Linear-Nonlinear Fusion Neural Operator for Partial Differential Equations

2026-03-25

AI Summary

Traditional numerical methods struggle to efficiently solve partial differential equations (PDEs), while existing neural operators remain limited in expressivity and learning efficiency. This work proposes the Linear-Nonlinear Fusion Neural Operator (LNF-NO), which explicitly decouples and multiplicatively fuses linear and nonlinear components to construct a lightweight, interpretable architecture that accommodates multi-function inputs and arbitrary geometric domains. LNF-NO can be seamlessly embedded into or used to enhance established frameworks such as DeepONet and FNO. Across multiple PDE benchmark tasks, LNF-NO achieves comparable or higher accuracy than current methods while significantly accelerating training—evidenced by a 2.7× speedup on the 3D Poisson-Boltzmann problem.

Technology Category

• Machine Learning: Mixture of Experts (MoE)
• Search and Optimization: Non-convex Optimization
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Architectures and workflows that use LLMs for crowd work
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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 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.

Problem

• neural operator
• partial differential equations
• linear-nonlinear decoupling
• operator learning
• efficient inference

Innovation

• neural operator
• linear-nonlinear decoupling
• multiplicative fusion
• PDE learning
• operator learning