Learned Geometry Selector Overview

• Learned Geometry Selector is a mechanism that dynamically selects scales, points, or representational geometries based on local data characteristics.
• It replaces fixed geometric heuristics with adaptive strategies in applications like UDF reconstruction and attention-based operator learning.
• The approach spans diverse implementations—including per-query radius estimation, attention modules, and masked domain gating—yielding improved model performance.

A learned geometry selector is a learned mechanism that determines which geometric neighborhood, relational structure, or domain subset should influence a model’s computation. In recent work, the term spans several technically distinct settings: per-query support-radius prediction for local-patch unsigned distance field reconstruction, activation-function biases that favor either output geometry or input geometry in hidden representations, attention modules that weight geometry points in operator learning, masked attention that gates in-domain points in PDE preconditioning, and anchor-based relative embeddings for comparing the latent geometries produced by forecasting models (Ogawa et al., 17 Jun 2026, Alleman et al., 2024, Chen et al., 12 Feb 2026, Versano et al., 2024, Kucukahmetler et al., 17 Feb 2026). Taken together, these works suggest that geometry selection is not a single algorithmic primitive but a recurring design pattern in which geometric evidence is mapped to a structural decision.

1. Conceptual scope and recurring forms

A plausible synthesis is that learned geometry selectors appear in three recurrent forms. First, they may select a scale, as in per-query support-radius estimation for local-patch UDF reconstruction. Second, they may select points or relations, as in attention mechanisms that query a geometry point cloud or masked attention that suppresses invalid grid locations. Third, they may select a representational geometry, as in activation functions that bias learning toward label-aligned or input-preserving hidden states (Ogawa et al., 17 Jun 2026, Alleman et al., 2024, Chen et al., 12 Feb 2026, Versano et al., 2024).

Context	Selector mechanism	Selection target
UDF-based point cloud reconstruction	learned per-query radius selector	best local support radius
One-hidden-layer classification	activation function as a geometry selector	output/label geometry or input geometry
Operator learning on arbitrary domains	self-, cross-, or hybrid-attention	geometric information needed for a query
Geometry-aware Helmholtz preconditioning	masked self-attention with domain mask	grid points that belong to the domain
Neural forecasting analysis	anchor-based relative embeddings	shared relational comparison of latent geometry

This scope matters because several of the cited papers reject the assumption that geometry can be reduced to a fixed handcrafted descriptor. Fixed radii, scalar curvature summaries, low-dimensional geometry parameterizations, or absolute latent coordinates are each described as insufficient in at least one setting. The common alternative is to infer the relevant geometry from data rather than prescribing it a priori.

2. Per-query support-scale selection in UDF reconstruction

In local-patch UDF reconstruction, the selector is an explicit geometry-aware front end. "Learned Radius Estimation for UDF-Based Point Cloud Reconstruction" replaces a hand-crafted radius choice with a learned per-query radius selector that predicts a continuous support radius for the patch passed to a frozen LoSF-UDF backbone (Ogawa et al., 17 Jun 2026). The paper’s motivation is that local-patch UDF accuracy depends strongly on support radius: too large a radius introduces unrelated nearby geometry, causing multi-surface contamination, oversmoothing, and loss of fine details, whereas too small a radius provides insufficient context, making the UDF estimate unstable or incomplete. The problem is particularly acute for thin structures, sharp boundaries, occlusion edges, sparse indoor scan regions, and mixed-density areas in consumer captures.

The paper argues that neither a fixed radius nor a one-dimensional curvature heuristic is adequate. A fixed radius assumes similar geometric scale across neighborhoods, even though dense regions need smaller support, sparse regions need larger support, high-curvature regions need tighter context, and flat or noisy regions may need more context. The earlier GeoLA approach adapts radius using a scalar curvature descriptor, but the paper states that curvature alone cannot express sampling density variation, anisotropy, patch incompleteness, multiple nearby surfaces, or open-vs-closed boundary structure.

The selector receives a parent patch $P \in \mathbb{R}^{N \times 3}$ and encodes it using a masked ResNet-PointNet feature encoder, the point count $\log(1+N)$, and a radial density histogram $\mathbf{h} \in \mathbb{R}^6$. A three-layer MLP with sigmoid output predicts a continuous radius ratio $\rho$, yielding

$\hat{r} = \rho \cdot R_{\text{parent}}.$

The paper states that $\rho \in [\rho_{\min}, \rho_{\max}]$. This makes the choice geometry-aware in the specific sense that radius is inferred from local patch geometry and density statistics rather than chosen by a handwritten rule.

