# Calibrated Filtered Reduced Order Modeling

## TL;DR

The paper introduces a calibrated filtered reduced order model (CF-ROM) framework that enhances nonlinear PDE simulations by combining spatial filtering with a calibration process to model unresolved interactions.

## Contribution

It presents a novel CF-ROM framework that uses explicit filtering and calibration to close reduced order models for nonlinear PDEs, applicable across various equations.

## Key findings

- Effective closure of filtered ROMs achieved through calibration.
- Framework demonstrated in fluid dynamics but applicable broadly.
- Optimization-based coefficient determination improves model accuracy.

## Abstract

We propose a calibrated filtered reduced order model (CF-ROM) framework for the numerical simulation of general nonlinear PDEs that are amenable to reduced order modeling. The novel CF-ROM framework consists of two steps: (i) In the first step, we use explicit ROM spatial filtering of the nonlinear PDE to construct a filtered ROM. This filtered ROM is low-dimensional, but is not closed (because of the nonlinearity in the given PDE). (ii) In the second step, we use a calibration procedure to close the filtered ROM, i.e., to model the interaction between the resolved and unresolved modes. To this end, we use a linear or quadratic ansatz to model this interaction and close the filtered ROM. To find the new coefficients in the closed filtered ROM, we solve an optimization problem that minimizes the difference between the full order model data and our ansatz. Although we use a fluid dynamics setting to illustrate how to construct and use the CF-ROM framework, we emphasize that it is built on general ideas of spatial filtering and optimization and is independent of (restrictive) phenomenological arguments. Thus, the CF-ROM framework can be applied to a wide variety of PDEs.