arXiv:2507.06038v3 Announce Type: replace-cross 
Abstract: Building on our previous work introducing Fredholm Neural Networks (Fredholm NNs/ FNNs) for solving integral equations, we extend the framework to tackle forward and inverse problems for linear and semi-linear elliptic partial differential equations. The proposed scheme consists of a deep neural network (DNN) which is designed to represent the iterative process of fixed-point iterations for the solution of elliptic PDEs using the boundary integral method within the framework of potential theory. The number of layers, weights, biases and hyperparameters are computed in an explainable manner based on the iterative scheme, and we therefore refer to this as the Potential Fredholm Neural Network (PFNN). We show that this approach ensures both accuracy and explainability, achieving small errors in the interior of the domain, and near machine-precision on the boundary. We provide a constructive proof for the consistency of the scheme and provide explicit error bounds for both the interior and boundary of the domain, reflected in the layers of the PFNN. These error bounds depend on the approximation of the boundary function and the integral discretization scheme, both of which directly correspond to components of the Fredholm NN architecture. In this way, we provide an explainable scheme that explicitly respects the boundary conditions. We assess the performance of the proposed scheme for the solution of both the forward and inverse problem for linear and semi-linear elliptic PDEs in two dimensions.

تم تقديم إطار عمل جديد يوسع الشبكات العصبية فريدولم (FNN) لمعالجة المشاكل الأمامية والعكسية في المعادلات التفاضلية الجزئية (PDE) البيضاوية الخطية وشبه الخطية. تستخدم هذه الطريقة شبكة عصبية عميقة مصممة لتمثيل عمليات التكرار الثابتة لحل PDE البيضاوية من خلال طريقة التكامل الحدودي، والتي تُسمى شبكة فريدولم العصبية المحتملة (PFNN).

Se ha introducido un nuevo marco que extiende las Redes Neuronales de Fredholm (FNN) para abordar problemas directos e inversos en ecuaciones diferenciales parciales (EDP) elípticas lineales y semilineales. Este enfoque utiliza una red neuronal profunda diseñada para representar iteraciones de punto fijo para resolver EDP elípticas mediante el método de integral de frontera, denominado Red Neuronal de Fredholm Potencial (PFNN).

Un nouveau cadre a été introduit, étendant les Réseaux de Neurones de Fredholm (FNN) pour traiter les problèmes directs et inverses dans les équations aux dérivées partielles (EDP) elliptiques linéaires et semi-linéaires. Cette approche utilise un réseau de neurones profond conçu pour représenter les itérations de point fixe pour résoudre les EDP elliptiques par la méthode d'intégrale de frontière, appelée Réseau de Neurones de Fredholm Potentiel (PFNN).

A new framework has been introduced that extends Fredholm Neural Networks (FNNs) to address forward and inverse problems in linear and semi-linear elliptic partial differential equations (PDEs). This approach utilizes a deep neural network designed to represent fixed-point iterations for solving elliptic PDEs through the boundary integral method, termed the Potential Fredholm Neural Network (PFNN).

Fredholm Neural Networks for forward and inverse problems in elliptic PDEs

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