Lagrangian Data-Driven Reduced Order Modeling of Finite Time Lyapunov Exponents

TL;DR

This paper introduces new Lagrangian inner products for reduced order modeling that significantly improve the accuracy of approximating both Lagrangian and Eulerian fields in complex systems without additional closure modeling.

Contribution

The paper proposes novel Lagrangian inner products for ROM basis construction, leading to substantially more accurate Lagrangian ROMs compared to standard Eulerian approaches.

Findings

Lagrangian ROMs are orders of magnitude more accurate than Eulerian ROMs.

The new ROMs accurately approximate both Lagrangian and Eulerian fields.

No closure modeling is needed for the improved accuracy.

Abstract

There are two main strategies for improving the projection-based reduced order model (ROM) accuracy: (i) improving the ROM, i.e., adding new terms to the standard ROM; and (ii) improving the ROM basis, i.e., constructing ROM bases that yield more accurate ROMs. In this paper, we use the latter. We propose new Lagrangian inner products that we use together with Eulerian and Lagrangian data to construct new Lagrangian ROMs. We show that the new Lagrangian ROMs are orders of magnitude more accurate than the standard Eulerian ROMs, i.e., ROMs that use standard Eulerian inner product and data to construct the ROM basis. Specifically, for the quasi-geostrophic equations, we show that the new Lagrangian ROMs are more accurate than the standard Eulerian ROMs in approximating not only Lagrangian fields (e.g., the finite time Lyapunov exponent (FTLE)), but also Eulerian fields (e.g., the streamfunction). We emphasize that the new Lagrangian ROMs do not employ any closure modeling to model the effect of discarded modes (which is standard procedure for low-dimensional ROMs of complex nonlinear systems). Thus, the dramatic increase in the new Lagrangian ROMs' accuracy is entirely due to the novel Lagrangian inner products used to build the Lagrangian ROM basis.