Numerical solution of inverse problems by weak adversarial networks
Numerical solution of inverse problems by weak adversarial networks
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DOI：
10.1088/1361-6420/abb447
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发表时间：
2020-02
期刊：
影响因子：
2.1
通讯作者：
Gang Bao;X. Ye;Yaohua Zang;Haomin Zhou
中科院分区：
文献类型：
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作者：
Gang Bao;X. Ye;Yaohua Zang;Haomin Zhou
本文提出了一种弱对抗网络方法来数值求解一类反问题，包括电阻抗层析成像和动态电阻抗层析成像问题。利用给定反问题的偏微分方程的弱公式，其中解和测试函数被参数化为深度神经网络。然后，弱形式和边界条件导致网络参数的鞍函数的极大极小问题。随着参数的交替更新，网络逐渐逼近逆问题的解。理论证明了该算法的收敛性。该方法是完全无网格的，不需要任何空间离散，特别适用于高维和低正则性的解决方案。对多种测试反问题的数值实验表明，该方法具有良好的精度和效率。
In this paper, a weak adversarial network approach is developed to numerically solve a class of inverse problems, including electrical impedance tomography and dynamic electrical impedance tomography problems. The weak formulation of the partial differential equation for the given inverse problem is leveraged, where the solution and the test function are parameterized as deep neural networks. Then, the weak formulation and the boundary conditions induce a minimax problem of a saddle function of the network parameters. As the parameters are alternatively updated, the network gradually approximates the solution of the inverse problem. Theoretical justifications are provided on the convergence of the proposed algorithm. The proposed method is completely mesh-free without any spatial discretization, and is particularly suitable for problems with high dimensionality and low regularity on solutions. Numerical experiments on a variety of test inverse problems demonstrate the promising accuracy and efficiency of this approach.