Approximation Theory of Laplacian-Based Neural Operators for Reaction-Diffusion System
Abstract
Neural operators provide a framework for learning solution operators of partial differential equations (PDEs), enabling efficient surrogate modeling for complex systems. While universal approximation results are now well understood, quantitative approximation analysis specific to nonlinear reaction-diffusion systems remains limited. In this paper, we study Laplacian-based neural operators applied to the solution mapping from initial conditions to time-dependent solutions of nonlinear reaction-diffusion systems. By exploiting the Laplacian spectral representation of the diffusion semigroup, we obtain polynomial complexity bounds on the network depth, width, and spectral rank, demonstrating that Laplacian eigenfunction-based neural operators can exploit the structure of the underlying PDE while avoiding the curse of parametric complexity encountered in generic operator learning. Numerical experiments support the theoretical findings and illustrate the benefit of Laplacian eigenfunction representations compared with Fourier-based spectral representations.
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BibTeX
@misc{furuya2026approximation,
title = {{Approximation Theory of Laplacian-Based Neural Operators for Reaction-Diffusion System}},
author = {Takashi Furuya and Ryo Ozawa and Jenn-Nan Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.12025},
eprint = {2605.12025},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.12025},
}
RIS
TY - UNPB
TI - Approximation Theory of Laplacian-Based Neural Operators for Reaction-Diffusion System
AU - Takashi Furuya
AU - Ryo Ozawa
AU - Jenn-Nan Wang
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2605.12025
UR - https://arxiv.org/abs/2605.12025
AB - Neural operators provide a framework for learning solution operators of partial differential equations (PDEs), enabling efficient surrogate modeling for complex systems. While universal approximation results are now well understood, quantitative approximation analysis specific to nonlinear reaction-diffusion systems remains limited. In this paper, we study Laplacian-based neural operators applied to the solution mapping from initial conditions to time-dependent solutions of nonlinear reaction-diffusion systems. By exploiting the Laplacian spectral representation of the diffusion semigroup, we obtain polynomial complexity bounds on the network depth, width, and spectral rank, demonstrating that Laplacian eigenfunction-based neural operators can exploit the structure of the underlying PDE while avoiding the curse of parametric complexity encountered in generic operator learning. Numerical experiments support the theoretical findings and illustrate the benefit of Laplacian eigenfunction representations compared with Fourier-based spectral representations.
ER -