arXiv:2511.13734v1 Announce Type: new 
Abstract: The accurate solution of nonlinear hyperbolic partial differential equations (PDEs) remains a central challenge in computational science due to the presence of steep gradients, discontinuities, and multiscale structures that make conventional discretization-based solvers computationally demanding. Physics-Informed Neural Networks (PINNs) embed the governing equations into the learning process, enabling mesh-free solution of PDEs, yet they often struggle to capture steep gradients, discontinuities, and complex nonlinear wave interactions. To address these limitations, this study employs the Extended Physics-Informed Neural Network (XPINN) framework to solve the nonlinear Buckley-Leverett equation with a nonconvex flux function, which models immiscible two-phase flow in porous media. The computational domain is dynamically decomposed in space and time into evolving pre-shock and post-shock regions, allowing localized subnetworks to efficiently learn distinct flow behaviors. Coupling between subnetworks is achieved through the Rankine-Hugoniot jump condition, which enforces physically consistent flux continuity across the moving shock interface. Numerical experiments demonstrate that the proposed XPINN approach accurately captures discontinuous saturation fronts and compound wave interactions without requiring artificial diffusion or entropy corrections. Compared to standard PINNs, the XPINN framework achieves superior stability, faster convergence, and enhanced resolution of nonlinear wave dynamics using smaller, domain-specific models with fewer trainable parameters, establishing it as an effective and scalable tool for solving challenging hyperbolic PDEs in multiphase flow problems. The code of this work is available on github.com/saifkhanengr/XPINN-for-Buckley-Leverett.

تقدم الدراسة إطار شبكة عصبية مستنيرة بالفيزياء الموسعة (XPINN) تهدف إلى حل المعادلات التفاضلية الجزئية غير الخطية ذات الطابع الهيبروليكي، وخاصة معادلة باكلي-ليفريت، التي تُعَد نموذجًا لتدفق مرحلتين غير القابلة للامتزاج في الوسائط المسامية. تواجه الطرق التقليدية تحديات بسبب التدرجات الحادة والانقطاعات. يسمح نهج XPINN بحل بدون شبكة من خلال تقسيم المجال الحسابي ديناميكيًا إلى مناطق متطورة، مما يعزز القدرة على التقاط التفاعلات الموجية المعقدة بشكل فعال.

El estudio presenta un marco de Red Neuronal Informada por la Física Extendida (XPINN) destinado a resolver ecuaciones diferenciales parciales hiperbólicas no lineales, en particular la ecuación de Buckley-Leverett, que modela el flujo de dos fases inmiscibles en medios porosos. Los métodos tradicionales enfrentan desafíos debido a gradientes pronunciados y discontinuidades. El enfoque XPINN permite una solución sin malla al descomponer dinámicamente el dominio computacional en regiones evolutivas, mejorando la capacidad para capturar interacciones de ondas complejas.

L'étude présente un cadre de Réseau de Neurones Informés par la Physique Étendu (XPINN) visant à résoudre des équations aux dérivées partielles hyperboliques non linéaires, en particulier l'équation de Buckley-Leverett, qui modélise l'écoulement de deux phases immiscibles dans des milieux poreux. Les méthodes traditionnelles rencontrent des difficultés en raison de gradients abrupts et de discontinuités. L'approche XPINN permet une solution sans maillage en décomposant dynamiquement le domaine de calcul en régions évolutives, améliorant ainsi la capacité à capturer des interactions d'ondes com…

The study presents an Extended Physics-Informed Neural Network (XPINN) framework aimed at solving nonlinear hyperbolic partial differential equations (PDEs), particularly the Buckley-Leverett equation, which models immiscible two-phase flow in porous media. Traditional methods face challenges due to steep gradients and discontinuities. The XPINN approach allows for a mesh-free solution by dynamically decomposing the computational domain into evolving regions, enhancing the ability to capture complex wave interactions effectively.

Extended Physics Informed Neural Network for Hyperbolic Two-Phase Flow in Porous Media

arXiv:2507.10241v2 Announce Type: replace 
Abstract: Physics-informed machine learning frameworks such as Physics-Informed Neural Networks (PINNs) and Physics-Informed Extreme Learning Machines (PI-ELMs) have shown great promise for solving partial differential equations (PDEs) but struggle with localized sharp gradients and singularly perturbed regimes, PINNs due to spectral bias and PI-ELMs due to their single-shot, non-adaptive formulation. We propose the Kernel-Adaptive Physics-Informed Extreme Learning Machine (KAPI-ELM), which performs Bayesian optimization over a low-dimensional, physically interpretable hyperparameter space governing the distribution of Radial Basis Function (RBF) centers and widths. This converts high-dimensional weight optimization into a low-dimensional distributional search, enabling targeted kernel refinement in regions with sharp gradients while also improving baseline solutions in smooth-flow regimes by tuning RBF supports. KAPI-ELM is validated on benchmark forward and inverse problems (1D convection-diffusion and 2D Poisson) involving PDEs with sharp gradients. It accurately resolves steep layers, improves smooth-solution fidelity, and recovers physical parameters robustly, matching or surpassing advanced methods such as the extended Theory of Functional Connections (X-TFC) with nearly an order of magnitude fewer tunable parameters. An extension to nonlinear problems is demonstrated by a curriculum-based solution of the steady Navier-Stokes equations via successive linearizations, yielding stable solutions for benchmark lid-driven cavity flow up to Re=100. These results indicate that KAPI-ELM provides an efficient and unified approach for forward and inverse PDEs, particularly in challenging sharp-gradient regimes.

