Linear-Nonlinear Fusion Neural Operator for Partial Differential Equations

TL;DR

The paper introduces LNF-NO, a neural operator model that decouples linear and nonlinear effects for efficient PDE solution approximation, achieving faster training and comparable accuracy.

Contribution

It proposes a novel linear-nonlinear fusion neural operator architecture that improves learning efficiency and interpretability for PDE operator learning tasks.

Findings

LNF-NO trains faster than baseline models on PDE benchmarks.

LNF-NO achieves comparable or better accuracy across various PDE problems.

LNF-NO is effective on both regular and irregular geometries.

Abstract

Neural operator learning directly constructs the mapping relationship from the equation parameter space to the solution space, enabling efficient direct inference in practical applications without the need for repeated solution of partial differential equations (PDEs) -- an advantage that is difficult to achieve with traditional numerical methods. In this work, we find that explicitly decoupling linear and nonlinear effects within such operator mappings leads to improved learning efficiency. This yields a novel network structure, namely the Linear-Nonlinear Fusion Neural Operator (LNF-NO), which models operator mappings via the multiplicative fusion of a linear component and a nonlinear component, thus achieving a lightweight and interpretable representation. This linear-nonlinear decoupling enables efficient capture of complex solution features at the operator level while maintaining stability and generality. LNF-NO naturally supports multiple functional inputs and is applicable to both regular grids and irregular geometries. Across a diverse suite of PDE operator-learning benchmarks, including nonlinear Poisson-Boltzmann equations and multi-physics coupled systems, LNF-NO is typically substantially faster to train than several representative neural operator baselines, while achieving comparable or improved accuracy across most tested cases. On the tested 3D Poisson-Boltzmann case, LNF-NO achieves strong accuracy while requiring substantially less training time than the three-dimensional Fourier Neural Operator and Transolver baselines.