physics.flu-dynOct 30, 2026

OPINE: Orthogonal Projection in Invertible Nonlinear Embeddings for Turbulent Flows

Authors: Anqiao Ouyang, Hongyi Ke, Qi Wang

Organizations: Faculty of Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada · Department of Aerospace Engineering, San Diego State University, San Diego, CA 92139, USA

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Abstract

Proper orthogonal decomposition (POD) represents physical fields with orthogonal modes ordered by variance, so that one decomposition gives reduced coordinates in every dimension, but it needs many modes when the data lie near a curved manifold. Nonlinear autoencoders are more compact, but their information loss happens inside a nonlinear encoder, and a model trained with one latent dimension gives no model with a smaller one. We introduce Orthogonal Projection in Invertible Nonlinear Embeddings (OPINE), which learns an invertible nonlinear change of coordinates and performs POD in the transformed coordinates. Because the map is invertible, all information loss occurs in the POD projection, and ordering its basis after training gives reduced coordinates of every smaller dimension from a single model. Training alternates gradient updates of the map with a damped update that tracks the evolving POD subspace, and we prove that this update converges locally to the exact POD subspace. We evaluate OPINE on synthetic manifolds, chaotic Kuramoto-Sivashinsky dynamics, forced Burgers turbulence, two-dimensional Kolmogorov flow, and three-dimensional minimal channel flow. OPINE is more accurate than POD at every tested retained dimension above one, and on the Kuramoto-Sivashinsky and Burgers dynamics, it is more accurate than convolutional and fully connected autoencoders by more than two orders of magnitude at the largest retained dimension.

Figures & tables

Figure 1: OPINE for a scalar field: only the rank- rr POD projection of z=vec⁡fθ(x)z=\operatorname{vec}f_{\theta}(x) loses information, and fθf_{\theta} and its exact inverse share θ\theta . Training alternates the two steps at bottom right (Section 3.4 ). For multi-component fields, S\mathcal{S} is omitted and fθ=Gθf_{\theta}=G_{\theta} .
  same invertible map, projection onto:
 PODfirst DD entriesFourier basisOPINE
Torus4.8×10−34.8{\times}10^{-3}5.3×10−15.3{\times}10^{-1}7.1×10−37.1{\times}10^{-3}1.9×10−5\mathbf{1.9{\times}10^{-5}}
Front1.0×10−21.0{\times}10^{-2}2.4×10−12.4{\times}10^{-1}3.9×10−53.9{\times}10^{-5}6.0×10−6\mathbf{6.0{\times}10^{-6}}
Pulse3.8×10−13.8{\times}10^{-1}3.4×10−13.4{\times}10^{-1}4.1×10−44.1{\times}10^{-4}1.3×10−4\mathbf{1.3{\times}10^{-4}}
Pulse width2.5×10−12.5{\times}10^{-1}5.6×10−15.6{\times}10^{-1}9.6×10−49.6{\times}10^{-4}6.8×10−4\mathbf{6.8{\times}10^{-4}}
Table 1: Bottleneck ablation at D=4D=4 : median test NMSE. POD projects the raw data. The other three columns retrain the same invertible map with a different projection: keeping the first DD entries of zz and setting the others to their training mean, onto a fixed Fourier basis, or onto the POD subspace of the transformed field, which is OPINE. Best in bold. Figure 2 shows sample reconstructions.
Figure 2: Sample reconstructions for the bottleneck ablation of Table 1 at D=4D=4 . Top: a test profile (grey) and its reconstructions; the profile is the test sample with the median OPINE error. Bottom: pointwise error on a logarithmic scale. POD misses the shape of each profile, and keeping the first DD entries of zz gives noisy profiles. The error of OPINE is about two orders of magnitude below that of POD, and on Front and Torus it is also well below that of the fixed Fourier basis.
Figure 3: Kuramoto–Sivashinsky (a) and forced Burgers turbulence (b). Left: a test trajectory in space and time. Middle: pointwise error of each method at the DD marked by the dashed line on the right, on a shared logarithmic scale (darker is worse); titles give the test NMSE of the run shown. Right: test NMSE versus DD (median and range over three seeds).
Figure 4: Kolmogorov flow. Left: test NMSE versus DD (and train NMSE of the learned models). Right: vorticity of a test snapshot and its reconstructions at D=64D=64 ; POD blurs the vortex cores and filaments that the CNN and OPINE retain.
Figure 5: Decoded reduced coordinates for Kolmogorov flow at D=64D=64 , each activated at one training standard deviation. POD and OPINE directions form phase-shifted pairs ( jj : index in variance order), and OPINE’s are sharper. CNN latent coordinates have no ordering or pairing, so for each pair we show the two most correlated ones (#: latent index).
Figure 6: Three-dimensional minimal channel. (a) Test NMSE versus retained dimension DD , with the train NMSE of the CNN and OPINE dashed. (b) Streamwise fluctuation in a wall-parallel plane of a test snapshot and its reconstructions at D=1024D=1024 . (c) QQ -criterion isosurfaces at a common isovalue for D=128D=128 in another test snapshot, and (d) streamwise-fluctuation isosurfaces u′/uτ=±4.29u^{\prime}/u_{\tau}=\pm 4.29 ( uτu_{\tau} : friction velocity) at D=1024D=1024 in the snapshot of (b); color indicates wall-normal position.
Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

