# A Reduced Order Modeling technique to study bifurcating phenomena: application to the Gross-Pitaevskii equation

## TL;DR

This paper introduces a computationally efficient reduced order modeling framework that combines continuation, Newton's method, and hyper-reduction to study bifurcations in nonlinear PDEs, demonstrated on the Gross-Pitaevskii equation.

## Contribution

The paper presents a novel ROM-based approach that significantly accelerates bifurcation analysis for nonlinear PDEs, especially for the Gross-Pitaevskii equation.

## Key findings

- 60 times faster bifurcation diagram construction compared to full order methods
- Effective tracing of steady solution branches with reduced computational cost
- Applicable to multi-parameter bifurcation studies

## Abstract

We propose a computationally efficient framework to treat nonlinear partial differential equations having bifurcating solutions as one or more physical control parameters are varied. Our focus is on steady bifurcations. Plotting a bifurcation diagram entails computing multiple solutions of a parametrized, nonlinear problem, which can be extremely expensive in terms of computational time. In order to reduce these demanding computational costs, our approach combines a continuation technique and Newton's method with a Reduced Order Modeling (ROM) technique, suitably supplemented with a hyper-reduction method. To demonstrate the effectiveness of our ROM approach, we trace the steady solution branches of a nonlinear Schr\"{o}dinger equation, called Gross-Pitaevskii equation, as one or two physical parameters are varied. In the two parameter study, we show that our approach is 60 times faster in constructing a bifurcation diagram than a standard Full Order Method.