Linear–Nonlinear Fusion Neural Operator for Partial Differential Equations

Heng Wu^1,2, Junjie Wang^1,2, Benzhuo Lu^1,2,*
1 State Key Laboratory of Mathematical Sciences (SKLMS), Institute of Computational Mathematics and Scientific/Engineering Computing (ICMSEC), National Center for Mathematics and Interdisciplinary Sciences (NCMIS), Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China
2 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China
* Corresponding Author: Benzhuo Lu. Email: email

Computer Modeling in Engineering & Sciences https://doi.org/10.32604/cmes.2026.084608

Received 26 April 2026; Accepted 02 July 2026; Published online 27 July 2026

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 investigate a two-path formulation that combines affine and nonlinear computational components within such operator mappings to improve 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 structurally transparent representation. This two-path formulation is designed to capture complex solution features at the operator level while retaining architectural simplicity. LNF-NO naturally supports multiple functional inputs and is applicable to both regular grids and fixed irregular-node discretizations. 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 three-dimensional Poisson–Boltzmann case, LNF-NO achieves competitive accuracy while requiring substantially less training time than the three-dimensional Fourier Neural Operator and Transolver baselines.