反问题的弱对抗网络数值解
Numerical Solution of Inverse Problems by Weak Adversarial Networks
摘要：
我们考虑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。我们在给定的反问题中利用PDE的弱公式，并将解和检验函数参数化为深度神经网络。弱形式和边界条件导致网络参数的鞍函数的极小极大问题。随着参数的交替更新，网络逐渐逼近逆问题的解。我们提供了理论上的理由所提出的算法的收敛性。我们的方法是完全无网格的，没有任何空间离散化，特别适合于高维和低正则性的解决方案的问题。各种测试反问题的数值实验表明，我们的方法有前途的精度和效率。
英文摘要：
We consider a weak adversarial network approach to numerically solve a class of inverse problems, including electrical impedance tomography and dynamic electrical impedance tomography problems. We leverage the weak formulation of PDE in the given inverse problem, and parameterize the solution and the test function as deep neural networks. 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. We provide theoretical justifications on the convergence of the proposed algorithm. Our 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 our approach.
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