发表机构
SISSA; KTH Royal Institute of Technology（的里雅斯特国际高等研究学校; 皇家理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对形状优化中高维FFD参数空间问题，提出主动子空间引导降维与Sinkhorn自编码器学习潜在分布的两阶段降阶建模方法，在热方程和阻力预测任务中显著降低测试误差。
AI 中文摘要
几何参数化偏微分方程出现在形状优化中，其中以可管理的成本在许多变形域上求解偏微分方程是一个核心挑战。自由形变（FFD）通过控制点网格参数化形状变化，但所得参数空间通常是高维的，且大多数方向对感兴趣量（QoI）几乎没有影响。我们通过一个两阶段降阶流程来解决这一问题。一个主动子空间（AS）引导的结构化控制点选择识别出驱动QoI的FFD权重的低维子空间。然后，在降阶权重上训练一个Sinkhorn自编码器（SAE），通过最小化聚合后验与先验之间的Wasserstein距离，在紧凑的潜在空间中学习其分布。我们比较了三种非侵入式降阶模型（ROM），分别将完整FFD权重、AS降阶权重或SAE潜在编码映射到QoI，每种模型使用随机森林、高斯过程和K近邻。我们在变形的斯坦福兔子上具有高斯随机Robin边界数据的非线性热方程，以及DTC Hull球鼻在FFD变形下的阻力预测上测试了该方法。在两种情况下，生成模型在可比支撑上捕获了QoI分布的主体，使设计目标对面的肩部变薄，而其在设计相关方向上的极值与FFD样本在数值和采样不确定性范围内匹配。在SAE潜在空间中学习的ROM在大多数回归器-度量组合中比在完整FFD权重上学习的ROM产生更小的测试误差，而AS降阶权重没有给出可比的增益。只有潜在表示达到了正预测系数$Q^2$。
英文摘要
Geometrically parametrised PDEs arise in shape optimisation, where solving a PDE over many deformed domains at manageable cost is a central challenge. Free-form Deformation (FFD) parametrises shape variations through a lattice of control points, but the resulting parameter space is typically high-dimensional, with most directions barely affecting the quantity of interest (QoI). We address this with a two-stage reduction pipeline. An Active Subspace (AS)-guided structured control-point selection identifies the low-dimensional subspace of FFD weights driving the QoI. A Sinkhorn Autoencoder (SAE) is then trained on the reduced weights, learning their distribution in a compact latent space by minimising the Wasserstein distance between aggregated posterior and prior. We compare three non-intrusive Reduced-order Models (ROMs), mapping full FFD weights, AS-reduced weights, or SAE latent codes to the QoI, each using Random Forests, Gaussian Processes, and K-Nearest Neighbours. We test the methodology on a non-linear heat equation with Gaussian random Robin boundary data on a deformed Stanford Bunny, and on drag prediction for the DTC Hull bulb under FFD deformations. In both cases, the generative model captures the bulk of the QoI distribution over a comparable support, thinning the shoulder opposite the design objective, while its extremes in the design-relevant direction match the FFD sample within numerical and sampling uncertainty. ROMs learned in the SAE latent space yield smaller test errors than those on full FFD weights in most regressor-metric combinations, whereas AS-reduced weights give no comparable gain. Only the latent representation attains a positive predictive coefficient $Q^2$.
Comments22 pages, 11 figures
