Point-based Neural Operators

• Point-based neural operators are models that operate directly on point cloud data by learning mappings from geometric and physical inputs to PDE solutions.
• They use continuous formulations discretized on arbitrarily sampled points, ensuring robustness for irregular, adaptive, or variable domains.
• Architectural families employing attention mechanisms, spectral evaluations, and tokenization demonstrate efficacy in surrogate modeling and geometry processing.

Searching arXiv for papers on point-based neural operators and closely related operator-learning methods on point clouds, arbitrary point sets, and geometric domains. Point-based neural operators are neural operator models that operate directly on point clouds or coordinate samples of a domain rather than on structured grids or purely graph-discretized meshes. In the PDE setting, they are used to approximate solution operators on arbitrary geometries, often through a continuous formulation that is later discretized on sampled points. A common abstraction is the solution map

$\mathcal{G}^{\dagger} : (a,\Omega)\mapsto u,$

where $a$ denotes parameter fields or forcing data on a domain $\Omega$ and $u$ is the corresponding solution field; in discrete engineering settings, the queried output may be written as

$u(x_q) = M(x_q, F, G, S, p),$

with $x_q$ a query coordinate, $F$ sampled forcing data, $G$ sampled boundary or initial data, $S$ a geometry point cloud, and $p$ an optional parametric design vector (Zeng et al., 24 Jan 2025, Zhong et al., 7 Oct 2025).

1. Conceptual scope and operator-learning setting

The defining characteristic of point-based neural operators is that geometry enters through sampled coordinates and pointwise features rather than through a fixed Cartesian lattice. This is central when the domain is complex, variable, adaptively sampled, or topologically changing. In the engineering benchmark literature, point-based neural operators are described as models that operate directly on point clouds or coordinate samples of the PDE domain and learn mappings from geometric and physical inputs to field outputs on arbitrary $a$0D geometries (Zhong et al., 7 Oct 2025).

A recurrent formulation begins from an abstract PDE $a$1, with neural operator training based on paired samples $a$2 and an empirical objective

$a$3

This perspective emphasizes that the learned object is a map between function spaces, not merely a predictor on one fixed discretization. The stated motivation is robustness to arbitrary discretizations, irregular or adaptive meshes, variable geometries, and topology changes (Zeng et al., 24 Jan 2025).

This functional viewpoint also explains why point-based neural operators appear in several distinct subfields. Some works address surrogate modeling for parametric PDEs on point clouds, some learn differential operators directly on point sets for geometry processing, some extend spectral neural operators to arbitrary non-equispaced samples, and some use pointwise information to calibrate spectral or physics-informed models. The common denominator is that the operator acts on sampled fields and geometry without requiring a rigid grid.

2. Continuous formulations and discretization on point sets

A major design principle is to define the operator in a continuous setting and discretize only afterward. In the analysis literature, neural operators are written as maps $a$4 between Banach or Sobolev spaces, for example $a$5 and $a$6, with stability expressed as

$a$7

The same work makes its universality argument by discretizing the domain into points $a$8 and approximating $a$9 from the sampled values $\Omega$0, which places sampled-point representations directly inside the theoretical operator-learning framework (Le et al., 2024).

An early operator-theoretic prototype is Point Convolutional Neural Networks, where a point cloud function is lifted to a volumetric function, processed by an ordinary Euclidean operator, and then sampled back on the point cloud. The resulting point-cloud operator is written as

$\Omega$1

with extension and restriction providing the two outer maps. In this construction, point convolution is the pull-back of Euclidean volumetric convolution through an extension-restriction mechanism, and the paper explicitly interprets the framework as a prototype for point-based neural operators (Atzmon et al., 2018).

A different continuous-to-discrete strategy appears in the Resolution Independent Neural Operator. Instead of feeding raw sampled values to a branch net, it learns a continuous dictionary $\Omega$2 parameterized as implicit neural representations and projects arbitrarily sampled input functions onto coefficient vectors,

$\Omega$3

with coefficients obtained by regularized projection. DeepONet then acts on the coefficient vector $\Omega$4 rather than on a resolution-dependent sensor vector. This directly removes the requirement that all input functions be observed at identical locations, while preserving the operator-learning structure of branch and trunk networks (Bahmani et al., 2024).

These constructions suggest a useful unifying interpretation: point-based neural operators differ less in the abstract learning target than in how they realize lifting, aggregation, and projection for irregular samples. Some volumetrize the data, some infer continuous basis coefficients, and some discretize integral or differential operators directly on the point set.

