arXiv:2511.18260v1 Announce Type: new 
Abstract: Parametric PDEs power modern simulation, design, and digital-twin systems, yet their many-query workloads still hinge on repeatedly solving large finite-element systems. Existing operator-learning approaches accelerate this process but often rely on opaque learned trunks, require extensive labeled data, or break down when boundary and source data vary independently from physical parameters. We introduce RB-DeepONet, a hybrid operator-learning framework that fuses reduced-basis (RB) numerical structure with the branch-trunk architecture of DeepONet. The trunk is fixed to a rigorously constructed RB space generated offline via Greedy selection, granting physical interpretability, stability, and certified error control. The branch network predicts only RB coefficients and is trained label-free using a projected variational residual that targets the RB-Galerkin solution. For problems with independently varying loads or boundary conditions, we develop boundary and source modal encodings that compress exogenous data into low-dimensional coordinates while preserving accuracy. Combined with affine or empirical interpolation decompositions, RB-DeepONet achieves a strict offline-online split: all heavy lifting occurs offline, and online evaluation scales only with the RB dimension rather than the full mesh. We provide convergence guarantees separating RB approximation error from statistical learning error, and numerical experiments show that RB-DeepONet attains accuracy competitive with intrusive RB-Galerkin, POD-DeepONet, and FEONet while using dramatically fewer trainable parameters and achieving significant speedups. This establishes RB-DeepONet as an efficient, stable, and interpretable operator learner for large-scale parametric PDEs.

تم تقديم إطار جديد يسمى RB-DeepONet لتحسين كفاءة حل المعادلات التفاضلية الجزئية (PDEs) البارامترية مع بيانات الحدود والمصدر المتغيرة بشكل مستقل. يجمع هذا النموذج الهجين لتعلم المشغلين بين الهياكل العددية ذات الأساس المخفض مع بنية DeepONet، مما يسمح بتحسين التفسير الفيزيائي والاستقرار مع ضمان التحكم المعتمد في الأخطاء.

Se ha introducido un nuevo marco llamado RB-DeepONet para mejorar la eficiencia en la resolución de ecuaciones diferenciales parciales (EDP) paramétricas con datos de frontera y fuente que varían independientemente. Este modelo híbrido de aprendizaje de operadores combina estructuras numéricas de base reducida con la arquitectura DeepONet, permitiendo una mejor interpretabilidad física y estabilidad, al tiempo que garantiza un control de error certificado.

Un nouveau cadre appelé RB-DeepONet a été introduit pour améliorer l'efficacité de la résolution des équations aux dérivées partielles (EDP) paramétriques avec des données de frontière et de source variant indépendamment. Ce modèle hybride d'apprentissage d'opérateurs combine des structures numériques à base réduite avec l'architecture DeepONet, permettant une meilleure interprétabilité physique et stabilité tout en garantissant un contrôle d'erreur certifié.

A new framework called RB-DeepONet has been introduced to enhance the efficiency of solving parametric partial differential equations (PDEs) with independently varying boundary and source data. This hybrid operator-learning model combines reduced-basis numerical structures with the DeepONet architecture, allowing for improved physical interpretability and stability while ensuring certified error control.

Reduced-Basis Deep Operator Learning for Parametric PDEs with Independently Varying Boundary and Source Data

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