Neural Eigenfluid Bases: Spectral Objectives, Gauge, and Topology
Abstract
Reduced-order models simplify fluid simulation by representing complex flows with a small number of spatial modes. Building on Laplacian Eigenfluids, we introduce Neural Eigenfluid Bases (NEB), a neural representation whose basis functions can be evaluated anywhere in space. We enforce incompressibility and impermeable boundaries directly through differential lifts and potential boundary conditions. Geometry-local orthogonalization and Rayleigh minimization then learn an ordered basis. We connect learning in potential and velocity spaces, and derive a spectral-variance relation explaining why these objectives weight approximate modes differently, even though their exact positive spectra agree. Gauge fixing reduces the number of spatial derivatives needed for potential learning, while a Dirichlet energy loss learns independent circulation fields around obstacles. With 1024 modes, both objectives keep velocity errors below 1% in periodic vortex merger. For random flows around obstacles, potential-based learning captures more initial energy and gives lower final errors than numerical Laplacian bases of the same size. The reduced state is about a thousand times smaller than the full reference model. Three-dimensional tests separate spatial reconstruction errors from errors in coefficient evolution. Precomputed Galerkin operators allow millisecond-scale updates when the same basis is reused across simulations. This provides a physically constrained basis for continuous flow reconstruction and efficient reduced-order simulation.
Related
- Non-Hermitian Spectral Topology Shaped by Memory
- ConQUESTA: Topology-Aware Post-Training Structural Pruning of Quantum Neural Networks
- Topology Obstructs Pure Foundation Neural Quantum States
- Neural topology optimization of ship structures under propulsion machinery vibrations
- Neural-network-based material identification in photon-counting computed tomography using region-of-interest spectral features
BibTeX
@misc{zhang2026neural,
title = {{Neural Eigenfluid Bases: Spectral Objectives, Gauge, and Topology}},
author = {Baiming Zhang and Shiying Xiong and Zhecheng Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33925},
eprint = {2610.33925},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33925},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Neural Eigenfluid Bases: Spectral Objectives, Gauge, and Topology
AU - Baiming Zhang
AU - Shiying Xiong
AU - Zhecheng Wang
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33925
UR - https://arxiv.org/abs/2610.33925
AB - Reduced-order models simplify fluid simulation by representing complex flows with a small number of spatial modes. Building on Laplacian Eigenfluids, we introduce Neural Eigenfluid Bases (NEB), a neural representation whose basis functions can be evaluated anywhere in space. We enforce incompressibility and impermeable boundaries directly through differential lifts and potential boundary conditions. Geometry-local orthogonalization and Rayleigh minimization then learn an ordered basis. We connect learning in potential and velocity spaces, and derive a spectral-variance relation explaining why these objectives weight approximate modes differently, even though their exact positive spectra agree. Gauge fixing reduces the number of spatial derivatives needed for potential learning, while a Dirichlet energy loss learns independent circulation fields around obstacles. With 1024 modes, both objectives keep velocity errors below 1% in periodic vortex merger. For random flows around obstacles, potential-based learning captures more initial energy and gives lower final errors than numerical Laplacian bases of the same size. The reduced state is about a thousand times smaller than the full reference model. Three-dimensional tests separate spatial reconstruction errors from errors in coefficient evolution. Precomputed Galerkin operators allow millisecond-scale updates when the same basis is reused across simulations. This provides a physically constrained basis for continuous flow reconstruction and efficient reduced-order simulation.
ER -