On Geometry Regularization in Autoencoder Reduced-Order Models with Latent Neural ODE Dynamics

Mikhail Osipov

cs.LGMar 3, 2026

On Geometry Regularization in Autoencoder Reduced-Order Models with Latent Neural ODE Dynamics

Authors: Mikhail Osipov

Organizations: Independent Researcher, Italy

Abstract

We investigate geometric regularization strategies for learned latent representations in encoder--decoder reduced-order models. In a fixed experimental setting for the advection--diffusion--reaction (ADR) equation, we model latent dynamics using a neural ODE and evaluate four regularization approaches applied during autoencoder pre-training: (a) near-isometry regularization of the decoder Jacobian, (b) a stochastic decoder gain penalty based on random directional gains, (c) a second-order directional curvature penalty, and (d) Stiefel projection of the first decoder layer. Across multiple seeds, we find that (a)--(c) often produce latent representations that make subsequent latent-dynamics training with a frozen autoencoder more difficult, especially for long-horizon rollouts, even when they improve local decoder smoothness or related sensitivity proxies. In contrast, (d) consistently improves conditioning-related diagnostics of the learned latent dynamics and tends to yield better rollout performance. We discuss the hypothesis that, in this setting, the downstream impact of latent-geometry mismatch outweighs the benefits of improved decoder smoothness.

Figures & tables

Method	$\mathcal{L}_{\text{val}}$	$\mathrm{CI}_{95}(\mathcal{L}_{\text{val}})$	$ep_{\text{target}}$	$\mathrm{CI}_{95}(ep_{\text{target}})$
Stiefel projection	$8.6\cdot 10^{-4}$	$1.0\cdot 10^{-4}$	13.5	1.6
Vanilla baseline	$8.8\cdot 10^{-4}$	$1.2\cdot 10^{-4}$	14.1	1.4
Curvature penalty	$1.1\cdot 10^{-3}$	$1.4\cdot 10^{-4}$	14.5	1.2
Gain penalty	$1.4\cdot 10^{-3}$	$1.8\cdot 10^{-4}$	18.1	2.5
Isometry penalty	$1.4\cdot 10^{-3}$	$1.8\cdot 10^{-4}$	19.3	3.4

Table 1: Latent NODE training metadata for different regularization methods

Figure 1: Maximum relative error over rollout horizon values

\in\{80,160,240,320\}

on fine grid for different regularization methods and ”vanilla” baseline with no regularization. Uncertainty is reported as a 95% confidence interval.

Figure 2: Mean relative error over rollout horizon values

\in\{80,160,240,320\}

on fine grid for different regularization methods and ”vanilla” baseline with no regularization. Uncertainty is reported as a 95% confidence interval.

Method	(A) Paired $\Delta\varepsilon_{\text{max}}$	(B) Paired $\Delta\varepsilon_{\text{mean}}$
Stiefel projection	0.006 $\pm 4.2\cdot 10^{-3}$	0.005 $\pm 3.1\cdot 10^{-3}$
Curvature penalty	-0.028 $\pm 8.7\cdot 10^{-3}$	-0.024 $\pm 5.1\cdot 10^{-3}$
Gain penalty	-0.035 $\pm 7.6\cdot 10^{-3}$	-0.028 $\pm 4.6\cdot 10^{-3}$
Isometry penalty	-0.036 $\pm 7.9\cdot 10^{-3}$	-0.028 $\pm 4.5\cdot 10^{-3}$

Table 2: Paired rollout-error differences relative to the vanilla baseline

Method	Dyn. conditioning	Decoder gain	Latent error
Stiefel projection	124.2 $\pm 11.5$	4.428 $\pm 9.7\cdot 10^{-2}$	0.988 $\pm 0.022$
Vanilla	145.3 $\pm 13.9$	3.380 $\pm 3.0\cdot 10^{-3}$	1.156 $\pm 0.043$
Curvature penalty	316.6 $\pm 18.9$	1.570 $\pm 1.6\cdot 10^{-2}$	3.434 $\pm 0.119$
Isometry penalty	407.2 $\pm 19.8$	1.000 $\pm 7.0\cdot 10^{-5}$	4.066 $\pm 0.122$
Gain penalty	424.3 $\pm 21.6$	1.017 $\pm 7.2\cdot 10^{-3}$	4.217 $\pm 0.133$

Table 3: Conditioning of the learned latent dynamics, decoder gain, latent error

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Method	$\Delta\varepsilon_{\text{max}}$	Std	95% CI	$p$ -value
Stiefel projection	0.009	0.019	0.010	0.058
Isometry penalty	-0.012	0.020	0.010	0.988
Gain penalty	-0.014	0.018	0.010	0.997
Curvature penalty	-0.016	0.020	0.011	0.997
Method	$\Delta\varepsilon_{\text{mean}}$	Std	95% CI	$p$ -value
Stiefel projection	0.005	0.012	0.006	0.080

Appendix

Table 4: Paired rollout-error differences relative to the vanilla baseline

Reduced-Order Modeling

Reduced-order modeling builds compact approximations of complex systems to make simulation, prediction, and control more efficient.

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Geometric regularization of autoencoders via observed stochastic dynamics

Sean Hill, Felix X. -F. Ye

Department of Mathematics and Statistics, University at Albany, SUNY

Latent Dynamics Modeling Autoencoders Manifold Learning

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Stochastic dynamical systems with slow or metastable behavior evolve, on long time scales, on an unknown low-dimensional manifold in high-dimensional ambient space. Building a reduced simulator from short-burst ambient ensembles is a long-standing problem: local-chart methods like ATLAS suffer from exponential landmark scaling and per-step reprojection, while autoencoder alternatives leave tangent-bundle geometry poorly constrained, and the errors propagate into the learned drift and diffusion. We observe that the ambient covariance~

Λ

already encodes coordinate-invariant tangent-space information, its range spanning the tangent bundle. Using this, we construct a tangent-bundle penalty and an inverse-consistency penalty for a three-stage pipeline (chart learning, latent drift, latent diffusion) that learns a single nonlinear chart and the latent SDE. The penalties induce a function-space metric, the

ρ

-metric, strictly weaker than the Sobolev

H^1

norm yet achieving the same chart-quality generalization rate up to logarithmic factors. For the drift, we derive an encoder-pullback target via Itô's formula on the learned encoder and prove a bias decomposition showing the standard decoder-side formula carries systematic error for any imperfect chart. Under a

W^{2,\infty}

chart-convergence assumption, chart-level error propagates controllably to weak convergence of the ambient dynamics and to convergence of radial mean first-passage times. Experiments on four surfaces embedded in up to

201

ambient dimensions reduce radial MFPT error by

50

70\%

under rotation dynamics and achieve the lowest inter-well MFPT error on most surface--transition pairs under metastable Müller--Brown Langevin dynamics, while reducing end-to-end ambient coefficient errors by up to an order of magnitude relative to an unregularized autoencoder.

Abstract

Reduced-Order Modeling

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