Training uses an offline procedure based on cached UDF error curves. For each query $q$, the method evaluates $K$ candidate radii $\{r_1,\ldots,r_K\}$, passes each candidate patch through the frozen LoSF-UDF backbone, and records a per-query error vector $\mathbf{e} \in \mathbb{R}^K$. The best discrete candidate is

$\log(1+N)$0

To avoid hard classification onto a discrete candidate set, the paper fits a local parabola through $\log(1+N)$1 and uses the vertex to define an off-grid target radius $\log(1+N)$2. If the minimum lies on the boundary or the local neighborhood is non-convex, the target falls back to $\log(1+N)$3. Supervision is provided by a confidence-weighted normalized $\log(1+N)$4 loss,

$\log(1+N)$5

The weighting emphasizes queries for which radius choice matters most.

Integration is deliberately lightweight. The pipeline extracts a parent patch around a query point, predicts $\log(1+N)$6, forms a patch at radius $\log(1+N)$7, feeds that patch to the frozen LoSF-UDF backbone, and uses the resulting UDF estimate for mesh extraction with DCUDF. The backbone is not retrained. This supports the paper’s claim that the selector functions as a compact adaptation layer on top of an existing UDF reconstructor.

Empirically, the method is evaluated on ShapeNet-Cars, DeepFashion3D, and ScanNet, with ScanNet held out entirely from training. On ScanNet, relative to the curvature-based GeoLA baseline, [email protected] improves from 0.645 to 0.691 and Chamfer Distance improves from 0.795 to 0.692. Across all three datasets, the method achieves the best CD and [email protected]. The paper notes, however, that fixed-radius LoSF-UDF can retain slightly higher coarse metrics in some cases, specifically [email protected] and Normal Consistency. The stated interpretation is therefore specific: the learned selector is especially beneficial for fine-detail reconstruction rather than uniformly dominant on every metric.

3. Activation functions as selectors of representational geometry

"Task structure and nonlinearity jointly determine learned representational geometry" treats the activation function itself as a geometry selector (Alleman et al., 2024). In one-hidden-layer networks trained on classification tasks, the paper’s central claim is that learned representational geometry depends jointly on input geometry, label/output geometry, and nonlinearity. The main empirical pattern is that Tanh networks tend to produce more target-aligned, disentangled representations, whereas ReLU networks tend to preserve more of the raw input structure and often yield more heterogeneous, feature-specialized hidden units.

The paper formalizes the comparison through tasks in which the alignment between input geometry and label geometry can be controlled. In a broader parameterization, with centered representation matrices $\log(1+N)$8 and kernels $\log(1+N)$9, $\mathbf{h} \in \mathbb{R}^6$0, kernel alignment is measured by

$\mathbf{h} \in \mathbb{R}^6$1

The authors then study target alignment $\mathbf{h} \in \mathbb{R}^6$2, input alignment $\mathbf{h} \in \mathbb{R}^6$3, and input-output alignment $\mathbf{h} \in \mathbb{R}^6$4, where $\mathbf{h} \in \mathbb{R}^6$5 denotes the hidden representation. Complementary measures are linear decodability, parallelism score (PS), and cross-condition generalization performance (CCGP).

In the basic four-cluster binary task, both Tanh and ReLU learn the trained dichotomy, but they differ in what else remains encoded. Tanh networks show high target alignment, high PS, and high CCGP; ReLU networks show high input alignment and better decodability of untrained dichotomies. The distinction is thus not whether the trained task is solved, but whether the hidden representation inherits label geometry or retains cluster-specific input geometry.

The mechanistic explanation is phrased in terms of the hidden-weight update,

$\mathbf{h} \in \mathbb{R}^6$6

For Tanh, symmetric two-sided saturation causes gradients to shrink as units become strongly tuned, damping indefinite specialization and pushing weights toward inter-class directions. For ReLU, only inputs with positive pre-activation contribute to learning, producing an asymmetric dynamic in which units specialize for different regions of input space. The paper explicitly concludes that two-sided asymptotic saturation, rather than behavior near the origin, is the crucial factor behind the Tanh–ReLU difference.

The effect depends on task geometry. When the separability parameter satisfies $\mathbf{h} \in \mathbb{R}^6$7, corresponding to XOR geometry, the paper states that no one-hidden-layer model can produce a strongly target-aligned disentangled representation; both Tanh and ReLU exhibit low target alignment, low PS, and low CCGP. For $\mathbf{h} \in \mathbb{R}^6$8, Tanh shifts rapidly toward target-aligned abstract representations, while ReLU changes more gradually and continues to preserve more input structure. Increasing training noise makes Tanh behave more like ReLU, reducing its bias toward abstraction. A common misconception is therefore addressed directly: the more disentangled representation is not an invariant property of an architecture alone, but depends on the interaction between nonlinearity and task structure.

