aiqbench analysis
This paper advances the theoretical understanding of neural operators by establishing polynomial complexity bounds for Laplacian-based architectures applied to nonlinear reaction-diffusion systems. By leveraging spectral representations of the diffusion semigroup, the authors demonstrate a method to mitigate parametric complexity while maintaining high fidelity in PDE surrogate modeling. The work provides critical insights into optimizing network depth and width for scientific machine learning tasks involving partial differential equations.
Approximation Theory of Laplacian-Based Neural Operators for Reaction-Diffusion System
arXiv:2605.12025v2 Announce Type: replace 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.
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.
Machine Learning (cs.LG); Machine Learning (stat.ML)
arXiv:2605.12025 [cs.LG]
(or arXiv:2605.12025v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2605.12025 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 225 words