Neural Manifold Gluing Explained

• Neural manifold gluing is a geometric design principle that builds a global manifold from locally valid neural representations using trainable transition operators.
• It employs techniques such as graph-harmonic scattering, asymptotic analytic patch gluing, and probabilistic chart averaging to enforce compatibility across local charts.
• The approach has shown empirical success in neuroscience, differential equations, topology recovery, and multi-domain graph pre-training.

Neural manifold gluing is a family of geometric constructions in which locally valid neural or analytic representations are combined into a coherent global object. In recent arXiv literature, the term has been used in at least four technically distinct but conceptually related senses: gluing structural and functional brain manifolds through graph-harmonic scattering on the SPD manifold of functional connectivity; gluing asymptotic analytic solution patches by replacing integration constants with neural coefficient functions; gluing multiple degenerate normalizing-flow charts to recover non-trivial manifold topology; and gluing local graph-induced Riemannian pieces into a unified manifold for multi-domain graph pre-training (Dan et al., 2024, Kim et al., 9 Jan 2026, Yu et al., 30 May 2025, Sun et al., 28 Feb 2026). Across these settings, the common pattern is local chart construction, compatibility enforcement on overlaps or interfaces, and a neural mechanism that supplies the transition structure.

1. Conceptual scope and definitions

In differential-geometric terms, manifold gluing means constructing a global manifold from local pieces by specifying compatibility on overlaps. The recent neural variants retain this chart-based logic, but replace explicit transition maps or hand-designed matching rules with trainable operators, coefficient functions, probabilistic responsibilities, or discrete tangent transports.

In "Exploring the Enigma of Neural Dynamics Through A Scattering-Transform Mixer Landscape for Riemannian Manifold" (Dan et al., 2024), neural manifold gluing denotes the coherent coupling of structural connectivity and functional connectivity through a common Riemannian and graph-harmonic framework. Functional connectivity matrices are treated as points on the SPD manifold, while structural connectivity provides graph harmonics and scattering operators that constrain all manifold mappings.

In "GlueNN: gluing patchwise analytic solutions with neural networks" (Kim et al., 9 Jan 2026), the phrase denotes gluing asymptotic analytic charts of a differential equation into a global solution. Classical integration constants are promoted to scale-dependent neural functions, and the full governing equation is enforced over the domain so that transitions are learned rather than fixed by a hand-chosen matching point.

In "Learning geometry and topology via multi-chart flows" (Yu et al., 30 May 2025), gluing refers to building an atlas of degenerate normalizing flows, each chart covering a portion of a manifold that may have non-trivial topology. The gluing mechanism is probabilistic: chart overlap is handled by posterior component responsibilities, and geometric quantities are aggregated across charts.

In "Multi-Domain Riemannian Graph Gluing for Building Graph Foundation Models" (Sun et al., 28 Feb 2026), neural manifold gluing is a theory and neural construction for merging heterogeneous graph datasets into one unified, smooth Riemannian manifold. Local metrics are learned from perturbation-induced tangent frames, then glued by metric-preserving edge transports, holonomy regularization, and curvature-inspired smoothness.

These formulations do not define a single canonical method. Rather, they identify a common research program: represent local structure in a geometry-aware way, then learn globally consistent transition behavior.

2. Local charts, metrics, and coordinate systems

The local objects being glued differ across domains, but all four frameworks are organized around chart-like representations.

For brain connectivity, each functional connectivity matrix is an SPD matrix

$X \in \mathrm{Sym}_+^N,$

often computed from whole-brain BOLD time courses $H \in \mathbb{R}^{N \times T}$ by

$X = H H^\top \in \mathrm{Sym}_+^N.$

The SPD manifold is therefore the neural manifold on which functional states live, while structural connectivity defines a graph Laplacian

$L = D - A, \qquad L = U \Lambda U^\top,$

whose eigenvectors $U$ provide graph harmonics (Dan et al., 2024).

