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

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.