arXiv:2603.13496v1 Announce Type: new 
Abstract: Constructing reduced-order models (ROMs) capable of efficiently predicting the evolution of high-dimensional, parametric systems is crucial in many applications in engineering and applied sciences. A popular class of projection-based ROMs projects the high-dimensional full-order model (FOM) dynamics onto a low-dimensional manifold. These projection-based ROMs approaches often rely on classical model reduction techniques such as proper orthogonal decomposition (POD) or, more recently, on neural network architectures such as autoencoders (AEs). In the case that the ROM is constructed by the POD, one has approximation guaranteed based based on the singular values of the problem at hand. However, POD-based techniques can suffer from slow decay of the singular values in transport- and advection-dominated problems. In contrast to that, AEs allow for better reduction capabilities than the POD, often with the first few modes, but at the price of theoretical considerations. In addition, it is often observed, that AEs exhibits a plateau of the projection error with the increment of the dimension of the trial manifold. In this work, we propose a deep invertible AE architecture, named inv-AE, that improves upon the stagnation of the projection error typical of traditional AE architectures, e.g., convolutional, and the reconstructions quality. Inv-AE is composed of several invertible neural network layers that allows for gradually recovering more information about the FOM solutions the more we increase the dimension of the reduced manifold. Through the application of inv-AE to a parametric 1D Burgers' equation and a parametric 2D fluid flow around an obstacle with variable geometry, we show that (i) inv-AE mitigates the issue of the characteristic plateau of (convolutional) AEs and (ii) inv-AE can be combined with popular projection-based ROM approaches to improve their accuracy.

قدمت دراسة جديدة Autoencoders القابلة للعكس العميقة لتقليل الأبعاد للأنظمة الديناميكية، حيث تركز على بناء نماذج منخفضة الترتيب (ROM) يمكنها التنبؤ بكفاءة بتطور الأنظمة البارامترية عالية الأبعاد. يتناقض هذا النهج مع التقنيات التقليدية مثل التحليل المتعامد المناسب (POD)، حيث يقدم قدرات تقليل أفضل من خلال هياكل الشبكات العصبية.

Un nuevo estudio ha presentado autoencoders invertibles profundos para la reducción de dimensionalidad de sistemas dinámicos, centrándose en la construcción de modelos de orden reducido (ROM) que pueden predecir de manera eficiente la evolución de sistemas paramétricos de alta dimensión. Este enfoque contrasta con técnicas tradicionales como la descomposición ortogonal propia (POD), ofreciendo mejores capacidades de reducción a través de arquitecturas de redes neuronales.

Une nouvelle étude a introduit des autoencodeurs inversibles profonds pour la réduction de dimension des systèmes dynamiques, se concentrant sur la construction de modèles d'ordre réduit (ROM) capables de prédire efficacement l'évolution de systèmes paramétriques de haute dimension. Cette approche contraste avec les techniques traditionnelles comme la décomposition orthogonale propre (POD), offrant de meilleures capacités de réduction grâce à des architectures de réseaux neuronaux.

A new study has introduced Deep Invertible Autoencoders for Dimensionality Reduction of Dynamical Systems, focusing on constructing reduced-order models (ROMs) that can efficiently predict the evolution of high-dimensional, parametric systems. This approach contrasts with traditional techniques like proper orthogonal decomposition (POD), offering improved reduction capabilities through neural network architectures.