يقدم نظام Kernel-Adaptive Physics-Informed Extreme Learning Machine (KAPI-ELM) حلاً للتحديات المتعلقة بحل المعادلات التفاضلية الجزئية (PDEs) ذات التدرجات الحادة، مما يعزز من قدرات أطر التعلم الآلي المستندة إلى الفيزياء مثل PINNs وPI-ELMs. تستخدم هذه الطريقة الجديدة تحسين بايزي لتعديل معلمات دالة القاعدة الشعاعية (RBF)، مما يحسن الأداء في كل من السيناريوهات ذات التدرجات الحادة والانسياب السلس.

La introducción del Kernel-Adaptive Physics-Informed Extreme Learning Machine (KAPI-ELM) aborda los desafíos en la resolución de ecuaciones en derivadas parciales (EDP) con gradientes agudos, mejorando las capacidades de los marcos de aprendizaje automático informados por la física como los PINNs y PI-ELMs. Este nuevo enfoque utiliza la optimización bayesiana para refinar los parámetros de la función de base radial (RBF), mejorando el rendimiento tanto en escenarios de gradientes agudos como en flujos suaves.

L'introduction du Kernel-Adaptive Physics-Informed Extreme Learning Machine (KAPI-ELM) répond aux défis de la résolution des équations aux dérivées partielles (EDP) avec des gradients aigus, améliorant ainsi les capacités des cadres d'apprentissage automatique informés par la physique tels que les PINNs et les PI-ELMs. Cette nouvelle approche utilise l'optimisation bayésienne pour affiner les paramètres de la fonction de base radiale (RBF), améliorant les performances dans les scénarios à gradients aigus et à flux lisse.

The introduction of the Kernel-Adaptive Physics-Informed Extreme Learning Machine (KAPI-ELM) addresses challenges in solving partial differential equations (PDEs) with sharp gradients, enhancing the capabilities of existing physics-informed machine learning frameworks like PINNs and PI-ELMs. This new approach utilizes Bayesian optimization to refine Radial Basis Function (RBF) parameters, improving performance in both sharp gradient and smooth-flow scenarios.

Kernel-Adaptive PI-ELMs for Forward and Inverse Problems in PDEs with Sharp Gradients

arXiv:2409.00651v2 Announce Type: replace-cross 
Abstract: In this study, we explore the application of Physics-Informed Neural Networks (PINNs) to the analysis of bifurcation phenomena in ecological migration models. By integrating the fundamental principles of diffusion-advection-reaction equations with deep learning techniques, we address the complexities of species migration dynamics, particularly focusing on the detection and analysis of Hopf bifurcations. Traditional numerical methods for solving partial differential equations (PDEs) often involve intricate calculations and extensive computational resources, which can be restrictive in high-dimensional problems. In contrast, PINNs offer a more flexible and efficient alternative, bypassing the need for grid discretization and allowing for mesh-free solutions. Our approach leverages the DeepXDE framework, which enhances the computational efficiency and applicability of PINNs in solving high-dimensional PDEs. We validate our results against conventional methods and demonstrate that PINNs not only provide accurate bifurcation predictions but also offer deeper insights into the underlying dynamics of diffusion processes. Despite these advantages, the study also identifies challenges such as the high computational costs and the sensitivity of PINN performance to network architecture and hyperparameter settings. Future work will focus on optimizing these algorithms and expanding their application to other complex systems involving bifurcations. The findings from this research have significant implications for the modeling and analysis of ecological systems, providing a powerful tool for predicting and understanding complex dynamical behaviors.

أظهرت دراسة حديثة تطبيق الشبكات العصبية المستندة إلى الفيزياء (PINNs) للكشف عن ظواهر التفرع في نماذج الهجرة البيئية، مع التركيز بشكل خاص على تفرعات هوف. من خلال دمج معادلات الانتشار والتدفق والتفاعل مع تقنيات التعلم العميق، تقدم هذه الدراسة بديلاً أكثر كفاءة للطرق العددية التقليدية لحل المعادلات التفاضلية الجزئية (PDEs)، التي غالبًا ما تتطلب موارد حسابية كبيرة.

Un estudio reciente ha demostrado la aplicación de Redes Neuronales Informadas por la Física (PINNs) para detectar fenómenos de bifurcación en modelos de migración ecológica, centrándose específicamente en las bifurcaciones de Hopf. Al integrar ecuaciones de difusión-advención-reacción con técnicas de aprendizaje profundo, esta investigación ofrece una alternativa más eficiente a los métodos numéricos tradicionales para resolver ecuaciones en derivadas parciales (EDPs), que a menudo requieren recursos computacionales extensos.

Une étude récente a démontré l'application des réseaux de neurones informés par la physique (PINNs) pour détecter les phénomènes de bifurcation dans les modèles de migration écologique, en se concentrant spécifiquement sur les bifurcations de Hopf. En intégrant des équations de diffusion-advectio-réaction avec l'apprentissage profond, cette recherche offre une alternative plus efficace aux méthodes numériques traditionnelles pour résoudre les équations aux dérivées partielles (EDP), qui nécessitent souvent des ressources computationnelles importantes.

A recent study has demonstrated the application of Physics-Informed Neural Networks (PINNs) to detect bifurcation phenomena in ecological migration models, specifically focusing on Hopf bifurcations. By integrating diffusion-advection-reaction equations with deep learning, this research offers a more efficient alternative to traditional numerical methods for solving partial differential equations (PDEs), which often require extensive computational resources.

Extended Physics Informed Neural Network for Hyperbolic Two-Phase Flow in Porous Media

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