基于有限测试与神经代理的边界数据材料系数潜空间反演
Latent Inversion of Material Coefficients from Boundary Data via Finite Tests and Neural Surrogates
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中文总结 AI 辅助
本文提出一种潜在空间反演方法，通过有限边界激励和神经代理重建椭圆方程中的材料系数，在保证局部稳定性的同时降低计算成本。
中文摘要 AI 辅助
我们通过有限次边界激励（每次激励产生完整的Dirichlet迹）重建标量椭圆方程中的空间变化材料系数。为缓解不适定性并降低重复求解PDE的成本，我们将系数限制在低维族中（该族可通过解析方式指定或从样本中学习），并在其潜在坐标中求解逆问题。我们考虑具有m个潜在坐标且在参考点处导数为满秩的C¹参数化。若连续线性化的Neumann-to-Dirichlet映射在相应切空间上是单射的，则至多m次激励足以保证局部单射性和Lipschitz稳定性。系数灵敏度的收敛性随后将此稳定性传递到协调有限元离散化。对于足够细的网格，稳定性常数和邻域可以独立于网格尺寸进行选择。在局部残差比较条件下，代理前向映射的一致精度可得到系数误差界，该误差界将表示误差、数据噪声、有限元误差和代理误差分开。导数精度额外保持了代理自身的局部稳定性。所有稳定性陈述均相对于参考系数局部成立。二维数值实验结合了夹杂物和裂纹状系数的解析与学习表示以及神经前向代理。实验展示了潜在空间重建、降低的在线成本以及可选FEM细化带来的进一步改进。
英文摘要
We reconstruct a spatially varying material coefficient in a scalar elliptic equation from finitely many boundary excitations, each producing a full Dirichlet trace. To mitigate the ill-posedness and the cost of repeated PDE solves, we restrict the coefficient to a low-dimensional family, specified analytically or learned from samples, and solve the inverse problem in its latent coordinates. We consider a \(C^1\) parametrization with \(m\) latent coordinates and full-rank derivative at a reference point. If the continuous linearized Neumann-to-Dirichlet map is injective on the corresponding tangent space, at most \(m\) excitations suffice for local injectivity and Lipschitz stability. Convergence of the coefficient sensitivities then transfers this stability to conforming finite element discretizations. For sufficiently fine meshes, the stability constant and neighborhood can be chosen independently of the mesh size. Under a local residual-comparison condition, uniform accuracy of the surrogate forward map yields coefficient-error bounds separating representation error, data noise, finite element error, and surrogate error. Derivative accuracy additionally preserves the surrogate's own local stability. All stability statements are local to a reference coefficient. Two-dimensional numerical experiments combine analytic and learned representations of inclusions and crack-like coefficients with neural forward surrogates. They illustrate latent-space reconstruction, reduced online cost, and further improvement from optional FEM-based refinement.
