Neural Operator Splitting for Fluid Dynamics in Complex Geometry
Abstract
Solving dynamical systems in complex and diverse geometries remains challenging for both numerical solvers and learning-based surrogate models. We propose Neural Operator Splitting, a framework for learning fluid dynamics while preserving the physical structure of fractional-step numerical schemes. Instead of learning the full-step evolution monolithically, we decompose each time step into physically interpretable steps, and learn each operator separately using intermediate states from the underlying numerical solver. Guided by the linearity properties of the fractional steps, we introduce a conditional-linear Fourier neural operator (CL-FNO) whose linear state mappings are modulated by nonlinear conditioning on inputs such as boundary geometry. We further introduce two complementary strategies, symmetry sampling and corrective cycling, to mitigate error accumulation during autoregressive rollout. Experiments on 2D and 3D fluid flows with varying geometries and initial conditions show that our full model reduces mean relative L2 rollout error by 68\% in 2D and 40\% in 3D compared with a single full-step neural operator. At comparable error levels, neural-operator inference is approximately 103 times faster per step than the Monte Carlo solver.
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BibTeX
@misc{li2026neural,
title = {{Neural Operator Splitting for Fluid Dynamics in Complex Geometry}},
author = {Jianxiang Li and Xi Deng and Hong Chul Nam and Anima Anandkumar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27671},
eprint = {2610.27671},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27671},
}
RIS
TY - UNPB
TI - Neural Operator Splitting for Fluid Dynamics in Complex Geometry
AU - Jianxiang Li
AU - Xi Deng
AU - Hong Chul Nam
AU - Anima Anandkumar
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27671
UR - https://arxiv.org/abs/2610.27671
AB - Solving dynamical systems in complex and diverse geometries remains challenging for both numerical solvers and learning-based surrogate models. We propose Neural Operator Splitting, a framework for learning fluid dynamics while preserving the physical structure of fractional-step numerical schemes. Instead of learning the full-step evolution monolithically, we decompose each time step into physically interpretable steps, and learn each operator separately using intermediate states from the underlying numerical solver. Guided by the linearity properties of the fractional steps, we introduce a conditional-linear Fourier neural operator (CL-FNO) whose linear state mappings are modulated by nonlinear conditioning on inputs such as boundary geometry. We further introduce two complementary strategies, symmetry sampling and corrective cycling, to mitigate error accumulation during autoregressive rollout. Experiments on 2D and 3D fluid flows with varying geometries and initial conditions show that our full model reduces mean relative L^2 rollout error by 68\% in 2D and 40\% in 3D compared with a single full-step neural operator. At comparable error levels, neural-operator inference is approximately 10^3 times faster per step than the Monte Carlo solver.
ER -