For GlueNN, the local charts are asymptotic analytic solutions $f_j$ valid in different patches $p_j$ of a domain. If a patchwise solution is written as

$$f_j(x; c_j^{(1)}, c_j^{(2)}, \dots) = \sum_i c_j^{(i)} \phi_j^{(i)}(x),$$

then the constants $c_j^{(i)}$ are the chart coordinates. Neural manifold gluing begins when these constants are promoted to smooth functions $c_j^{(i)}(x;\theta)$, yielding the global ansatz

$H \in \mathbb{R}^{N \times T}$0

This turns local asymptotic coordinates into learned transition variables (Kim et al., 9 Jan 2026).

For multi-chart flows, each chart is a degenerate normalizing flow

$H \in \mathbb{R}^{N \times T}$1

where $H \in \mathbb{R}^{N \times T}$2 is a latent-space flow and $H \in \mathbb{R}^{N \times T}$3 embeds latent coordinates into $H \in \mathbb{R}^{N \times T}$4. Each chart induces the pullback metric

$H \in \mathbb{R}^{N \times T}$5

so each local map carries its own Riemannian geometry (Yu et al., 30 May 2025).

For GraphGlue, each graph sample yields a local manifold piece $H \in \mathbb{R}^{N \times T}$6 around an embedding $H \in \mathbb{R}^{N \times T}$7. An Adaptive Orthogonal Frame $H \in \mathbb{R}^{N \times T}$8 is constructed from perturbation-induced tangent directions, and the local metric is

$H \in \mathbb{R}^{N \times T}$9

In the learned frame coordinates, the metric is diagonal SPD, and its log-determinant

$X = H H^\top \in \mathrm{Sym}_+^N.$0

acts as a volume-density scalar field (Sun et al., 28 Feb 2026).

A plausible implication is that neural manifold gluing is less about one specific network architecture than about how local geometry is parameterized before compatibility is imposed.

3. Gluing mechanisms

The central technical question is how local pieces are made compatible. The answer differs substantially across the four lines of work.

In the brain-connectivity setting, gluing is realized by imposing structural graph geometry on every mapping of the functional SPD manifold. For harmonic index $X = H H^\top \in \mathrm{Sym}_+^N.$1 and region $X = H H^\top \in \mathrm{Sym}_+^N.$2, a localized harmonic wavelet is defined by

$X = H H^\top \in \mathrm{Sym}_+^N.$3

with spectral filter $X = H H^\top \in \mathrm{Sym}_+^N.$4. Collecting these wavelets gives a block matrix $X = H H^\top \in \mathrm{Sym}_+^N.$5, and the core positive scattering map is

$X = H H^\top \in \mathrm{Sym}_+^N.$6

After block-wise max pooling and symmetrization,

$X = H H^\top \in \mathrm{Sym}_+^N.$7

the output remains in $X = H H^\top \in \mathrm{Sym}_+^N.$8. Structural connectivity is therefore glued onto functional connectivity by forcing manifold mappings to be built from SC-derived harmonic operators (Dan et al., 2024).

In GlueNN, gluing is accomplished by neural coefficient functions trained under a composite loss

$X = H H^\top \in \mathrm{Sym}_+^N.$9

The data term enforces initial or boundary conditions, the differential-equation term enforces the full governing equation on collocation points, and the patch term suppresses a patch’s contribution outside its domain of validity. The coefficients become approximately constant in patch interiors and vary smoothly in transition zones. The paper explicitly characterizes this as a neural partition-of-unity-like gluing (Kim et al., 9 Jan 2026).

In multi-chart flows, the gluing mechanism is probabilistic rather than explicit. A collection of chart densities is combined as

$L = D - A, \qquad L = U \Lambda U^\top,$0

and overlapping charts are glued through responsibilities

$L = D - A, \qquad L = U \Lambda U^\top,$1

A manifold point associated with an ambient point $L = D - A, \qquad L = U \Lambda U^\top,$2 is reconstructed by responsibility-weighted averaging,

$L = D - A, \qquad L = U \Lambda U^\top,$3

Tangent vectors and other geometric quantities are combined analogously. The transition maps are therefore implicit, encoded by overlapping reconstructions and posterior weights (Yu et al., 30 May 2025).

