFunction (biology)Partial derivativeBoundary value problemDifferential equationMathematical optimizationMathematical analysisArtificial intelligenceBiologyEvolutionary biologyModel Reduction and Neural NetworksMachine Learning and ELMAdvancements in Semiconductor Devices and Circuit Design