arXiv:2504.18091v3 Announce Type: replace 
Abstract: Physics-informed neural networks have attracted significant attention in scientific machine learning for their capability to solve forward and inverse problems governed by partial differential equations. However, the accuracy of PINN solutions is often limited by the treatment of boundary conditions. Conventional penalty-based methods, which incorporate boundary conditions as penalty terms in the loss function, cannot guarantee exact satisfaction of the given boundary conditions and are highly sensitive to the choice of penalty parameters. This paper demonstrates that distance functions, specifically R-functions, can be leveraged to enforce boundary conditions, overcoming these limitations. R-functions provide normalized distance fields, enabling flexible representation of boundary geometries, including non-convex domains, and facilitating various types of boundary conditions. Nevertheless, distance functions alone are insufficient for accurate inverse analysis in PINNs. To address this, we propose an integrated framework that combines the normalized distance field with bias-corrected adaptive weight tuning to improve both accuracy and efficiency. Numerical results show that the proposed method provides more accurate and efficient solutions to various inverse problems than penalty-based approaches, even in the presence of non-convex geometries with complex boundary conditions. This approach offers a reliable and efficient framework for inverse analysis using PINNs, with potential applications across a wide range of engineering problems.

تسلط دراسة حديثة الضوء على التقدم في الشبكات العصبية المستندة إلى الفيزياء (PINNs) لحل المشكلات المعقدة في التعلم الآلي العلمي. تتناول هذه البحث التحدي الشائع المتمثل في تطبيق شروط الحدود بدقة، والتي كانت تاريخياً تحد من فعالية PINNs. من خلال تقديم دوال المسافة العادية وضبط الوزن التكيفي، تقترح الدراسة نهجًا أكثر موثوقية وكفاءة للتحليل العكسي. هذا مهم لأنه يعزز من إمكانيات PINNs في تطبيقات متنوعة، مما يمهد الطريق لحلول محسنة في المجالات التي تعتمد على المعادلات التفاضلية الجزئية.

Un estudio reciente destaca los avances en redes neuronales informadas por la física (PINNs) para resolver problemas complejos en el aprendizaje automático científico. Esta investigación aborda el desafío común de aplicar con precisión las condiciones de contorno, que históricamente ha limitado la efectividad de las PINNs. Al introducir funciones de distancia normalizadas y ajuste de peso adaptativo, el estudio propone un enfoque más confiable y eficiente para el análisis inverso. Esto es significativo ya que mejora el potencial de las PINNs en diversas aplicaciones, allanando el camino para soluciones mejoradas en campos que dependen de ecuaciones en derivadas parciales.

Une étude récente met en lumière les avancées des réseaux de neurones informés par la physique (PINNs) pour résoudre des problèmes complexes en apprentissage automatique scientifique. Cette recherche aborde le défi commun d'appliquer avec précision les conditions aux limites, qui a historiquement limité l'efficacité des PINNs. En introduisant des fonctions de distance normalisées et un réglage de poids adaptatif, l'étude propose une approche plus fiable et efficace pour l'analyse inverse. Cela est significatif car cela améliore le potentiel des PINNs dans diverses applications, ouvrant la voie à de meilleures solutions dans des domaines dépendant des équations aux dérivées partielles.

A recent study highlights the advancements in physics-informed neural networks (PINNs) for solving complex problems in scientific machine learning. This research addresses the common challenge of accurately applying boundary conditions, which has historically limited the effectiveness of PINNs. By introducing normalized distance functions and adaptive weight tuning, the study proposes a more reliable and efficient approach to inverse analysis. This is significant as it enhances the potential of PINNs in various applications, paving the way for improved solutions in fields reliant on partial differential equations.

Reliable and efficient inverse analysis using physics-informed neural networks with normalized distance functions and adaptive weight tuning

arXiv:2511.11048v2 Announce Type: replace-cross 
Abstract: 4D flow magnetic resonance imaging (MRI) is a reliable, non-invasive approach for estimating blood flow velocities, vital for cardiovascular diagnostics. Unlike conventional MRI focused on anatomical structures, 4D flow MRI requires high spatiotemporal resolution for early detection of critical conditions such as stenosis or aneurysms. However, achieving such resolution typically results in prolonged scan times, creating a trade-off between acquisition speed and prediction accuracy. Recent studies have leveraged physics-informed neural networks (PINNs) for super-resolution of MRI data, but their practical applicability is limited as the prohibitively slow training process must be performed for each patient. To overcome this limitation, we propose PINGS-X, a novel framework modeling high-resolution flow velocities using axes-aligned spatiotemporal Gaussian representations. Inspired by the effectiveness of 3D Gaussian splatting (3DGS) in novel view synthesis, PINGS-X extends this concept through several non-trivial novel innovations: (i) normalized Gaussian splatting with a formal convergence guarantee, (ii) axes-aligned Gaussians that simplify training for high-dimensional data while preserving accuracy and the convergence guarantee, and (iii) a Gaussian merging procedure to prevent degenerate solutions and boost computational efficiency. Experimental results on computational fluid dynamics (CFD) and real 4D flow MRI datasets demonstrate that PINGS-X substantially reduces training time while achieving superior super-resolution accuracy. Our code and datasets are available at https://github.com/SpatialAILab/PINGS-X.

تم تقديم إطار عمل جديد يسمى PINGS-X لتحسين كفاءة الدقة الفائقة في التصوير بالرنين المغناطيسي بتدفق 4D، وهو أمر حاسم لتقدير سرعات تدفق الدم بدقة في تشخيص القلب والأوعية الدموية. تستخدم هذه الطريقة تمثيلات غاوسية موحدة مستندة إلى الفيزياء مع محاذاة المحاور لمعالجة التحديات المتعلقة بأوقات المسح المطولة والتوازن بين سرعة الاكتساب ودقة التنبؤ.

Se ha introducido un nuevo marco llamado PINGS-X para mejorar la eficiencia de la super-resolución en la IRM 4D, que es crucial para la estimación precisa de las velocidades del flujo sanguíneo en el diagnóstico cardiovascular. Este enfoque utiliza un splatting gaussiano normalizado informado por la física con alineación de ejes para abordar los desafíos de los tiempos de escaneo prolongados y el compromiso entre la velocidad de adquisición y la precisión de las predicciones.

Un nouveau cadre nommé PINGS-X a été introduit pour améliorer l'efficacité de la super-résolution en IRM 4D, qui est cruciale pour une estimation précise des vitesses de flux sanguin dans le diagnostic cardiovasculaire. Cette approche utilise un splatting gaussien normalisé informé par la physique avec alignement des axes pour relever les défis des temps de scan prolongés et du compromis entre la vitesse d'acquisition et la précision des prédictions.

A novel framework named PINGS-X has been introduced to enhance the efficiency of super-resolution in 4D flow MRI, which is crucial for accurate blood flow velocity estimation in cardiovascular diagnostics. This approach utilizes physics-informed normalized Gaussian splatting with axes alignment to address the challenges of prolonged scan times and the trade-off between acquisition speed and prediction accuracy.

Reliable and efficient inverse analysis using physics-informed neural networks with normalized distance functions and adaptive weight tuning

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