Detecting hidden geometrical structures from surface measurements under electromagnetic, acoustic, or mechanical loading is the goal of noninvasive imaging techniques in medical and industrial applications. Solving the inverse problem can be challenging due to the unknown topology and geometry, the sparsity of the data, and the complexity of the physical laws. Physics-informed neural networks (PINNs) have shown promise as a simple-yet-powerful tool for problem inversion, but they have yet to be applied to general problems with a priori unknown topology. Here, we introduce a topology optimization framework based on PINNs that solves geometry detection problems without prior knowledge of the number or types of shapes. We allow for arbitrary solution topology by representing the geometry using a material density field that approaches binary values thanks to a novel eikonal regularization. We validate our framework by detecting the number, locations, and shapes of hidden voids and inclusions in linear and nonlinear elastic bodies using measurements of outer surface displacement from a single mechanical loading experiment. Our methodology opens a pathway for PINNs to solve various engineering problems targeting geometry optimization.

翻译：从表面测量中检测隐藏在电磁、声学或机械载荷下的几何结构，是医学和工业应用中非侵入性成像技术的目标。由于未知的拓扑与几何、数据的稀疏性以及物理定律的复杂性，求解逆问题可能具有挑战性。物理信息神经网络（PINNs）作为一种简单而强大的问题反演工具已展现出潜力，但尚未应用于先验未知拓扑的通用问题。本文提出一种基于PINNs的拓扑优化框架，该框架无需预先知道形状的数量或类型即可解决几何检测问题。我们通过材料密度场表示几何结构，并利用新颖的程函正则化使密度场趋近于二值化，从而允许任意解拓扑。通过单一机械加载实验中外表面位移的测量，我们在线性和非线性弹性体中检测隐藏空洞与夹杂物的数量、位置及形状，验证了该框架的有效性。我们的方法为PINNs解决面向几何优化的各类工程问题开辟了路径。