arXiv:2510.21592v2 Announce Type: replace 
Abstract: Data-driven deep learning methods like neural operators have advanced in solving nonlinear temporal partial differential equations (PDEs). However, these methods require large quantities of solution pairs\u2014the solution functions and right-hand sides (RHS) of the equations. These pairs are typically generated via traditional numerical methods, which need thousands of time steps iterations far more than the dozens required for training, creating heavy computational and temporal overheads. To address these challenges, we propose a novel data generation algorithm, called HOmologous Perturbation in Solution Space (HOPSS), which directly generates training datasets with fewer time steps rather than following the traditional approach of generating large time steps datasets. This algorithm simultaneously accelerates dataset generation and preserves the approximate precision required for model training. Specifically, we first obtain a set of base solution functions from a reliable solver, usually with thousands of time steps, and then align them in time steps with training datasets by downsampling. Subsequently, we propose a "homologous perturbation" approach: by combining two solution functions (one as the primary function, the other as a homologous perturbation term scaled by a small scalar) with random noise, we efficiently generate comparable-precision PDE data points. Finally, using these data points, we compute the variation in the original equation's RHS to form new solution pairs. Theoretical and experimental results show HOPSS lowers time complexity. For example, on the Navier-Stokes equation, it generates 10,000 samples in approximately 10% of traditional methods' time, with comparable model training performance.

تُحقق التطورات الأخيرة في أساليب التعلم العميق المدفوعة بالبيانات، وخاصة المشغلين العصبيين، تقدمًا كبيرًا في حل المعادلات التفاضلية الجزئية الزمنية غير الخطية. هذا أمر بالغ الأهمية لأن هذه الأساليب تعتمد تقليديًا على توليد كميات كبيرة من أزواج الحلول من خلال تقنيات عددية تقليدية، والتي يمكن أن تستغرق وقتًا طويلاً. من خلال تسريع هذه العملية، يمكن للباحثين تحسين كفاءة وفعالية تدريب النماذج، مما يؤدي في النهاية إلى حلول أسرع وأكثر دقة في مختلف التطبيقات العلمية والهندسية.

Los recientes avances en métodos de aprendizaje profundo impulsados por datos, especialmente los operadores neuronales, están logrando avances significativos en la resolución de ecuaciones diferenciales parciales temporales no lineales (EDP). Esto es crucial porque estos métodos dependen tradicionalmente de la generación de grandes cantidades de pares de soluciones a través de técnicas numéricas convencionales, que pueden ser muy lentas. Al acelerar este proceso de generación de datos, los investigadores pueden mejorar la eficiencia y efectividad del entrenamiento de modelos, lo que lleva a soluciones más rápidas y precisas en diversas aplicaciones científicas y de ingeniería.

Les avancées récentes dans les méthodes d'apprentissage profond basées sur les données, en particulier les opérateurs neuronaux, font des progrès significatifs dans la résolution des équations aux dérivées partielles (EDP) temporelles non linéaires. Cela est crucial car ces méthodes dépendent traditionnellement de la génération de grandes quantités de paires de solutions par des techniques numériques conventionnelles, ce qui peut prendre beaucoup de temps. En accélérant ce processus de génération de données, les chercheurs peuvent améliorer l'efficacité et l'efficacité de l'entraînement des modèles, conduisant finalement à des solutions plus rapides et plus précises dans diverses applications scientifiques et techniques.

Recent advancements in data-driven deep learning methods, particularly neural operators, are making significant strides in solving nonlinear temporal partial differential equations (PDEs). This is crucial because these methods traditionally rely on generating large quantities of solution pairs through conventional numerical techniques, which can be time-consuming. By accelerating this data generation process, researchers can enhance the efficiency and effectiveness of training models, ultimately leading to faster and more accurate solutions in various scientific and engineering applications.

Accelerating Data Generation for Nonlinear temporal PDEs via homologous perturbation in solution space

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Accelerating Data Generation for Nonlinear temporal PDEs via homologous perturbation in solution space

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