In GraphGlue, gluing is discrete and Riemannian. For an edge $L = D - A, \qquad L = U \Lambda U^\top,$4, the tangent translation

$L = D - A, \qquad L = U \Lambda U^\top,$5

acts as the metric geometric mean map. It is stated to be the optimal isometry between tangent spaces in Frobenius norm and induces boundary isometries between local patches. Around cycles, compatibility is enforced by the holonomy map

$L = D - A, \qquad L = U \Lambda U^\top,$6

with triangle-based holonomy loss

$L = D - A, \qquad L = U \Lambda U^\top,$7

Curvature smoothness is encouraged through log-determinant variation,

$L = D - A, \qquad L = U \Lambda U^\top,$8

where $L = D - A, \qquad L = U \Lambda U^\top,$9 (Sun et al., 28 Feb 2026).

Taken together, these mechanisms show that “gluing” may mean constrained operator composition, learned coefficient interpolation, Bayesian chart averaging, or discrete parallel transport, depending on the problem class.

4. Neural architectures and training procedures

The architectures implementing neural manifold gluing are tailored to the geometry of the target problem.

DeepHoloBrain is described as a Riemannian analogue of an MLP-Mixer, where tokens are brain regions and channels are harmonic frequencies. The mapping $U$0 is interpreted as column-wise and row-wise scattering, and separate MLPs modulate the scaling parameter $U$1 for the column and row sides. A Mixer layer is written as

$U$2

After several layers, the representation is projected to the tangent space of the SPD manifold and passed to a fully connected head. The scale parameter is learned under a cross-entropy objective with non-negativity regularization on $U$3 (Dan et al., 2024).

GlueNN uses a head-trunk architecture. A shared head network processes the input coordinate, and multiple trunk networks output the scalar coefficient functions $U$4. In the chemical kinetics example, the ansatz is

$U$5

with a head of 2 hidden layers and output width 100, and two trunk networks with 1 hidden layer of width 100. In the inflationary vector example, the ansatz is

$U$6

with a head of 5 hidden layers with widths $U$7 and output width 50, plus three trunk networks of hidden width 50 (Kim et al., 9 Jan 2026).

The multi-chart flow framework trains a mixture of degenerate flows. For a chart $U$8, the regularized component objective is

$U$9

Training can proceed by direct MLE or by EM. In the EM formulation, the E-step computes responsibilities by a stopped-gradient softmax over chart log-densities, and the M-step optimizes the responsibility-weighted objective

$f_j$0

A balanced-usage regularizer can also be added to keep average responsibilities near the uniform prior (Yu et al., 30 May 2025).

GraphGlue uses a standard 2-layer GCN encoder in the reported experiments. Local geometry is generated by $f_j$1-sparse perturbation with $f_j$2 virtual nodes per sample, followed by QR-based AOF with length recovery. Dataset-level EMA prototypes are maintained as

$f_j$3

$f_j$4

Pre-training combines local self-supervision, a sample-prototype contrastive loss, and geometric regularization through $f_j$5 and $f_j$6. Adaptation uses a prompt matrix $f_j$7, a transfer graph to nearby prototypes, and a Riemannian mixture-of-experts whose aligned metric is

$f_j$8

The downstream representation is

$f_j$9

and is trained with

$p_j$0

(Sun et al., 28 Feb 2026)

5. Empirical manifestations and application domains

The empirical role of neural manifold gluing is clearest when local models are individually adequate but global consistency is otherwise difficult to achieve.

