# A DeepONet multi-fidelity approach for residual learning in reduced order modeling

## TL;DR

This paper introduces a multi-fidelity DeepONet approach to improve reduced order models by learning residual errors, combining model reduction techniques with neural networks for enhanced accuracy in real-time simulations.

## Contribution

It presents a novel framework integrating DeepONets with multi-fidelity data to learn residual errors in reduced order models, improving their accuracy and efficiency.

## Key findings

- Enhanced reduced order models with learned residuals.
- Successful application to Navier-Stokes and benchmark functions.
- Demonstrated improved accuracy over traditional methods.

## Abstract

In the present work, we introduce a novel approach to enhance the precision of reduced order models by exploiting a multi-fidelity perspective and DeepONets. Reduced models provide a real-time numerical approximation by simplifying the original model. The error introduced by the such operation is usually neglected and sacrificed in order to reach a fast computation. We propose to couple the model reduction to a machine learning residual learning, such that the above-mentioned error can be learned by a neural network and inferred for new predictions. We emphasize that the framework maximizes the exploitation of high-fidelity information, using it for building the reduced order model and for learning the residual. In this work, we explore the integration of proper orthogonal decomposition (POD), and gappy POD for sensors data, with the recent DeepONet architecture. Numerical investigations for a parametric benchmark function and a nonlinear parametric Navier-Stokes problem are presented.