4. Attention-based geometry selection in operator learning

In operator learning, learned geometry selection is realized through attention over geometry point clouds. "ArGEnT: Arbitrary Geometry-encoded Transformer for Operator Learning" proposes a Transformer-based geometry encoder that treats geometry itself as an input object rather than reducing it to a handcrafted low-dimensional parameterization (Chen et al., 12 Feb 2026). Geometry is represented as a point cloud

$\mathbf{h} \in \mathbb{R}^6$9

where each geometry point may include coordinates and optionally an SDF value,

$\rho$0

Query locations are similarly represented as $\rho$1. The model first projects these through MLPs and then applies attention, with rotary position embeddings used so that attention depends on relative geometry.

The paper defines three variants. In self-attention ArGEnT, query, key, and value come from the same input point set; geometry is encoded implicitly from the query point distribution and any attached features such as SDF. In cross-attention ArGEnT, the geometry point cloud $\rho$2 provides keys and values while query points provide queries, so the query explicitly attends to geometry points. In hybrid-attention ArGEnT, cross-attention first injects geometry information and self-attention then refines the resulting query-conditioned representation. The standard cross-attention form is

$\rho$3

Among the three, the cross-attention variant is presented as the clearest learned selector of geometric information relevant to each query location.

ArGEnT is integrated into DeepONet by replacing the trunk network with the geometry encoder. This preserves DeepONet’s branch–trunk factorization while making the trunk geometry conditioned. The branch network continues to encode non-geometric parameters $\rho$4, such as Reynolds number, freestream velocity, or load conditions, through coefficients $\rho$5. The trunk output $\rho$6 is no longer a function of coordinates alone; it becomes a geometry-conditioned feature map. This is central to the paper’s claim that geometry-dependent operator learning need not rely on an explicit geometry parameterization.

The treatment of signed distance functions is especially diagnostic. The paper states that SDF is useful or even necessary in standard DeepONet and in many self-attention settings, but that cross-attention and hybrid-attention can reduce reliance on SDF because they can infer geometry directly from the geometry point-cloud keys and values. In the supplementary SDF study, adding SDF improves standard DeepONet and self-attention models, while cross-attention and hybrid-attention show little or no gain.

Empirical validation spans laminar airfoil flow, turbulent airfoil flow, lid-driven cavity flow with parametrized cutouts, a 2D redox flow battery with varying rod configurations and inlet profiles, and a 3D jet engine bracket problem with nonparametric geometries and multiple load directions. The paper reports improved prediction accuracy and generalization relative to standard DeepONet, Point-DeepONet, and other baseline surrogates. It also states that DeepONet can fail on extrapolation geometries outside the span of its chosen parameterization, whereas ArGEnT continues to produce reasonable predictions. An important corrective point is also made: point-cloud representation alone is not sufficient. The paper notes that Point-DeepONet is not always better than DeepONet when explicit low-dimensional geometry parameters exist; the improvement is attributed to learned attention-based geometry selection rather than to unordered point clouds by themselves.

5. Masked domain selection in geometry-aware PDE preconditioning

"Attention-based hybrid solvers for linear equations that are geometry aware" instantiates learned geometry selection as masked domain gating inside a hybrid iterative solver for the Helmholtz equation (Versano et al., 2024). The PDE is

$\rho$7

with absorbing boundary conditions on $\rho$8. After discretization, the system is $\rho$9, and the learned operator is used as a nonlinear preconditioner in the HINTS framework. The paper emphasizes reuse: a geometry-aware DeepONet is trained on a simple geometry and then applied to unseen geometries without fine-tuning.

The geometry-aware mechanism combines signed-distance encoding with masked self-attention. The model operates on a fixed $\hat{r} = \rho \cdot R_{\text{parent}}.$0 grid on $\hat{r} = \rho \cdot R_{\text{parent}}.$1, flattened into $\hat{r} = \rho \cdot R_{\text{parent}}.$2. Its inputs are the grid coordinates $\hat{r} = \rho \cdot R_{\text{parent}}.$3, signed distance-to-boundary values $\hat{r} = \rho \cdot R_{\text{parent}}.$4, the source term $\hat{r} = \rho \cdot R_{\text{parent}}.$5 with values outside the domain set to zero, a masking matrix $\hat{r} = \rho \cdot R_{\text{parent}}.$6, and an evaluation point $\hat{r} = \rho \cdot R_{\text{parent}}.$7. The mask is defined so that entries corresponding to outside-domain points receive $\hat{r} = \rho \cdot R_{\text{parent}}.$8, forcing their post-softmax attention weight to vanish. The paper defines

$\hat{r} = \rho \cdot R_{\text{parent}}.$9

This is the precise geometry-selector mechanism: the network learns through attention, but only over locations that belong to $\rho \in [\rho_{\min}, \rho_{\max}]$0.