In the neuroscience setting, DeepHoloBrain is evaluated on task recognition and disease diagnosis. On HCP-A, task recognition accuracy is reported as $p_j$1, compared with SPDNet $p_j$2 and DeepO2P $p_j$3. On OASIS, AD versus CN accuracy is reported as $p_j$4, compared with SPDNet $p_j$5 and DeepO2P $p_j$6. The paper also reports that pretraining on HCP-A and fine-tuning on ADNI improves classification, with OURS+ accuracy $p_j$7 versus SPDNet+ $p_j$8. Attention analyses are described as highlighting sensorimotor regions for VISMOTOR, visual regions for FACENAME, and default mode network regions for AD diagnosis, while frequency-specific attention differs by task and remains consistent across ADNI and OASIS for disease-related signatures (Dan et al., 2024).

GlueNN is demonstrated on two 1D ODE problems. In the freeze-out example, the governing equation is

$p_j$9

with $$f_j(x; c_j^{(1)}, c_j^{(2)}, \dots) = \sum_i c_j^{(i)} \phi_j^{(i)}(x),$$0 and $$f_j(x; c_j^{(1)}, c_j^{(2)}, \dots) = \sum_i c_j^{(i)} \phi_j^{(i)}(x),$$1. Training uses $$f_j(x; c_j^{(1)}, c_j^{(2)}, \dots) = \sum_i c_j^{(i)} \phi_j^{(i)}(x),$$2 points on $$f_j(x; c_j^{(1)}, c_j^{(2)}, \dots) = \sum_i c_j^{(i)} \phi_j^{(i)}(x),$$3 with $$f_j(x; c_j^{(1)}, c_j^{(2)}, \dots) = \sum_i c_j^{(i)} \phi_j^{(i)}(x),$$4, and $$f_j(x; c_j^{(1)}, c_j^{(2)}, \dots) = \sum_i c_j^{(i)} \phi_j^{(i)}(x),$$5 points on $$f_j(x; c_j^{(1)}, c_j^{(2)}, \dots) = \sum_i c_j^{(i)} \phi_j^{(i)}(x),$$6 with $$f_j(x; c_j^{(1)}, c_j^{(2)}, \dots) = \sum_i c_j^{(i)} \phi_j^{(i)}(x),$$7. In the inflationary vector example, parameters are $$f_j(x; c_j^{(1)}, c_j^{(2)}, \dots) = \sum_i c_j^{(i)} \phi_j^{(i)}(x),$$8, $$f_j(x; c_j^{(1)}, c_j^{(2)}, \dots) = \sum_i c_j^{(i)} \phi_j^{(i)}(x),$$9, $c_j^{(i)}$0, with $c_j^{(i)}$1, $c_j^{(i)}$2, $c_j^{(i)}$3, $c_j^{(i)}$4, and a patch-suppression term on $c_j^{(i)}$5 with $c_j^{(i)}$6 and $c_j^{(i)}$7. In both cases, the learned solution is described as closely following the ground truth across regimes, while manual $c_j^{(i)}$8 or $c_j^{(i)}$9 matching remains sensitive to ±20% shifts in the matching location (Kim et al., 9 Jan 2026).

For multi-chart flows, the key empirical result is topological recovery. A single-chart degenerate flow is stated to be topologically restricted because its image is diffeomorphic to $c_j^{(i)}(x;\theta)$0. On the circle, single-chart models exhibit a gap and yield line-like persistent homology, whereas multi-chart models recover a strong 1-dimensional hole. On sphere and torus data, 4-chart models improve reconstruction, Wasserstein distance, exponential-map errors, and distance errors. On triangular meshes, multi-chart flows better approximate geodesic distances, and on Mocap1 only the multi-chart model produces approximately equal pairwise geodesic segments among five uniformly selected points, consistent with circular topology (Yu et al., 30 May 2025).

