论文标题
Lippmann-Schwinger方程的机器学习正则解决方案
Machine Learning Regularized Solution of the Lippmann-Schwinger Equation
论文作者
论文摘要
空间频域中离散的Lippmann-Schwinger方程的解决方案涉及散射势指定的线性操作员的反转。为了使这个不可避免的条件问题正常，我们提出了一种机器学习方法：具有长期记忆（LSTM）的复发性神经网络，以及在复发路径上Lippmann-Schninger kernel的空空间投影。学习方法是使用典型散射电位及其相应散射场的示例训练的。我们在两种情况下测试了提出的方法：电磁物体通过介电物体进行电磁散射，而通过多个筛选的库仑电位进行电子散射。在这两种情况下，测试示例的解决方案（与训练集的脱节）的迭代率更少，并且与线性求解器相比是准确的。我们还观察到了令人惊讶的概括能力：在电磁情况下，经过介电球体排列的LSTM能够为一般拓扑相似的物体（例如多边形）获得正确的溶液。这表明LSTM成功地将散射物理学纳入了反转算法。
Solution of the discretized Lippmann-Schwinger equation in the spatial frequency domain involves the inversion of a linear operator specified by the scattering potential. To regularize this inevitably ill-conditioned problem, we propose a machine learning approach: a recurrent neural network with long short-term memory (LSTM) and with the null space projection of the Lippmann-Schwinger kernel on the recurrence path. The learning method is trained using examples of typical scattering potentials and their corresponding scattered fields. We test the proposed method in two cases: electromagnetic scattering by dielectric objects, and electron scattering by multiple screened Coulomb potentials. In both cases the solutions to test examples, disjoint from the training set, were obtained with fewer iterations and were accurate compared to linear solvers. We also observed surprising generalization ability: in the electromagnetic case, an LSTM trained with random arrangements of dielectric spheres was able to obtain the correct solutions for general topologically similar objects, such as polygons. This suggests that the LSTM successfully incorporates the physics of scattering into the inversion algorithm.
