We consider the inverse acoustic obstacle problem for sound-soft star-shaped obstacles in two dimensions wherein the boundary of the obstacle is determined from measurements of the scattered field at a collection of receivers outside the object. One of the standard approaches for solving this problem is to reformulate it as an optimization problem: finding the boundary of the domain that minimizes the $L^2$ distance between computed values of the scattered field and the given measurement data. The optimization problem is computationally challenging since the local set of convexity shrinks with increasing frequency and results in an increasing number of local minima in the vicinity of the true solution. In many practical experimental settings, low frequency measurements are unavailable due to limitations of the experimental setup or the sensors used for measurement. Thus, obtaining a good initial guess for the optimization problem plays a vital role in this environment. We present a neural network warm-start approach for solving the inverse scattering problem, where an initial guess for the optimization problem is obtained using a trained neural network. We demonstrate the effectiveness of our method with several numerical examples. For high frequency problems, this approach outperforms traditional iterative methods such as Gauss-Newton initialized without any prior (i.e., initialized using a unit circle), or initialized using the solution of a direct method such as the linear sampling method. The algorithm remains robust to noise in the scattered field measurements and also converges to the true solution for limited aperture data. However, the number of training samples required to train the neural network scales exponentially in frequency and the complexity of the obstacles considered. We conclude with a discussion of this phenomenon and potential directions for future research.
翻译:我们认为,声软星形障碍的反声障碍在两个层面存在反向的声障问题,在这两个层面,障碍的界限是通过在物体外接收器收集的接收器中测量分散场的频率来确定的。解决该问题的标准方法之一是将这个问题重新定位为一个优化问题:找到将分散场的计算值与给定测量数据之间距离最小化为2美元的域界线。优化问题具有计算上的挑战性,因为本地的混凝土群越来越频繁地缩小,结果使得真正解决方案附近的当地微粒数量越来越多。在许多实际的实验环境中,由于实验设置或测量所用传感器的局限性,低频测量无法使用。因此,对优化问题进行良好的初步猜测在这种环境中起着关键作用。我们提出了一个神经网络的热源启动方法,通过经过训练的神经网络网络网络来初步测测测测优化问题。我们用几个数字实例来证明我们的方法的有效性。对于高频问题,这种方法超越了传统的内置方法,因为实验设置了用于模拟或测量所使用的传感器传感器。在初始的深度和直径分析方法中,我们用一个原始的直径分析方法,然后用直成一个直成直径的直径分析方法。