用于学习多输入多输出算子的核方法

中文摘要 AI 辅助

学习无限维对象之间的映射是科学机器学习中的核心挑战。我们提出一种通用的基于核的编码器-解码器框架用于算子学习，该框架将观测、表示、学习与重构过程分离开来。我们将该框架拓展至多输入多输出算子学习场景，其中算子在可能不同的函数空间的乘积之间进行映射。我们的逼近理论表明，尽管输入和输出的数量可能增加，但收敛速率由最具挑战性的组成逼近问题决定，而非整体问题维度。该框架可导出具有闭式训练与推理的实用核方法，兼具数学可处理性与计算效率。我们进一步将该方法专门用于多算子学习，提出KernelMO，这是一类具有互补算子值与乘积空间形式的核方法。在五类参数化偏微分方程上，所提方法实现了有竞争力或最优的预测精度，同时相较于神经算子架构及基于深度学习的模型降低了训练与推理成本，提供了一种高效且轻量的替代方案。

英文摘要

Learning mappings between infinite-dimensional objects is a central challenge in scientific machine learning. We introduce a general kernel-based encoder-decoder framework for operator learning that separates observation, representation, learning, and reconstruction. We develop this framework for multi-input, multi-output operator learning, where operators map between products of potentially distinct function spaces. Our approximation theory shows that, although the number of inputs and outputs can increase, the convergence rate is governed by the most challenging constituent approximation problem rather than the overall problem dimension. The framework leads to practical kernel methods with closed-form training and inference, combining mathematical tractability with computational efficiency. We further specialize the approach to multiple operator learning by introducing KernelMO, a family of kernel methods with complementary operator-valued and product-space formulations. Across five families of parametric partial differential equations, the proposed methods achieve competitive or state-of-the-art predictive accuracy while reducing training and inference costs relative to neural operator architectures and deep learning based models, offering an efficient and lightweight alternative.

发表机构

• University of California Los Angeles（加州大学洛杉矶分校）
• University of Arkansas（阿肯色大学）
• Johns Hopkins University（约翰斯·霍普金斯大学）

机构由 AI 辅助整理，请以论文原文为准。