Linear-Nonlinear Fusion Neural Operator for Partial Differential Equations

Heng Wu; Junjie Wang; Benzhuo Lu

cs.LGMar 25, 2026

Linear-Nonlinear Fusion Neural Operator for Partial Differential Equations

Authors: Heng Wu, Junjie Wang, Benzhuo Lu

Organizations: SKLMS, ICMSEC, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China · School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

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.

Figures & tables

Fig. 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.

Equation	Model	#Params	Epochs	Training Time (s)	Test rel. $\ell_{2}$
Laplace	DeepONet	973,313	5000	1470.08	1.16e-02
	FNO	9,133,866	500	999.22	1.66e-02
	UNO	9,207,722	500	1393.00	2.05e-02
	Transolver	2,158,538	500	7322.38	1.04e-02
	LNF-NO	2,891,956	500	257.57	7.43e-03
Burgers	DeepONet	938,497	5000	1760.09	1.05e-01

Table 1: Baseline benchmarks on regular grids (single-input operators). We report the test mean relative

\ell_{2}

error and wall-clock training time under the unified protocol.

Fig. 2: Convergence comparison on the source-free Poisson–Boltzmann benchmark (

k=1

). Left: test relative

\ell_{2}

error versus training epoch. Right: test relative

\ell_{2}

error versus wall-clock training time. LNF-NO exhibits consistently faster error reduction than the compared baselines, especially when measured against actual training time.

Equation	Model	#Params	Epochs	Training Time (s)	Test rel. $\ell_{2}$
PB with source ( $k=1$ )	DeepONet	4,096,769	5000	2549.46	2.64e-01
	FNO	11,138,330	500	3751.44	3.71e-02
	UNO	11,212,186	500	5683.71	3.63e-02
	Transolver	4,228,986	500	28264.52	4.85e-02
	LNF-NO	9,662,852	500	988.78	1.47e-02
PNP system	DeepONet	1,841,667	5000	12299.65	6.01e-02

Table 2: Baseline benchmarks on regular grids (multi-input operators). Each task takes multiple function-valued inputs (e.g., boundary traces and a source field) and predicts fields on a fixed grid.

Equation	Model	#Params	Epochs	Training Time (s)	Test rel. $\ell_{2}$
PB ( $k=0.01$ )	DeepONet	1,024,513	5000	1821.46	5.72e-02
	FNO	11,138,266	500	3747.64	6.24e-02
	UNO	11,212,122	500	5862.01	6.76e-02
	Transolver	4,228,730	500	28642.89	4.27e-02
	LNF-NO	8,027,156	500	668.07	1.65e-02
PB ( $k=1$ )	DeepONet	1,024,513	5000	1685.50	6.89e-02

Table 3: Poisson–Boltzmann regimes with varying nonlinear strengths (source-free). We vary the nonlinearity strength

k

in ( 5 ) with

f=0

and report test mean relative

\ell_{2}

error and training time.

Table 4: Performance on irregular domains for the Poisson–Boltzmann equation. The problems are posed on non-rectangular geometries with complex boundaries and discretized using unstructured node-based representations, for which grid-based spectral operators such as FNO are not directly applicable.

Model	#Params	Epochs	Time (s)	Rel. $\ell_{2}$
DeepONet	2.5M	5000	1349	3.34e-01
FNO (3D)	15.0M	500	7559	7.71e-02
UNO	13.0M	500	10612	8.53e-02
Transolver	11.8M	500	53311	3.66e-02
LNF-NO	21.7M	500	3253	4.32e-02

Table 5: Results on the 3D Poisson–Boltzmann benchmark on a

33^{3}

grid. We report test mean relative

\ell_{2}

error and wall-clock training time under the unified protocol.

Fig. 3: 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.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Fig. A1: Representative geometries and corresponding finite-element meshes for the irregular-domain Poisson–Boltzmann benchmarks.

Seed	Best test rel. $\ell_{2}$	Total training time (s)
0	1.99e-02	671.30
1	1.88e-02	588.14
2	1.88e-02	559.15
3	2.06e-02	680.43
4	2.09e-02	678.09
Mean $\pm$ Std.	$1.98\times 10^{-2}\pm 8.8\times 10^{-4}$	$635.42\pm 51.35$

Appendix

Table A1: Seed stability on the source-free Poisson–Boltzmann benchmark (

k=1

). We repeat the LNF-NO experiment across five random seeds and report the best test relative

\ell_{2}

error and total training time.

Panel of Figure 11 — (a) Laplace Equation: Prediction of the harmonic function on a regular grid.

Fig. A3: Darcy Flow (Continuous Coefficients). From left to right: Input Permeability $a(x,y)$ , Reference Solution $u(x,y)$ , Prediction, and Absolute Error.

Fig. A4: Darcy Flow (Discontinuous Coefficients). The model effectively handles sharp transitions in the solution field caused by jump coefficients.

Fig. A5: Poisson–Boltzmann (High Nonlinearity $k=100$ ). The model captures the sharp boundary layers induced by the strong sinh nonlinearity.

Fig. A6: Poisson–Boltzmann (Source-Driven). Includes the variable source term $f(x,y)$ (first column) as input.

Fig. A7: Geometric Generalization (Irregular PB). Predictions on complex non-simply-connected domains, including the “Flower” shape with elliptical holes. Since this task uses the node-based architecture, results are plotted directly on the finite-element mesh triangulation. LNF-NO accurately captures the solution field along the intricate curved boundaries, confirming its capability to handle unstructured data and complex topologies without regular grid interpolation.

Fig. A8: Poisson–Nernst–Planck (PNP) System. Simultaneous prediction of three strongly coupled fields: Electrostatic Potential $\phi$ (top), Cation Concentration $c_{+}$ (middle), and Anion Concentration $c_{-}$ (bottom). The input boundary conditions (not shown) drive complex internal distributions. The error maps (right column) demonstrate that LNF-NO maintains physical consistency across the variables and accurately resolves the sharp concentration gradients (Debye layers) near the boundaries, which are typical in electro-diffusion problems.

PDE Operator Learning

Partial differential equation (PDE) operator learning builds models that map inputs such as coefficients and initial conditions to solution fields.

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