Linear-Nonlinear Fusion Neural Operator for Partial Differential Equations
This work addresses the challenge of efficient and accurate operator learning for PDEs, which is important for applications in physics and engineering, but it appears incremental as it builds on existing neural operator methods like DeepONet and FNO.
The paper tackles the problem of learning neural operators for partial differential equations by proposing a Linear-Nonlinear Fusion Neural Operator (LNF-NO) that decouples linear and nonlinear effects, resulting in improved training efficiency and accuracy. For example, on a 3D Poisson-Boltzmann case, LNF-NO achieved the best accuracy and trained about 2.7 times faster than a Fourier Neural Operator baseline.
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 markedly 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 Deep Operator Networks (DeepONet) and Fourier Neural Operators (FNO), while achieving comparable or better accuracy in most cases. On the tested 3D Poisson-Boltzmann case, LNF-NO attains the best accuracy among the compared models and trains approximately 2.7x faster than a 3D FNO baseline.