论文标题
Cauchy逆问题的人工神经网络近似
An artificial neural network approximation for Cauchy inverse problems
论文作者
论文摘要
提出了一种新型的人工神经网络方法来解决Cauchy逆问题。它允许多个具有任意宽度和深度的隐藏层，从理论上讲，这对反问题产生了更好的近似值。在这项研究中，证明存在和收敛性为CAUCHY逆问题建立了神经网络方法的良好性，并提出了各种数值示例，以说明其准确性和稳定性。数值示例来自不同的观点，包括时间依赖性和与时间无关的情况，高空间尺寸案例高达8D，以及具有嘈杂边界数据和奇异计算域的情况。此外，数值结果还表明，具有较宽和更深层的隐藏层的神经网络可能会导致Cauchy逆问题更好地近似。
A novel artificial neural network method is proposed for solving Cauchy inverse problems. It allows multiple hidden layers with arbitrary width and depth, which theoretically yields better approximations to the inverse problems. In this research, the existence and convergence are shown to establish the well-posedness of neural network method for Cauchy inverse problems, and various numerical examples are presented to illustrate its accuracy and stability. The numerical examples are from different points of view, including time-dependent and time-independent cases, high spatial dimension cases up to 8D, and cases with noisy boundary data and singular computational domain. Moreover, numerical results also show that neural networks with wider and deeper hidden layers could lead to better approximation for Cauchy inverse problems.
