Fredholm神经网络
Fredholm Neural Networks
AI总结：
Fredholm神经网络利用固定点迭代方案求解线性和非线性Fredholm积分方程，提供高精度的数值解法，适用于求解微分方程和反问题，推动科学机器学习与数值分析的交叉发展。
摘要：
在可解释机器学习的家族中，我们提出了Fredholm神经网络（Fredholm NNs）：受固定点迭代方案启发的深度神经网络（DNN）架构，用于求解线性和非线性第二类Fredholm积分方程（FIEs）。我们还展示了所提出框架如何用于求解反问题。FIEs的应用包括求解常微分方程（ODEs）和偏微分方程（PDEs）等。我们首先证明Fredholm NNs能够提供准确的解。然后通过直接连接其值到底层数学理论，提供了DNN超参数和可训练/可解释权重和偏置的洞察。为了演示，我们使用Fredholm NNs解决线性和非线性问题，包括椭圆PDEs和边界值问题。我们展示了所提出的方法在域和边界上都实现了显著的数值近似精度。所提出的方法提供了神经网络与经典数值方法之间的联系，并认为它可以在不确定性量化（UQ）和可解释人工智能（XAI）等领域有应用。因此，我们相信它将在科学机器学习和数值分析的交集上触发进一步的进步。
英文摘要：
Within the family of explainable machine-learning, we present Fredholm neural networks (Fredholm NNs): deep neural networks (DNNs) architectures motivated by fixed-point iteration schemes for the solution of linear and nonlinear Fredholm integral equations (FIEs) of the second kind. We also show how the proposed framework can be used for the solution of inverse problems. Applications of FIEs include the solution of ordinary, as well as partial differential equations (ODEs, PDEs) and many more. We first prove that Fredholm NNs provide accurate solutions. We then provide insight into the values of the hyperparameters and trainable/explainable weights and biases of the DNN, by directly connecting their values to the underlying mathematical theory. For our illustrations, we use Fredholm NNs to solve both linear and nonlinear problems, including elliptic PDEs and boundary value problems. We show that the proposed scheme achieves significant numerical approximation accuracy across both the domain and boundary. The proposed methodology provides insight into the connection between neural networks and classical numerical methods, and we posit that it can have applications in fields such as Uncertainty Quantification (UQ) and explainable artificial intelligence (XAI). Thus, we believe that it will trigger further advances in the intersection between scientific machine learning and numerical analysis.
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