arXiv:2510.05620v2 Announce Type: replace-cross 
Abstract: The Monte Carlo-type Neural Operator (MCNO) introduces a framework for learning solution operators of one-dimensional partial differential equations (PDEs) by directly learning the kernel function and approximating the associated integral operator using a Monte Carlo-type approach. Unlike Fourier Neural Operators (FNOs), which rely on spectral representations and assume translation-invariant kernels, MCNO makes no such assumptions. The kernel is represented as a learnable tensor over sampled input-output pairs, and sampling is performed once, uniformly at random from a discretized grid. This design enables generalization across multiple grid resolutions without relying on fixed global basis functions or repeated sampling during training, while an interpolation step maps between arbitrary input and output grids to further enhance flexibility. Experiments on standard 1D PDE benchmarks show that MCNO achieves competitive accuracy with efficient computational cost. We also provide a theoretical analysis proving that the Monte Carlo estimator yields a bounded bias and variance under mild regularity assumptions. This result holds in any spatial dimension, suggesting that MCNO may extend naturally beyond one-dimensional problems. More broadly, this work explores how Monte Carlo-type integration can be incorporated into neural operator frameworks for continuous-domain PDEs, providing a theoretically supported alternative to spectral methods (such as FNO) and to graph-based Monte Carlo approaches (such as the Graph Kernel Neural Operator, GNO).

تم تقديم مشغل الشبكة العصبية من نوع مونت كارلو (MCNO) كإطار جديد لتعلم مشغلات الحلول للمعادلات التفاضلية الجزئية (PDE) أحادية البعد. تستخدم هذه الطريقة المبتكرة أسلوب مونت كارلو لتعلم دالة النواة مباشرة، مما يسمح بالتقريب الفعال لمشغلات التكامل دون الافتراضات المتعلقة بالنوى الثابتة بالترجمة الموجودة في مشغلات الشبكة العصبية فورييه (FNO).

Se ha introducido el Monte Carlo-type Neural Operator (MCNO) como un nuevo marco para aprender operadores de solución de ecuaciones diferenciales parciales (EDP) unidimensionales. Este enfoque innovador utiliza un método de tipo Monte Carlo para aprender directamente la función núcleo, permitiendo una aproximación efectiva de los operadores integrales sin las suposiciones de núcleos invariantes por traslación que se encuentran en los Fourier Neural Operators (FNO).

Le Monte Carlo-type Neural Operator (MCNO) a été introduit comme un nouveau cadre pour apprendre les opérateurs de solution des équations différentielles partielles (EDP) unidimensionnelles. Cette approche innovante utilise une méthode de type Monte Carlo pour apprendre directement la fonction noyau, permettant une approximation efficace des opérateurs intégrals sans les hypothèses de noyaux invariants par translation trouvées dans les Fourier Neural Operators (FNO).

The Monte Carlo-type Neural Operator (MCNO) has been introduced as a new framework for learning solution operators of one-dimensional partial differential equations (PDEs). This innovative approach utilizes a Monte Carlo-type method to directly learn the kernel function, allowing for effective approximation of integral operators without the assumptions of translation-invariant kernels found in Fourier Neural Operators (FNOs).

Monte Carlo-Type Neural Operator for Differential Equations

arXiv:2512.07425v1 Announce Type: cross 
Abstract: Real-time monitoring of induced seismicity is crucial for mitigating operational hazards, relying on the rapid and accurate classification of microseismic events from continuous data streams. However, while many deep learning models excel at this task, their high computational requirements often limit their practical application in real-time monitoring systems. To address this limitation, a lightweight model based on the Fourier Neural Operator (FNO) is proposed for microseismic event classification, leveraging its inherent resolution-invariance and computational efficiency for waveform processing. In the STanford EArthquake Dataset (STEAD), a global and large-scale database of seismic waveforms, the FNO-based model demonstrates high effectiveness for trigger classification, with an F1 score of 95% even in the scenario of data sparsity in training. The new FNO model greatly decreases the computer power needed relative to current deep learning models without sacrificing the classification success rate measured by the F1 score. A test on a real microseismic dataset shows a classification success rate with an F1 score of 98%, outperforming many traditional deep-learning techniques. A combination of high success rate and low computational power indicates that the FNO model can serve as a methodology of choice for real-time monitoring of microseismicity for induced seismicity. The method saves computational resources and facilitates both post-processing and real-time seismic processing suitable for the implementation of traffic light systems to prevent undesired induced seismicity.

تم اقتراح نموذج خفيف الوزن يعتمد على مشغل فورييه العصبي (FNO) لتصنيف الأحداث الزلزالية الدقيقة في الوقت الحقيقي، مما يعالج المتطلبات الحاسوبية العالية للنماذج التقليدية للتعلم العميق. وقد أظهر هذا النموذج درجة F1 ملحوظة تبلغ 95% في مجموعة بيانات زلزال ستانفورد، حتى مع ندرة بيانات التدريب.

Se ha propuesto un modelo ligero basado en el Operador Neural de Fourier (FNO) para la clasificación en tiempo real de eventos micro sísmicos, abordando las altas demandas computacionales de los modelos de aprendizaje profundo tradicionales. Este modelo ha mostrado un notable puntaje F1 del 95% en el Conjunto de Datos de Terremotos de Stanford, incluso con datos de entrenamiento escasos.

Un modèle léger basé sur l'Opérateur Neural de Fourier (FNO) a été proposé pour la classification en temps réel des événements micro-sismiques, répondant aux exigences computationnelles élevées des modèles d'apprentissage profond traditionnels. Ce modèle a montré un score F1 remarquable de 95 % dans le jeu de données STanford EArthquake, même avec des données d'entraînement rares.

A lightweight model based on the Fourier Neural Operator (FNO) has been proposed for real-time classification of microseismic events, addressing the high computational demands of traditional deep learning models. This model has shown a remarkable F1 score of 95% in the STanford EArthquake Dataset, even with sparse training data.

Monte Carlo-Type Neural Operator for Differential Equations

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