Manifold-adapted radial basis functions for reduced-order modelling of chaotic flows
Abstract
Chaotic systems often evolve on a low-dimensional attractor whose geometry varies from one region to another. We propose a non-intrusive reduced-order model that reads this local geometry by clustering and uses it to shape a radial basis library whose kernels adapt to each region. Fitting the reduced velocity onto this library by one global least-squares solve gives an explicit, differentiable vector field that reproduces the long-term statistics without any use of the governing equations. A radial basis field decays away from the data and cannot by itself return an escaped state. The integration is therefore stabilised by a kinematic corrector, whose reported magnitude measures how far each result rests on the learned field. On Lorenz-63 the model recovers the attractor, its marginal densities and its Lyapunov spectrum. On Lorenz-96 its valid prediction time matches typical configurations of neural-network and reservoir-computing forecasters and trails their best-tuned ones, and the invariant measure is reproduced on both the full state and on a reduced observable. On the Kuramoto--Sivashinsky equation and the quasiperiodic Kolmogorov flow the model matches the energy distribution and spectrum of an intrusive quantised-local Galerkin model and improves on a global Galerkin projection of the same reduced dimension. The recovery is dictated by the distance from a state to its nearest kernels, not by the one-step regression error.
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BibTeX
@misc{cuadrado2026manifoldadapted,
title = {{Manifold-adapted radial basis functions for reduced-order modelling of chaotic flows}},
author = {Miguel Pérez Cuadrado and Giorgio Maria Cavallazzi and Alfredo Pinelli},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.08571},
eprint = {2607.08571},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.08571},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Manifold-adapted radial basis functions for reduced-order modelling of chaotic flows
AU - Miguel Pérez Cuadrado
AU - Giorgio Maria Cavallazzi
AU - Alfredo Pinelli
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2607.08571
UR - https://arxiv.org/abs/2607.08571
AB - Chaotic systems often evolve on a low-dimensional attractor whose geometry varies from one region to another. We propose a non-intrusive reduced-order model that reads this local geometry by clustering and uses it to shape a radial basis library whose kernels adapt to each region. Fitting the reduced velocity onto this library by one global least-squares solve gives an explicit, differentiable vector field that reproduces the long-term statistics without any use of the governing equations. A radial basis field decays away from the data and cannot by itself return an escaped state. The integration is therefore stabilised by a kinematic corrector, whose reported magnitude measures how far each result rests on the learned field. On Lorenz-63 the model recovers the attractor, its marginal densities and its Lyapunov spectrum. On Lorenz-96 its valid prediction time matches typical configurations of neural-network and reservoir-computing forecasters and trails their best-tuned ones, and the invariant measure is reproduced on both the full state and on a reduced observable. On the Kuramoto--Sivashinsky equation and the quasiperiodic Kolmogorov flow the model matches the energy distribution and spectrum of an intrusive quantised-local Galerkin model and improves on a global Galerkin projection of the same reduced dimension. The recovery is dictated by the distance from a state to its nearest kernels, not by the one-step regression error.
ER -