FrFNO:Injecting the analytic Mittag-Leffler propagator into a resolution-robust neural operator for space-time fractional PDEs
Abstract
Fractional partial differential equations couple a memory-dependent time derivative with a nonlocal fractional Laplacian, and their repeated solution under varying fractional orders, initial data, or diffusivity fields is computationally expensive. Neural operators offer a fast surrogate, but standard architectures must relearn the dominant linear fractional evolution from data. We propose the \emph{Fractional Fourier Neural Operator} (FrFNO), a resolution-robust conditional operator. It injects the analytic Mittag--Leffler response of the frozen linear part as a once-precomputed, resolution-independent propagator table, and trains a spectral convolutional network only on the residual induced by variable coefficients and nonlinear advection, conditioned continuously on both fractional orders. The central theoretical result is that under zero-shot super-resolution the injected propagator fills the out-of-band modes that a standard Fourier neural operator sets to zero, replacing the out-of-band full-field tail by the smaller residual tail (a K-independent constant-factor reduction under weak perturbation). The residual and the full solution carry the same Sobolev order, so the advantage is amplitude reduction rather than a steeper tail. Mesh refinement drives the error to a residual-controlled floor. On a nonlinear two-dimensional space--time fractional Burgers problem FrFNO achieves the lowest relative L2 error among six baselines (FNO, PINO, PDNO, CNO, DeepONet, U-Net) at the training resolution and under zero-shot super-resolution, yields the smallest spectral phase error, and remains the best at the integer-order limit and for long integration windows. The same construction extends to coupled fractional systems such as fractional Allen-Cahn and Navier--Stokes equations. Code, training scripts, and reference outputs are available at https://github.com/Derek2021Pang/FrFNO.
Related
- Continuity properties of the Laguerre operator and its propagator
- Analytic discs and compactness of the ∂̅-Neumann operator
- The differentiation and integration operator on weighted spaces of analytic functions in the weight sequence setting
- Bohr-type inequalities for certain integral transforms of operator-valued analytic functions on shifted disc
- Norms of the Hilbert Matrix Operator on Some Analytic Function Spaces
BibTeX
@misc{pang2026frfnoinjecting,
title = {{FrFNO:Injecting the analytic Mittag-Leffler propagator into a resolution-robust neural operator for space-time fractional PDEs}},
author = {Guofei Pang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21512},
eprint = {2609.21512},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21512},
}
RIS
TY - UNPB
TI - FrFNO:Injecting the analytic Mittag-Leffler propagator into a resolution-robust neural operator for space-time fractional PDEs
AU - Guofei Pang
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21512
UR - https://arxiv.org/abs/2609.21512
AB - Fractional partial differential equations couple a memory-dependent time derivative with a nonlocal fractional Laplacian, and their repeated solution under varying fractional orders, initial data, or diffusivity fields is computationally expensive. Neural operators offer a fast surrogate, but standard architectures must relearn the dominant linear fractional evolution from data. We propose the \emph{Fractional Fourier Neural Operator} (FrFNO), a resolution-robust conditional operator. It injects the analytic Mittag--Leffler response of the frozen linear part as a once-precomputed, resolution-independent propagator table, and trains a spectral convolutional network only on the residual induced by variable coefficients and nonlinear advection, conditioned continuously on both fractional orders. The central theoretical result is that under zero-shot super-resolution the injected propagator fills the out-of-band modes that a standard Fourier neural operator sets to zero, replacing the out-of-band full-field tail by the smaller residual tail (a K-independent constant-factor reduction under weak perturbation). The residual and the full solution carry the same Sobolev order, so the advantage is amplitude reduction rather than a steeper tail. Mesh refinement drives the error to a residual-controlled floor. On a nonlinear two-dimensional space--time fractional Burgers problem FrFNO achieves the lowest relative L^2 error among six baselines (FNO, PINO, PDNO, CNO, DeepONet, U-Net) at the training resolution and under zero-shot super-resolution, yields the smallest spectral phase error, and remains the best at the integer-order limit and for long integration windows. The same construction extends to coupled fractional systems such as fractional Allen-Cahn and Navier--Stokes equations. Code, training scripts, and reference outputs are available at https://github.com/Derek2021Pang/FrFNO.
ER -