MSINO: Curvature-Aware Sobolev Optimization for Manifold Neural Networks

Suresan Pareth

cs.LGFeb 26, 2026

MSINO: Curvature-Aware Sobolev Optimization for Manifold Neural Networks

Authors: Suresan Pareth

Abstract

We introduce Manifold Sobolev Informed Neural Optimization (MSINO), a curvature aware training framework for neural networks defined on Riemannian manifolds. The method replaces standard Euclidean derivative supervision with a covariant Sobolev loss that aligns gradients using parallel transport and improves stability via a Laplace Beltrami smoothness regularization term. Building on classical results in Riemannian optimization and Sobolev theory on manifolds, we derive geometry dependent constants that yield (i) a Descent Lemma with a manifold Sobolev smoothness constant, (ii) a Sobolev Polyak Lojasiewicz inequality giving linear convergence guarantees for Riemannian gradient descent and stochastic gradient descent under explicit step size bounds, and (iii) a two step Newton Sobolev method with local quadratic contraction in curvature controlled neighborhoods. Unlike prior Sobolev training in Euclidean space, MSINO provides training time guarantees that explicitly track curvature and transported Jacobians. Applications include surface imaging, physics informed learning settings, and robotics on Lie groups such as SO(3) and SE(3). The framework unifies value and gradient based learning with curvature aware convergence guarantees for neural training on manifolds.

Figures & tables

Task	Domain	Operator	Vertices
Cortical Thickness	3D Mesh ( $M$ )	Cotangent Laplacian	2562
Climate Temperature	$\mathbb{S}^{2}$	Spherical Laplacian	4096 grid points
Robot Geodesics	$\mathrm{SO}(3)$	Lie Algebra Log-Map	–

Table 1. Domains, operators, and discretizations used across MSINO tasks.

Figure 1. Unified MSINO metrics across all tasks . Columns show total Sobolev loss, adaptive scheduling $\lambda_{k}$ , contraction ratio $\rho_{k}$ , and the smoothness constant $L^{M}_{\mathrm{sob}}(\theta_{k})$ .

Figure 2. MSINO convergence on cortical mesh .

Epoch	Total	Val	Sob	Lap	$\lambda_{k}$	$\rho_{k}$
1	10.428062	9.827149	0.594605	6.308184	2.026	1.000
5	3.836373	0.163237	0.653500	3019.635010	2.267	0.974
10	3.533200	1.470795	0.761388	1301.016479	2.632	1.000
15	3.439175	0.670132	0.802080	1966.962280	2.768	1.000
20	3.381762	0.666484	0.743773	1971.504395	2.566	0.995
25	3.427012	0.980426	0.790914	1655.671753	2.730	1.000

Table 2. MSINO training statistics for cortical surface regression .

Figure 3. Predicted cortical thickness on the mesh.

Figure 4. Prediction vs ground truth vs error .

Epoch	Total	Val	Sob	$\lambda_{k}$	$\rho_{k}$
1	217.7687	214.5644	3.2043	10.000	0.973
5	161.9446	158.3310	3.6136	10.000	0.995
10	155.2852	151.2550	4.0303	10.000	0.993
15	154.7804	151.0713	3.7091	10.000	0.967
20	154.8004	151.0769	3.7234	10.000	0.991
25	154.1294	151.0634	3.0660	10.000	0.961

Table 3. MSINO training statistics for climate temperature regression on

\mathbb{S}^{2}

. Each row reports total Sobolev loss, validation loss, first-order Sobolev term, Laplacian penalty, adaptive

\lambda_{k}

, and contraction ratio

\rho_{k}

Figure 5. Climate experiment: convergence on $\mathbb{S}^{2}$ . (Left) Total Sobolev loss $L_{\mathrm{Sob},M}(\theta_{k})$ . (Right) Contraction ratio $\rho_{k}$ , illustrating stable behaviour compatible with Theorems 4.1 and 5.3 .

Figure 6. Learned temperature field on the sphere (Mollweide projection). MSINO recovers the dominant large-scale climate structures while respecting the intrinsic geometry of $\mathbb{S}^{2}$ .

Figure 7. Climate field: ground truth, MSINO prediction, and absolute error (RMSE = 12.29). High-frequency residuals are concentrated near sharp front-like transitions, and their magnitude is controlled by the curvature-aware smoothness constant in Definition 2.3 .

Epoch	Loss	$L^{M}_{\mathrm{sob}}(\theta_{k})$	lr_cap
1	0.9240	45.085	0.02218
5	0.5782	52.514	0.01904
10	0.9932	60.404	0.01656
15	0.9630	63.143	0.01584
20	$2.97\times 10^{12}$	68.577	0.01458

Table 4. MSINO training statistics for robot geodesic learning on

\mathrm{SO}(3)

. Each row reports geodesic loss, estimated Sobolev smoothness constant

L^{M}_{\mathrm{sob}}(\theta_{k})

, and the corresponding learning-rate cap

\text{lr\_cap}\approx 1/L^{M}_{\mathrm{sob}}

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Entropic Riemannian Neural Optimal Transport

Alessandro Micheli, Silvia Sapora, Anthea Monod, Samir Bhatt

Imperial College London London, UK · University of Oxford Oxford, UK · Statens Serum Institut Copenhagen, Denmark+1

Riemannian Geometry Optimal Transport

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Many machine learning problems involve data supported on curved spaces such as spheres, rotation groups, hyperbolic spaces, and general Riemannian manifolds, where Euclidean geometry can distort distances, averages, and the resulting optimal transport (OT) problem. Existing manifold OT methods have pursued amortized out-of-sample maps, while entropic regularization has made discrete OT more scalable, but these advantages have remained largely disjoint. We propose Entropic Riemannian Neural Optimal Transport (Entropic RNOT), a unified framework that combines intrinsic entropic OT with amortized out-of-sample evaluation on Riemannian manifolds. Our method learns a single target-side Schrödinger potential through a neural pullback parameterization, recovers the induced Gibbs coupling, and uses the resulting conditional laws to construct intrinsic transport surrogates. These include barycentric projections on Cartan-Hadamard manifolds and heat-smoothed conditional surrogates on stochastically complete manifolds, the latter turning possibly atomic target laws into absolutely continuous ones. For fixed regularization

\varepsilon>0

, we prove that the proposed hypothesis class recovers the entropic optimal coupling in strong probabilistic metrics. As consequences, barycentric surrogates converge in

L^2

, while heat-smoothed surrogates are stable at fixed heat time and asymptotically unbiased as the heat time vanishes. The guarantees hold for compactly supported data on possibly noncompact manifolds. Empirically, our method matches or improves over Euclidean, tangent-space, and log-Euclidean baselines on benchmarks over

\mathbb{S}^2

\mathrm{SO}(3)

\mathrm{SPD}(3)

\mathrm{SE}(3)

, and

\mathbb{H}^2

, scales favorably relative to discrete manifold Sinkhorn, and in a protein-ligand docking application, refines poses on

\mathrm{SE}(3)

without retraining or per-instance optimization.

Abstract

Riemannian Neural Networks

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