arXiv:2507.06038v4 Announce Type: replace-cross Abstract: Building on our previous work on Fredholm Neural Networks (Fredholm NNs/ FNNs) for solving integral equations, we extend the framework to inverse problems for linear and nonlinear elliptic partial differential equations. The proposed scheme consists of a custom-designed deep neural network (DNN) in which the number of layers, weights, biases and hyperparameters are computed in an explainable manner based on a fixed-point scheme, and we therefore refer to this as the Potential Fredholm Neural Network (PFNN). We first build the PFNN as a
The continuous advancements in AI and deep learning provide fertile ground for extending neural network applications to complex mathematical problems like inverse PDEs, building on prior successes in integral equations.
This work introduces a more explainable and potentially robust neural network architecture (PFNN) for solving inverse problems in elliptic PDEs, which are fundamental in many scientific and engineering domains.
The development of Potential Fredholm Neural Networks offers a new, potentially more efficient and interpretable method for tackling challenging inverse problems, moving beyond traditional numerical methods.
- · AI/ML researchers and developers
- · Engineering simulation and design
- · Scientific computing organizations
- · Academic research institutions
- · Traditional numerical methods that are less efficient or interpretable
- · Companies relying on less optimized solutions for inverse problems
Fredholm Neural Networks extend their applicability to a broader class of complex differential equations.
Improved efficiency in solving inverse problems could accelerate R&D cycles in fields like material science or medical imaging.
Enhanced explainability in AI-driven solutions could increase adoption and trust in critical applications where black-box models are problematic.
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