Manifold Navigation Learning Using Recurrent Networks: Efficient Training Techniques

Sarah Harkins Dayton; Sathya Rengaswami; Vasileios Maroulas; David L. Boothe; Ioannis D. Schizas

cs.CVOct 9, 2026

Manifold Navigation Learning Using Recurrent Networks: Efficient Training Techniques

Authors: Sarah Harkins Dayton, Sathya Rengaswami, Vasileios Maroulas, David L. Boothe, Ioannis D. Schizas

Organizations: Department of Math, University of Tennessee. · U.S. Army Combat Capabilities Development Command, Army Research Laboratory.

Abstract

Position tracking is a well-studied problem in Euclidean navigation, yet comparatively little attention has been given to navigation on general manifolds, where motion is governed by the underlying geometry of the space. In this work, we formalize the problem of manifold navigation as a velocity-driven tracking task and propose a coordinate-chart-based learning framework in which learning models operate entirely in a Euclidean representation. Predicted trajectories are mapped to the manifold via an immersion function prior to error computation, allowing geometric consistency to be evaluated without explicitly enforcing manifold constraints during learning. Within this framework, we focus on Gated Recurrent Units (GRUs) for sequential position prediction on curved manifolds after comparisons with alternative recurrent architectures. We evaluate our GRU models, trained with pertinent custom loss functions, against both a standard MSE loss function and a naive vector-addition baseline. We evaluate our approach on both analytically (and numerically) defined manifolds, as well as a surface reconstructed from a LiDAR point cloud. Our results show that recurrent models can learn to approximate manifold-constrained dynamics from velocity sequences and consistently outperform Euclidean baselines. Our results demonstrate that certain custom loss functions significantly improve long-horizon prediction accuracy compared to conventional point-wise loss functions.

Figures & tables

Strategy	Notation	Formula for $j^{th}$ term
Standard	$\mathcal{L}_{k}$	$j,\quad j\in[1,k]$
Exponential	exp_k	$\text{integer}(n_{\max}^{j/(k+1)})$
Mixed linear	mixed_p_q $\Delta=\left\lceil\frac{n_{\max}-p}{q}\right\rceil$	$\begin{cases}j,&j\in[1,p]\\ p+\Delta(j-p),&j\in(p,p+q]\end{cases}$

TABLE I : Strategies for Index Selection

Scheme	Indices
$\mathcal{L}_{50}$	1,…, 50
exp_10	1, 1, 2, 2, 4, 5, 8, 12, 17, 24, 35, 50
mixed_7_7	1, 2, 3, 4, 5, 6, 7, 14, 20, 26, 32, 38, 44, 50

TABLE II : The different loss functions and the corresponding indices aggregated.

	Plane		Sphere		Torus
Loss Func.	Brownian	AR(1)	Brownian	AR(1)	Brownian	AR(1)
Baseline	– (–)	– (–)	0.00005 (–)	0.00200 (–)	0.00387 (–)	0.00742 (–)
$\mathcal{L}_{50}$	0.00118 (0.00109)	0.00054 (0.00037)	0.00007 (0.00002)	0.00018 (0.00030)	0.00161 (0.00106)	0.00055 (0.00038)
exp_10	0.00030 (0.00048)	0.00009 (0.00006)	0.00009 (0.00012)	0.00013 (0.00010)	0.00054 (0.00033)	0.00025 (0.00015)
mixed_7_7	0.00027 (0.00022)	0.00007 (0.00002)	$0.00006\,(0.00003)^{*}$	0.00013 (0.00012)	0.00029 (0.00018)	0.00046 (0.00018)

TABLE III : Prediction errors (MSE over all 100 timesteps) of GRUs trained with different loss functions. Values shown are averaged over 10 runs, with standard deviation in parentheses. Bold signifies the lowest MSE, while an asterisk next to the result indicates the lowest recurrent-based MSE.

Fig. 1 : Mean squared error at each time step in the test trajectories for GRU models with different loss functions averaged across 10 independently trained models of each respective type against the vector addition baseline model.

Setup	Brownian	AR(1)
Baseline	0.01268 (NA)	0.00225 (NA)
$\mathcal{L}_{50}$	0.00291 (0.00112)	0.00032 (0.00011)
exp_10	0.00276 (0.00102)	0.00032 (0.00014)
mixed_7_7	0.00271 (0.00095)	0.00031 (0.00011)

TABLE IV : Prediction MSE on the reconstructed surface

Panel of Figure 6 — Fig. 2 : Predicton MSE for Brownian motion trajectories on the torus and the reconstructed surface with the $10^{th}$ time step of the velocity sequence missing.

Panel of Figure 7 — Fig. 3 : Mean squared error of predictions for Brownian motion trajectories on the torus and the reconstructed surface for different missing velocity portions.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

		Plane		Sphere		Torus
Loss Func.	Model	Brownian	AR(1)	Brownian	AR(1)	Brownian	AR(1)
	Baseline	– (–)	– (–)	0.00005 (–)	0.00200 (–)	0.00387 (–)	0.00742 (–)
$\mathcal{L}_{10}$	GRU	0.10376 (0.09999)	0.28024 (0.08625)	0.03291 (0.00406)	0.02627 (0.01133)	0.11175 (0.02301)	0.09435 (0.01313)
	LSTM	0.54111 (0.32620)	0.06560 (0.02796)	0.00936 (0.00489)	0.01819 (0.00631)	0.14782 (0.05111)	0.51796 (0.13962)
	RNN	2.26389 (6.02657)	0.00985 (0.02976)	0.00150 (0.00171)	0.00811 (0.00604)	0.08622 (0.16793)	0.10722 (0.27902)
$\mathcal{L}_{30}$	GRU	0.00574 (0.00411)	0.00608 (0.00385)	0.00149 (0.00041)	0.00019 (0.00017)	0.00334 (0.00155)	0.01392 (0.00225)

Appendix

TABLE V : Prediction error (MSE) for GRU, LSTM, and RNN with different versions of the standard loss function. The mean values and standard deviation (in parentheses) of the localization MSE are displayed.

Setup	Brownian	AR(1)
Baseline	0.01268 (NA)	0.00225 (NA)
GRU	0.00291 (0.00112)	0.00032 (0.00011)
LSTM	0.00264 (0.00094)	0.00036 (0.00010)
RNN	0.01611 (0.02645)	0.00112 (0.00034)

Appendix

TABLE VI : Prediction error (MSE) of recurrent models on the reconstructed surface with the

\mathcal{L}_{50}

loss function.

Setup	Brownian	AR(1)
Baseline	0.01268 (NA)	0.00225 (NA)
GRU	0.00271 (0.00095)	0.00031 (0.00011)
LSTM	0.00295 (0.00098)	0.00036 (0.00011)
RNN	0.06543 (0.07541)	0.01220 (0.02937)

Appendix

TABLE VII : Prediction error (MSE) of recurrent models on reconstructed surface with the

\text{mixed}\_{7\_7}

loss function.

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