Evolution of nonlinear reduced-order solutions for PDEs with conserved quantities

TL;DR

This paper introduces RONS, a unified framework for deriving nonlinear reduced-order models of PDEs that can incorporate conserved quantities and are applicable to a broad class of problems, demonstrated on three diverse equations.

Contribution

The paper presents RONS, a novel method for constructing nonlinear reduced-order models that generalize Galerkin projection and enforce conserved quantities.

Findings

RONS effectively models PDEs with nonlinear parameter dependence.

The framework can incorporate conserved quantities into reduced models.

Demonstrated success on advection-diffusion, nonlinear Schrödinger, and Euler equations.

Abstract

Reduced-order models of time-dependent partial differential equations (PDEs) where the solution is assumed as a linear combination of prescribed modes are rooted in a well-developed theory. However, more general models where the reduced solutions depend nonlinearly on time varying parameters have thus far been derived in an ad hoc manner. Here, we introduce Reduced-order Nonlinear Solutions (RONS): a unified framework for deriving reduced-order models that depend nonlinearly on a set of time-dependent parameters. The set of all possible reduced-order solutions are viewed as a manifold immersed in the function space of the PDE. The parameters are evolved such that the instantaneous discrepancy between reduced dynamics and the full PDE dynamics is minimized. This results in a set of explicit ordinary differential equations on the tangent bundle of the manifold. In the special case of linear parameter dependence, our reduced equations coincide with the standard Galerkin projection. Furthermore, any number of conserved quantities of the PDE can readily be enforced in our framework. Since RONS does not assume an underlying variational formulation for the PDE, it is applicable to a broad class of problems. We demonstrate the efficacy of RONS on three examples: an advection-diffusion equation, the nonlinear Schrodinger equation and Euler's equation for ideal fluids.