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

TL;DR

The paper introduces MNO, a neural operator architecture that learns solution mappings for multi-parameter nonlinear boundary value problems, enabling flexible handling of multiple varying parameters simultaneously.

Contribution

It proposes a novel multimodal neural operator architecture that generalizes existing methods to jointly learn multiple parameters of nonlinear BVPs.

Findings

Effective handling of multiple parameter variations in BVPs.

Superior performance on both linear and nonlinear BVPs.

Unified framework for multi-parameter solution operators.

Abstract

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.