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Linear-Nonlinear Fusion Neural Operator for Partial Differential Equations

Heng Wu, Junjie Wang, Benzhuo Lu

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.

Linear-Nonlinear Fusion Neural Operator for Partial Differential Equations

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.
Paper Structure (63 sections, 1 theorem, 27 equations, 11 figures, 6 tables)

This paper contains 63 sections, 1 theorem, 27 equations, 11 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Operator learning with input/output discretization
  4. LNF-NO architecture (multi-input, multi-output)
  5. Approximation Guarantee
  6. Optimization
  7. Results
  8. Baseline benchmarks on regular grids
  9. Single-input operators
  10. Multi-input operators
  11. Nonlinear Strength Regimes in Poisson--Boltzmann Equations
  12. Irregular geometries
  13. Extension to three-dimensional problems
  14. Discussion
  15. Dataset generation and preprocessing
  16. ...and 48 more sections

Key Result

Theorem 2.1

Assume $K_h\subset\mathbb{R}^{d_{\mathrm{in}}}$ is compact and $F_h^\star\in C(K_h;\mathbb{R}^{d_{\mathrm{out}}})$. If LNF-NO employs non-polynomial activations (e.g., ReLU or $\tanh$) in its feedforward components, then for any $\varepsilon>0$ there exist parameters $\theta$ such that

Figures (11)

  • Figure 1: Architecture of the proposed Linear--Nonlinear Fusion Neural Operator (LNF-NO). Each input component (typically a discretized function, e.g., boundary traces or source fields) is encoded separately and concatenated into a latent representation. An operator core then fuses a linear branch and a nonlinear branch via element-wise multiplication ($\odot$), producing a raw prediction that can be optionally refined by a lightweight decoder. Arrows indicate the data flow.
  • Figure 2: 3D Poisson--Boltzmann extension. Central cross-sections of the solution field $u(x,y,z)$ at $x=0.5$, $y=0.5$, and $z=0.5$. The prediction is reconstructed from boundary data by LNF-NO.
  • Figure A1: Representative geometries and corresponding finite-element meshes for the irregular-domain Poisson--Boltzmann benchmarks.
  • Figure : (a) Laplace Equation: Prediction of the harmonic function on a regular grid.
  • Figure A3: Darcy Flow (Continuous Coefficients). From left to right: Input Permeability $a(x,y)$, Reference Solution $u(x,y)$, Prediction, and Absolute Error.
  • ...and 6 more figures

Theorems & Definitions (1)

  • Theorem 2.1: Universal approximation