标题： MNO：用于参数化非线性边值问题的多模态神经算子 显示英文标题

标题： MNO : A Multi-modal Neural Operator for Parametric Nonlinear BVPs

摘要： 我们引入了一种新颖的多模态神经算子（MNO）架构，旨在学习多参数非线性边界值问题（BVPs）的解算子。 传统的神经算子主要将偏微分方程（PDE）的系数或源项独立地映射到解中，这限制了它们的灵活性和适用性。 相比之下，我们提出的MNO架构通过以统一的方式将包括PDE系数、源项和边界条件在内的多个参数映射到解空间，从而推广了这些方法。 我们的MNO受到快速多极方法（FMM）的层次嵌套基的启发，并通过三个关键组件系统地构建：一个参数高效的广义FMM（GFMM）块，一个基于GFMM块的单参数映射的单模态神经算子（UNO），以及最重要的是，一种多模态融合机制，将这些组件扩展以学习联合映射。 我们在线性和非线性BVPs上展示了我们方法的多模态泛化能力。 我们的实验表明，该网络能够有效处理PDE系数和源项或边界项的同时变化。 显示英文摘要

摘要： We introduce a novel Multimodal Neural Operator (MNO) architecture designed to learn solution operators for multi-parameter nonlinear boundary value problems (BVPs). Traditional neural operators primarily map either the PDE coefficients or source terms independently to the solution, limiting their flexibility and applicability. In contrast, our proposed MNO architecture generalizes these approaches by mapping multiple parameters including PDE coefficients, source terms, and boundary conditions to the solution space in a unified manner. Our MNO is motivated by the hierarchical nested bases of the Fast Multipole Method (FMM) and is constructed systematically through three key components: a parameter efficient Generalized FMM (GFMM) block, a Unimodal Neural Operator (UNO) built upon GFMM blocks for single parameter mappings, and most importantly, a multimodal fusion mechanism extending these components to learn the joint map. We demonstrate the multimodal generalization capacity of our approach on both linear and nonlinear BVPs. Our experiments show that the network effectively handles simultaneous variations in PDE coefficients and source or boundary terms.