Because the solution is complex-valued, the architecture uses separate real and imaginary DeepONets. The branch network takes the concatenation of $\rho \in [\rho_{\min}, \rho_{\max}]$1, $\rho \in [\rho_{\min}, \rho_{\max}]$2, and $\rho \in [\rho_{\min}, \rho_{\max}]$3, applies masked self-attention, and then passes the result through a dense network with widths $\rho \in [\rho_{\min}, \rho_{\max}]$4 and Tanh activations. The trunk network receives $\rho \in [\rho_{\min}, \rho_{\max}]$5 as a 3D input and maps it through a dense network $\rho \in [\rho_{\min}, \rho_{\max}]$6 with sine activations. Training uses $\rho \in [\rho_{\min}, \rho_{\max}]$7, 300 Gaussian-random-field source functions, a $\rho \in [\rho_{\min}, \rho_{\max}]$8 grid with $\rho \in [\rho_{\min}, \rho_{\max}]$9, 1000 epochs, batch size 64, and initial learning rate $q$0.

The hybrid solver alternates classical iterations and neural corrections, with the normalized HINTS update scaling residuals to match the training distribution. The paper reports transfer to unseen domains including irregular polygons, cracks, bumps, holes, and obstacle/exterior domains with one or multiple holes. On difficult geometries, especially those with abrupt boundary changes such as narrow cracks and irregular obstacles, the masked model often converges where vanilla DeepONet and the naive geometry-aware CNN baseline diverge. At the same time, the non-masked model can still work well on domains without severe geometric discontinuities and can sometimes outperform the masked version. This prevents an overly broad conclusion: masked attention is presented as a robustness improvement for geometry transfer, not as an unconditional replacement for all simpler mechanisms. The paper further states that each application of the network costs constant work per point and that, on sparse linear systems, a neural-preconditioned iteration step remains $q$1 in memory and computation.

6. Relative geometry as an evaluation framework

Not every treatment of learned geometry focuses on selection during inference; some focus on how the resulting latent geometry should be compared. "Relative Geometry of Neural Forecasters: Linking Accuracy and Alignment in Learned Latent Geometry" introduces anchor-based, geometry-agnostic relative embeddings for comparing latent spaces across forecasting models (Kucukahmetler et al., 17 Feb 2026). The core motivation is that raw latent coordinates are unstable under rotations, scalings, and related ambiguities. The paper therefore represents each latent vector $q$2 by its similarities to a shared anchor set $q$3,

$q$4

This yields a shared relational space in which pairwise representational alignment can be measured.

The main similarity score is cosine similarity on the relative embeddings, complemented by T1 similarity and rank similarity. The paper sweeps anchor counts $q$5, finds that alignment estimates stabilize for about $q$6, and uses $q$7 for the main experiments and $q$8 for stitching. Evaluation covers seven canonical systems: Lorenz-63, a stable limit cycle or spiral system, double pendulum, Hopf normal form, logistic map, POD wake, and a skew-product/random skew system. Model families include MLPs, GRU-based RNNs, Transformers, and ESNs, with Koopman and Neural ODE variants for several families.

The central empirical message is that family-level geometry is reproducible. MLPs align with MLPs, RNNs align with RNNs, Transformers are often less aligned with other families, and ESNs show weaker alignment in several systems. RNNs are the most consistently aligned. At the same time, the paper stresses that high forecast accuracy can coexist with low alignment, especially for Transformers and, in some settings, ESNs. Accuracy and alignment are generally positively correlated, but alignment is not a substitute for predictive loss.

This framework does not itself choose geometry in the same operational sense as the UDF, operator-learning, or PDE-preconditioning papers. A plausible implication is that it provides an evaluation lens for learned geometry selectors by separating representational geometry from task performance. It also addresses a common misconception: similar predictive performance does not imply similar latent geometry. That point is reinforced by the paper’s latent stitching experiments, where relative stitching works best within model families and cross-family stitching is weaker.