arXiv:2408.00681v2 Announce Type: replace 
Abstract: We introduce a novel deep operator network (DeepONet) framework that incorporates generalised variational inference (GVI) using R\'enyi's $\alpha$-divergence to learn complex operators while quantifying uncertainty. By incorporating Bayesian neural networks as the building blocks for the branch and trunk networks, our framework endows DeepONet with uncertainty quantification. The use of R\'enyi's $\alpha$-divergence, instead of the Kullback-Leibler divergence (KLD), commonly used in standard variational inference, mitigates issues related to prior misspecification that are prevalent in Variational Bayesian DeepONets. This approach offers enhanced flexibility and robustness. We demonstrate that modifying the variational objective function yields superior results in terms of minimising the mean squared error and improving the negative log-likelihood on the test set. Our framework's efficacy is validated across various mechanical systems, where it outperforms both deterministic and standard KLD-based VI DeepONets in predictive accuracy and uncertainty quantification. The hyperparameter $\alpha$, which controls the degree of robustness, can be tuned to optimise performance for specific problems. We apply this approach to a range of mechanics problems, including gravity pendulum, advection-diffusion, and diffusion-reaction systems. Our findings underscore the potential of $\alpha$-VI DeepONet to advance the field of data-driven operator learning and its applications in engineering and scientific domains.

تم تقديم إطار جديد لشبكة المشغلين العميقة، Alpha-VI DeepONet، الذي يستخدم الاستدلال التبايني العام مع تباين α لـ R'enyi لتعزيز قياس عدم اليقين في تعلم المشغلين المعقدين. يدمج هذا الإطار الشبكات العصبية البايزية في هيكله، مما يعالج مشكلات تحديد الأولويات الشائعة في الأساليب البايزية التباينية التقليدية.

Se ha introducido un nuevo marco de red de operadores profundos, Alpha-VI DeepONet, que utiliza inferencia variacional generalizada con la divergencia α de R'enyi para mejorar la cuantificación de la incertidumbre en el aprendizaje de operadores complejos. Este marco integra redes neuronales bayesianas en su arquitectura, abordando problemas de especificación errónea de priors comunes en enfoques bayesianos variacionales tradicionales.

Un nouveau cadre de réseau d'opérateurs profonds, Alpha-VI DeepONet, a été introduit, utilisant l'inférence variationnelle généralisée avec la divergence α de R'enyi pour améliorer la quantification de l'incertitude dans l'apprentissage d'opérateurs complexes. Ce cadre intègre des réseaux neuronaux bayésiens dans son architecture, abordant les problèmes de spécification erronée des priors courants dans les approches bayésiennes variationnelles traditionnelles.

A novel deep operator network framework, Alpha-VI DeepONet, has been introduced, utilizing generalized variational inference with R'enyi's α-divergence to enhance uncertainty quantification in learning complex operators. This framework integrates Bayesian neural networks into its architecture, addressing prior misspecification issues common in traditional variational Bayesian approaches.

Alpha-VI DeepONet: A prior-robust variational Bayesian approach for enhancing DeepONets with uncertainty quantification

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Alpha-VI DeepONet: A prior-robust variational Bayesian approach for enhancing DeepONets with uncertainty quantification

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