GraphGlue is evaluated on six main datasets: Arxiv, Computers, Reddit, FB15k_237, PROTEINS, and HIV. In few-shot adaptation, GraphGlue is reported to outperform baselines in both 1-shot and 5-shot settings. Specific examples include Computers 1-shot at $c_j^{(i)}(x;\theta)$1 versus the best baseline $c_j^{(i)}(x;\theta)$2, Reddit 1-shot at $c_j^{(i)}(x;\theta)$3 versus baseline values around $c_j^{(i)}(x;\theta)$4–$c_j^{(i)}(x;\theta)$5, and Reddit 5-shot at $c_j^{(i)}(x;\theta)$6 versus GCOPE $c_j^{(i)}(x;\theta)$7. The paper also reports a geometric scaling law: on Computers and Reddit, adding more pre-training datasets yields approximately logarithmic gains in 1-shot accuracy and logarithmic decreases in 1-shot transfer loss, with weaker gains in 5-shot settings (Sun et al., 28 Feb 2026).

These results suggest that neural manifold gluing is particularly valuable in four circumstances: when structure constrains dynamics, when asymptotic regimes must be blended, when global topology is non-trivial, and when heterogeneous domains must share a common geometric substrate.

6. Limitations, misconceptions, and open directions

A recurring misconception is that neural manifold gluing names a single model family. The literature instead uses the term for several mechanisms that share a chart-and-compatibility viewpoint but differ in their mathematical objects, losses, and target tasks (Dan et al., 2024, Kim et al., 9 Jan 2026, Yu et al., 30 May 2025, Sun et al., 28 Feb 2026).

Another misconception is that gluing is merely interpolation. In these papers, gluing is usually constrained by geometry or physics: SC-derived harmonics restrict functional mappings on the SPD manifold; ODE/PDE residuals constrain coefficient transitions; chart responsibilities are coupled to density modeling and reconstruction; and tangent transports are required to be metric-preserving while holonomy and curvature losses regulate global consistency. The gluing variables are therefore not arbitrary blending weights.

The limitations are also domain-specific. DeepHoloBrain depends on high-quality structural connectivity estimation, and the data note that errors in SC propagate to harmonics and scattering; parcellation choices such as AAL-90 affect the manifold charts; and explicit PDE-like formulations of wave propagation on graphs are not fully developed (Dan et al., 2024). GlueNN requires good asymptotic forms, exhibits hyperparameter sensitivity through $c_j^{(i)}(x;\theta)$8, is demonstrated only on 1D ODE examples, and depends on sufficient but not excessive network capacity (Kim et al., 9 Jan 2026). Multi-chart flows incur heavier training and geometric computation than single-chart flows, rely on imperfect overlaps, and may encounter geodesic solvers that converge to non-shortest paths (Yu et al., 30 May 2025). GraphGlue assumes a smooth manifold of bounded tangent dimension $c_j^{(i)}(x;\theta)$9, uses a diagonal metric in the AOF basis, approximates holonomy and curvature through triangles and adjacent edges, and depends strongly on the perturbation scheme used to construct local frames (Sun et al., 28 Feb 2026).

The forward-looking directions stated in the source materials are similarly diverse. The neuroscience work proposes multi-modal gluing, cross-subject gluing, task-manifold atlases, and broader structure-dynamics applications (Dan et al., 2024). GlueNN suggests gluing local dynamical system surrogates, coordinate charts on data manifolds, neural partitions of unity for domain decomposition, and hybrids with operator learning (Kim et al., 9 Jan 2026). The multi-chart flow framework points toward broader computational differential geometry on learned manifolds with non-trivial topology (Yu et al., 30 May 2025). GraphGlue highlights extensions beyond graphs, non-diagonal metrics, richer connections, learned curvature profiles, integration with LLMs, and more sophisticated manifold-based MoE routing (Sun et al., 28 Feb 2026).

Taken together, the current literature indicates that neural manifold gluing is best understood as a geometric design principle: build local representations that are intrinsically meaningful, and learn the compatibility structure needed to make them function as a coherent global manifold, solution, or representation space.