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：我们考虑二维空间中声软星形障碍物的逆声学问题，其中障碍物边界由物体外部一组接收器测得的散射场数据确定。解决该问题的标准方法之一是将问题重构为优化问题：通过最小化散射场计算值与给定测量数据之间的$L^2$距离来求解区域边界。该优化问题在计算上具有挑战性，因为局部凸性集随频率升高而收缩，导致真实解附近局部极小值的数量增加。在许多实际实验场景中，由于实验装置或测量传感器的限制，低频测量数据无法获取。因此，在此环境下获得良好的优化问题初始估计至关重要。本文提出一种神经网络暖启动方法求解逆散射问题，通过训练后的神经网络获取优化问题的初始估计。我们通过多个数值算例验证了该方法的有效性。针对高频问题，本方法优于传统迭代方法（如无先验初始化（即使用单位圆初始化）的高斯-牛顿法）或使用直接方法（如线性采样法）解初始化的方法。该算法对散射场测量噪声具有鲁棒性，且在有限孔径数据下仍能收敛到真实解。然而，训练神经网络所需的样本数量随频率和障碍物复杂度呈指数增长。我们最后讨论了该现象及未来研究的潜在方向。