arXiv:2511.16573v1 Announce Type: cross 
Abstract: Neural operators have demonstrated considerable effectiveness in accelerating the solution of time-dependent partial differential equations (PDEs) by directly learning governing physical laws from data. However, for PDEs governed by conservation laws(e.g., conservation of mass, energy, or matter), existing neural operators fail to satisfy conservation properties, which leads to degraded model performance and limited generalizability. Moreover, we observe that distinct PDE problems generally require different optimal neural network architectures. This finding underscores the inherent limitations of specialized models in generalizing across diverse problem domains.
  To address these limitations, we propose Exterior-Embedded Conservation Framework (ECF), a universal conserving framework that can be integrated with various data-driven neural operators to enforce conservation laws strictly in predictions. The framework consists of two key components: a conservation quantity encoder that extracts conserved quantities from input data, and a conservation quantity decoder that adjusts the neural operator's predictions using these quantities to ensure strict conservation compliance in the final output. Since our architecture enforces conservation laws, we theoretically prove that it enhances model performance. To validate the performance of our method, we conduct experiments on multiple conservation-law-constrained PDE scenarios, including adiabatic systems, shallow water equations, and the Allen-Cahn problem. These baselines demonstrate that our method effectively improves model accuracy while strictly enforcing conservation laws in the predictions.

تم اقتراح إطار الحفاظ على التكامل الخارجي (ECF) لتحسين قدرة المشغلين العصبيين على حل المعادلات التفاضلية الجزئية (PDEs) المعتمدة على الزمن مع الحفاظ على قوانين الحفظ. تواجه المشغلون العصبيون التقليديون صعوبات في الحفاظ على هذه القوانين، مما يؤثر على أدائهم وقابلية تعميمهم. يهدف ECF إلى الاندماج مع مجموعة متنوعة من المشغلين العصبيين المعتمدين على البيانات لضمان الالتزام الصارم بمبادئ الحفظ في التنبؤات.

Se propone el Marco de Conservación Integrado Exterior (ECF) para mejorar la capacidad de los operadores neuronales para resolver ecuaciones diferenciales parciales (EDP) dependientes del tiempo mientras se preservan las leyes de conservación. Los operadores neuronales tradicionales tienen dificultades para mantener estas leyes, lo que afecta su rendimiento y generalizabilidad. El ECF busca integrarse con varios operadores neuronales impulsados por datos para garantizar el cumplimiento estricto de los principios de conservación en las predicciones.

Le cadre de conservation intégré extérieur (ECF) est proposé pour améliorer la capacité des opérateurs neuronaux à résoudre des équations aux dérivées partielles (EDP) dépendant du temps tout en préservant les lois de conservation. Les opérateurs neuronaux traditionnels ont du mal à maintenir ces lois, ce qui affecte leur performance et leur généralisabilité. L'ECF vise à s'intégrer à divers opérateurs neuronaux basés sur des données pour garantir le respect strict des principes de conservation dans les prédictions.

The Exterior-Embedded Conservation Framework (ECF) is proposed to enhance neural operators' ability to solve time-dependent partial differential equations (PDEs) while preserving conservation laws. Traditional neural operators struggle to maintain these laws, which affects their performance and generalizability. The ECF aims to integrate with various data-driven neural operators to ensure strict adherence to conservation principles in predictions.

An Exterior-Embedding Neural Operator Framework for Preserving Conservation Laws

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

arXiv:2511.18777v1 Announce Type: new 
Abstract: Neural operators have shown great potential in solving a family of Partial Differential Equations (PDEs) by modeling the mappings between input and output functions. Fourier Neural Operator (FNO) implements global convolutions via parameterizing the integral operators in Fourier space. However, it often results in over-smoothing solutions and fails to capture local details and high-frequency components. To address these limitations, we investigate incorporating the spatial-frequency localization property of Wavelet transforms into the Transformer architecture. We propose a novel Wavelet Attention (WA) module with linear computational complexity to efficiently learn locality-aware features. Building upon WA, we further develop the Spectral Attention Operator Transformer (SAOT), a hybrid spectral Transformer framework that integrates WA's localized focus with the global receptive field of Fourier-based Attention (FA) through a gated fusion block. Experimental results demonstrate that WA significantly mitigates the limitations of FA and outperforms existing Wavelet-based neural operators by a large margin. By integrating the locality-aware and global spectral representations, SAOT achieves state-of-the-art performance on six operator learning benchmarks and exhibits strong discretization-invariant ability.

تم تقديم Spectral Attention Operator Transformer (SAOT) كإطار مبتكر يعزز قدرات المشغلين العصبيين في حل المعادلات التفاضلية الجزئية (PDEs). من خلال دمج الانتباه الموجي مع الانتباه القائم على فورييه، يتناول SAOT قيود الأساليب الحالية، لا سيما في التقاط التفاصيل المحلية والمكونات عالية التردد في الحلول.

Se ha presentado el Spectral Attention Operator Transformer (SAOT) como un marco innovador que mejora las capacidades de los operadores neuronales en la resolución de ecuaciones en derivadas parciales (EDP). Al integrar la Atención por Ondas con la Atención basada en Fourier, el SAOT aborda las limitaciones de los métodos existentes, especialmente en la captura de detalles locales y componentes de alta frecuencia en las soluciones.

Le Spectral Attention Operator Transformer (SAOT) a été présenté comme un cadre innovant qui améliore les capacités des opérateurs neuronaux dans la résolution des équations aux dérivées partielles (EDP). En intégrant l'attention par ondelettes avec l'attention basée sur Fourier, le SAOT répond aux limitations des méthodes existantes, en particulier dans la capture des détails locaux et des composants haute fréquence dans les solutions.

The Spectral Attention Operator Transformer (SAOT) has been introduced as an innovative framework that enhances the capabilities of neural operators in solving Partial Differential Equations (PDEs). By integrating Wavelet Attention with Fourier-based Attention, SAOT addresses the limitations of existing methods, particularly in capturing local details and high-frequency components in solutions.

An Exterior-Embedding Neural Operator Framework for Preserving Conservation Laws

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