自适应多级深度学习用于第二类剧烈振荡Fredholm积分方程
Adaptive Multi-Grade Deep Learning for Highly Oscillatory Fredholm Integral Equations of the Second Kind
AI总结：
本文提出了一种自适应多级深度学习方法，用于高效求解第二类剧烈振荡Fredholm积分方程，通过动态调整网络级别提升计算精度与稳定性。
摘要：
本文研究了多级深度学习（MGDL）在求解第二类剧烈振荡Fredholm积分方程中的应用。我们对连续和离散的MGDL模型进行了严谨的误差分析，证明在足够小的四次项误差下，离散模型保留了其连续对应项的收敛性和稳定性。我们识别出DNN训练误差是主要的近似误差来源，由此提出了一种新的自适应MGDL算法，该算法根据训练性能选择网络级别。针对剧烈振荡（包括波数500）和奇异解的数值实验验证了所提方法的准确性、有效性和鲁棒性。
英文摘要：
This paper studies the use of Multi-Grade Deep Learning (MGDL) for solving highly oscillatory Fredholm integral equations of the second kind. We provide rigorous error analyses of continuous and discrete MGDL models, showing that the discrete model retains the convergence and stability of its continuous counterpart under sufficiently small quadrature error. We identify the DNN training error as the primary source of approximation error, motivating a novel adaptive MGDL algorithm that selects the network grade based on training performance. Numerical experiments with highly oscillatory (including wavenumber 500) and singular solutions confirm the accuracy, effectiveness and robustness of the proposed approach.
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