Physics-driven proper orthogonal decomposition: A simulation methodology for partial differential equations

Alessandro Pulimeno; Graham Coates-Farley; Martin Veresko; Lin Jiang; Ming-Cheng Cheng; Yu Liu; Daqing Hou

doi:10.1016/j.mex.2023.102204

Physics-driven proper orthogonal decomposition: A simulation methodology for partial differential equations

MethodsX. 2023 Apr 28:10:102204. doi: 10.1016/j.mex.2023.102204. eCollection 2023.

Authors

Alessandro Pulimeno^{1

2}, Graham Coates-Farley³, Martin Veresko³, Lin Jiang³, Ming-Cheng Cheng³, Yu Liu³, Daqing Hou³

Affiliations

¹ Department of Mechanical and Aerospace Engineering, Clarkson University, Potsdam, NY 13699, USA.
² Current affiliation, Department of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands.
³ Department of Electrical and Computing Engineering, Clarkson University, Potsdam, NY 13699, USA.

Abstract

A simulation methodology derived from a learning algorithm based on Proper Orthogonal Decomposition (POD) is presented to solve partial differential equations (PDEs) for physical problems of interest. Using the developed methodology, a physical problem of interest is projected onto a functional space described by a set of basis functions (or POD modes) that are trained via the POD by solution data collected from direct numerical simulations (DNSs) of the PDE. The Galerkin projection of the PDE is then performed to account for physical principles guided by the PDE. The procedure to construct the physics-driven POD-Galerkin simulation methodology is presented in detail, together with demonstrations of POD-Galerkin simulations of dynamic thermal analysis on a microprocessor and the Schrödinger equation for a quantum nanostructure. The physics-driven methodology allows a reduction of several orders in degrees of freedom (DoF) while maintaining high accuracy. This leads to a drastic decrease in computational effort when compared with DNS. The major steps for implementing the methodology include:•Solution data collection from DNSs of the physical problem subjected to parametric variations of the system.•Calculations of POD modes and eigenvalues from the collected data using the method of snapshots.•Galerkin projection of the governing equation onto the POD space to derive the model.

Keywords: Galerkin projection; Heat transfer; Machine learning; Partial differential equations; Physics-driven simulation methodology based on Proper Orthogonal Decomposition enabled by Galerkin projection (POD-GP); Quantum nanostructures.