Reconstructing complex 3D interfaces from indirect measurements remains a grand challenge in scientific computing, particularly for ill-posed inverse problems like Electrical Impedance Tomography (EIT). Traditional shape optimization struggles with topological changes and regularization tuning, while emerging deep learning approaches often compromise physical fidelity or require prohibitive amounts of paired training data. We present a transformative ``solver-in-the-loop'' framework that bridges this divide by coupling a pre-trained 3D generative prior with a rigorous boundary integral equation (BIE) solver. Unlike Physics-Informed Neural Networks (PINNs) that treat physics as soft constraints, our architecture enforces the governing elliptic PDE as a hard constraint at every optimization step, ensuring strict physical consistency. Simultaneously, we navigate a compact latent manifold of plausible geometries learned by a differentiable neural shape representation, effectively regularizing the ill-posed problem through data-driven priors rather than heuristic smoothing. By propagating adjoint shape derivatives directly through the neural decoder, we achieve fast, stable convergence with dramatically reduced degrees of freedom. Extensive experiments on 3D high-contrast EIT demonstrate that this principled hybrid approach yields superior geometric accuracy and data efficiency which is difficult to achieve using traditional methods, establishing a robust new paradigm for physics-constrained geometric discovery.

翻译：从间接测量中重建复杂三维界面仍是科学计算中的重大挑战，尤其对于电阻抗成像（EIT）这类病态逆问题。传统形状优化方法难以处理拓扑变化且正则化参数调整困难，而新兴的深度学习方法往往牺牲物理保真度或需要海量配对训练数据。我们提出一种变革性"求解器在环"框架，通过将预训练的三维生成先验与严格的边界积分方程（BIE）求解器耦合，弥合了这一鸿沟。与将物理约束作为软约束的物理信息神经网络（PINNs）不同，我们的架构在每一步优化中将控制椭圆型偏微分方程作为硬约束强制执行，确保严格的物理一致性。同时，我们通过可微分神经形状表示学习的紧凑几何潜流形导航，利用数据驱动先验而非启发式平滑有效正则化病态问题。通过直接将伴随形状导数传播至神经解码器，我们实现了快速稳定的收敛，并大幅降低了自由度。在三维高对比度EIT上的大量实验表明，这种原则性混合方法在几何精度和数据效率上均优于传统方法，为物理约束几何发现建立了稳健的新范式。