DatasetDDPODCNNMLPOPINEOPINE-spectral
Torus113.8×10−13.8{\times}10^{-1}4.9×10−24.9{\times}10^{-2}4.1×10−2\mathbf{4.1{\times}10^{-2}}3.8×10−13.8{\times}10^{-1}3.7×10−13.7{\times}10^{-1}
 225.2×10−25.2{\times}10^{-2}1.2×10−2\mathbf{1.2{\times}10^{-2}}1.4×10−21.4{\times}10^{-2}1.3×10−21.3{\times}10^{-2}1.7×10−21.7{\times}10^{-2}
 332.8×10−22.8{\times}10^{-2}1.1×10−31.1{\times}10^{-3}7.6×10−4\mathbf{7.6{\times}10^{-4}}6.0×10−36.0{\times}10^{-3}6.3×10−36.3{\times}10^{-3}
 444.8×10−34.8{\times}10^{-3}1.6×10−5\mathbf{1.6{\times}10^{-5}}2.3×10−52.3{\times}10^{-5}1.9×10−51.9{\times}10^{-5}2.6×10−52.6{\times}10^{-5}
 661.2×10−31.2{\times}10^{-3}1.8×10−51.8{\times}10^{-5}2.4×10−52.4{\times}10^{-5}1.5×10−51.5{\times}10^{-5}5.5×10−6\mathbf{5.5{\times}10^{-6}}
Pulse117.3×10−17.3{\times}10^{-1}3.1×10−2\mathbf{3.1{\times}10^{-2}}5.8×10−25.8{\times}10^{-2}4.9×10−14.9{\times}10^{-1}7.0×10−17.0{\times}10^{-1}
Appendix
Table 2: Median test NMSE of the one-dimensional study. OPINE uses the spatial coupling, OPINE-spectral the spectral-split coupling described above. Best in bold; “–” marks runs not completed.
DatasetDDfirst DD entriesfixed Fourier basisOPINE
Torus333.3×10−13.3{\times}10^{-1}6.1×10−36.1{\times}10^{-3}6.0×10−3\mathbf{6.0{\times}10^{-3}}
Torus445.3×10−15.3{\times}10^{-1}7.1×10−37.1{\times}10^{-3}1.9×10−5\mathbf{1.9{\times}10^{-5}}
Pulse225.8×10−15.8{\times}10^{-1}6.1×10−16.1{\times}10^{-1}6.1×10−4\mathbf{6.1{\times}10^{-4}}
Pulse443.4×10−13.4{\times}10^{-1}4.1×10−44.1{\times}10^{-4}1.3×10−4\mathbf{1.3{\times}10^{-4}}
Pulse width224.5×10−14.5{\times}10^{-1}5.7×10−15.7{\times}10^{-1}1.1×10−2\mathbf{1.1{\times}10^{-2}}
Pulse width445.6×10−15.6{\times}10^{-1}9.6×10−49.6{\times}10^{-4}6.8×10−4\mathbf{6.8{\times}10^{-4}}
Appendix
Table 3: Bottleneck ablation with the same spatial invertible map: median test NMSE.
DatasetDDfrozenα=0.03\alpha=0.03α=0.3\alpha=0.3exacteigendecomposition
Torus336.1×10−36.1{\times}10^{-3}6.0×10−36.0{\times}10^{-3}5.7×10−35.7{\times}10^{-3}5.8×10−35.8{\times}10^{-3}4.7×10−3\mathbf{4.7{\times}10^{-3}}
Pulse221.7×10−31.7{\times}10^{-3}6.1×10−46.1{\times}10^{-4}1.5×10−41.5{\times}10^{-4}6.2×10−46.2{\times}10^{-4}3.7×10−5\mathbf{3.7{\times}10^{-5}}
Appendix
Table 4: Subspace-update ablation with the spatial invertible map: median test NMSE. The eigendecomposition variant was run with a CPU eigensolver; with a GPU eigensolver, all six runs failed.
DatasetcouplingLLwwparameterstest NMSE
