We propose a new model-order reduction framework to poorly reducible problems arising from parametric partial differential equations with geometric variability. In such problems, the solution manifold exhibits a slowly decaying Kolmogorov $N$-width, so that standard projection-based model order reduction techniques based on linear subspace approximations become ineffective. To overcome this difficulty, we introduce an optimal morphing strategy: For each solution sample, we compute a bijective morphing from a reference domain to the sample domain such that, when all the solution fields are pulled back to the reference domain, their variability is reduced. We formulate a global optimization problem on the morphings that maximizes the energy captured by the first $r$ modes of the mapped fields obtained from Proper Orthogonal Decomposition, thus maximizing the reducibility of the dataset. Finally, using a non-intrusive Gaussian Process regression on the reduced coordinates, we build a fast surrogate model that can accurately predict new solutions, highlighting the practical benefits of the proposed approach for many-query applications. The framework is general, independent of the underlying partial differential equation, and applies to scenarios with either parameterized or non-parameterized geometries.

翻译：本文提出一种新的模型降阶框架，用于处理具有几何可变性的参数偏微分方程所引发的难约简问题。此类问题的解流形表现出缓慢衰减的Kolmogorov $N$-宽度，使得基于线性子空间近似的标准投影型模型降阶方法失效。为克服这一困难，我们引入最优形变策略：针对每个解样本，计算从参考域到样本域的双射形变，使得所有解场被拉回至参考域时，其变异性显著降低。我们构建了关于形变的全局优化问题，以最大化通过本征正交分解所得映射场前$r$个模态捕获的能量，从而最大化数据集的约简潜力。最后，通过对降维坐标进行非侵入式高斯过程回归，构建能够精确预测新解的快速代理模型，突显了该方法在多查询应用中的实用价值。该框架具有通用性，不依赖于特定偏微分方程形式，适用于参数化或非参数化几何场景。