标题： 导数信息算子学习的降维：近似误差分析 显示英文标题

标题： Dimension reduction for derivative-informed operator learning: An analysis of approximation errors

摘要： 我们研究通过神经网络在无限维可分希尔伯特空间之间的非线性算子的导数信息学习。 这些算子可能来源于偏微分方程（PDE）的解，并用于科学和工程中的许多基于仿真的外层任务，如PDE约束优化、贝叶斯反问题和最优实验设计。 在这些情况下，神经网络近似可以作为代理模型，以加速外层任务的求解。 然而，由于无限维的外层任务通常需要了解底层几何结构，算子导数的近似精度也会显著影响代理模型的性能。 受此启发，我们在无限维高斯输入测度上分析了神经算子在索博列夫范数下的近似误差。 我们关注的是降阶基神经算子（RBNO），它使用定义在由降阶正交基集张成的主要输入/输出子空间上的线性编码器和解码器。 为此，我们研究了两种生成基的方法；主成分分析（PCA）和导数信息子空间（DIS），分别使用数据协方差或导数的主要特征向量作为降阶基。 然后，我们推导了来自维度约简和潜在神经网络近似的误差界限，包括PCA/DIS经验估计相关的采样误差。 我们的分析在椭圆PDE的数值实验中得到验证，结果表明由映射信息的基（即DIS或输出PCA）对于算子及其导数的重建和泛化误差都能提供准确的结果，而输入PCA在秩和训练样本数量足够大时才能表现良好。 显示英文摘要

摘要： We study the derivative-informed learning of nonlinear operators between infinite-dimensional separable Hilbert spaces by neural networks. Such operators can arise from the solution of partial differential equations (PDEs), and are used in many simulation-based outer-loop tasks in science and engineering, such as PDE-constrained optimization, Bayesian inverse problems, and optimal experimental design. In these settings, the neural network approximations can be used as surrogate models to accelerate the solution of the outer-loop tasks. However, since outer-loop tasks in infinite dimensions often require knowledge of the underlying geometry, the approximation accuracy of the operator's derivatives can also significantly impact the performance of the surrogate model. Motivated by this, we analyze the approximation errors of neural operators in Sobolev norms over infinite-dimensional Gaussian input measures. We focus on the reduced basis neural operator (RBNO), which uses linear encoders and decoders defined on dominant input/output subspaces spanned by reduced sets of orthonormal bases. To this end, we study two methods for generating the bases; principal component analysis (PCA) and derivative-informed subspaces (DIS), which use the dominant eigenvectors of the covariance of the data or the derivatives as the reduced bases, respectively. We then derive bounds for errors arising from both the dimension reduction and the latent neural network approximation, including the sampling errors associated with the empirical estimation of the PCA/DIS. Our analysis is validated on numerical experiments with elliptic PDEs, where our results show that bases informed by the map (i.e., DIS or output PCA) yield accurate reconstructions and generalization errors for both the operator and its derivatives, while input PCA may underperform unless ranks and training sample sizes are sufficiently large.