Torus ( D=3D=3 )spatial22323212,32812{,}3287.4×10−37.4{\times}10^{-3}
Torus ( D=3D=3 )spatial44323224,65624{,}6566.0×10−36.0{\times}10^{-3}
Torus ( D=3D=3 )spatial886464180,384180{,}3844.4×10−34.4{\times}10^{-3}
Torus ( D=3D=3 )spectral22323219,52419{,}5248.9×10−38.9{\times}10^{-3}
Torus ( D=3D=3 )spectral44323239,04839{,}0486.3×10−36.3{\times}10^{-3}
KS ( D=8D=8 )spatial44323224,65624{,}6563.8×10−53.8{\times}10^{-5}
Appendix
Table 5: Coupling-stack size: parameters and median test NMSE.
 D=1D=1D=2D=2D=3D=3D=4D=4D=5D=5D=6D=6
Torus, prefix3.8×10−13.8{\times}10^{-1}3.8×10−23.8{\times}10^{-2}1.9×10−21.9{\times}10^{-2}4.4×10−44.4{\times}10^{-4}2.3×10−42.3{\times}10^{-4}1.5×10−51.5{\times}10^{-5}
trained at DD3.8×10−13.8{\times}10^{-1}1.3×10−21.3{\times}10^{-2}6.0×10−36.0{\times}10^{-3}1.9×10−51.9{\times}10^{-5}–1.5×10−51.5{\times}10^{-5}
Pulse width, prefix9.3×10−19.3{\times}10^{-1}4.9×10−14.9{\times}10^{-1}2.4×10−12.4{\times}10^{-1}1.8×10−21.8{\times}10^{-2}7.3×10−37.3{\times}10^{-3}2.1×10−42.1{\times}10^{-4}
trained at DD5.2×10−15.2{\times}10^{-1}1.1×10−21.1{\times}10^{-2}1.9×10−21.9{\times}10^{-2}6.8×10−46.8{\times}10^{-4}–2.2×10−42.2{\times}10^{-4}
Appendix
Table 6: Ordered prefixes of one model trained at K=6K=6 versus models trained at each DD : median test NMSE (no model was trained at D=5D=5 ). The two rows come from separate training runs.
  D=32D=3264641281282562565125121024102420482048
Kolmogorovprefix0.3570.3570.1900.1900.07430.07430.01820.01820.001520.00152  
 trained at DD0.2090.2090.1080.1080.03400.03400.009230.009230.001520.00152  
 POD0.3580.3580.1950.1950.08040.08040.02610.02610.006100.00610  
Minimalprefix0.7100.7100.6340.6340.5470.5470.4600.4600.3690.3690.2800.2800.2040.204
channeltrained at DD0.6910.6910.6080.6080.5220.5220.4400.4400.3560.3560.2770.2770.2040.204
 POD0.7130.7130.6410.6410.5610.5610.4800.4800.3970.3970.3140.3140.2360.236
Appendix
Table 7: Test NMSE of the ordered prefixes of one OPINE model trained at the largest DD , of OPINE models trained at each DD , and of POD.
Figure 7: CNN autoencoder baseline, illustrated for a two-dimensional field with three velocity components: strided convolutions with boundary-aware padding compress the field to a DD -dimensional latent vector, and a mirrored decoder reconstructs it. The three-dimensional experiments use its volumetric counterpart.
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