3. Point-set and token-based architectural families

Within this common framework, several architectural families recur. In the engineering benchmark literature, a generic point-based operator is described as predicting the field at each point $\Omega$5 through a stacked point-operator composition,

$\Omega$6

where $\Omega$7 denotes point features and the attention rule depends on the model class (Zhong et al., 7 Oct 2025).

The simplest baseline is PointNet-style processing: shared MLPs followed by symmetric pooling on unordered point sets. This is easy to deploy, but the benchmark states that it is weaker at capturing local structure. More expressive point-based neural operators replace global pooling with nonlocal attention or learned tokenization. In GNOT, the update is globally nonlocal,

$\Omega$8

with learned $\Omega$9, $u$0, and $u$1. In Transolver, points are first pooled into “physics-aware slice tokens,”

$u$2

then processed globally, and finally projected back to points through

$u$3

This tokenization mechanism is a distinctive point-based aggregation-interpolation pattern: points are pooled into latent physics tokens, processed globally, and reconstructed pointwise (Zhong et al., 7 Oct 2025).

Architectural reuse has also been studied at the operator-block level. DPNO introduces a dual-path composition with a ResNet-like stream,

$u$4

and a DenseNet-like stream,

$u$5

For DeepONet, this dual-path mechanism is applied to the trunk net rather than the branch net, with the stated interpretation that it enriches the output-space basis family while keeping the input-function encoding unchanged (Wang et al., 17 Jul 2025).

4. Spectral and basis-based methods on arbitrary point distributions

A second major line of work adapts spectral neural operators to non-equispaced point sets. Direct Spectral Evaluation replaces FFT-based spectral layers with direct evaluation of a truncated spectral basis on arbitrary samples. For point coordinates $u$6, the transform is written as

$u$7

where $u$8 is a Vandermonde-like matrix of basis evaluations on the sampled points. The same idea is used with Fourier bases in Euclidean domains and spherical harmonics on spherical point clouds. Since only a small truncated set of modes is retained, the paper gives the complexity of DSE as $u$9, contrasting it with $u(x_q) = M(x_q, F, G, S, p),$0 FFTs and $u(x_q) = M(x_q, F, G, S, p),$1 brute-force DFTs (Lingsch et al., 2023).

This arbitrary-point spectralization is presented as a drop-in replacement for the spectral transform blocks in FNO, UFNO, FFNO, and SFNO. The stated practical consequence is that Fourier-based neural operators become compatible with irregular grids, lattices, point clouds, and spherical domains without interpolation onto a regular mesh (Lingsch et al., 2023).

A more adaptive variant appears in the Holistic Physics Solver, whose core block is described in the details as a Point-Calibrated Spectral Neural Operator. Here the spectral basis is not fixed globally. Instead, each point predicts frequency preferences through a gate

$u(x_q) = M(x_q, F, G, S, p),$2

and the truncated Laplace-Beltrami transform is replaced by a point-calibrated transform

$u(x_q) = M(x_q, F, G, S, p),$3

If $u(x_q) = M(x_q, F, G, S, p),$4, the construction reduces to the ordinary Laplace-Beltrami spectral transform, so classical spectral neural operators appear as a special case. The intended effect is to combine spectral continuity priors with point-level flexibility, especially on irregular geometries and under limited training data (Yue et al., 2024).

Taken together, these works show that “spectral” and “point-based” are not opposed categories. In current operator-learning practice, spectral processing can be executed directly on point sets, or the spectral basis itself can be made point-conditioned.

5. Geometry-native differential operators and learned geometric structure

Another large branch of point-based neural operators focuses on learning differential operators or geometric quantities directly on point clouds. The Neural Laplacian Operator is exemplary: it constructs a symmetric KNN graph, uses a GNN to produce per-point features, decodes edge weights $u(x_q) = M(x_q, F, G, S, p),$5 and masses $u(x_q) = M(x_q, F, G, S, p),$6, and trains the resulting operator not by matching matrix entries but by matching its action on probe functions,

$u(x_q) = M(x_q, F, G, S, p),$7

This functional imitation objective is introduced specifically to bypass the connectivity mismatch between the KNN graph and the mesh on which the cotangent Laplacian is defined. The method is reported to reduce MSE by roughly an order of magnitude relative to the best prior method on ShapeNet testing data and to support downstream geometry processing tasks such as heat diffusion, geodesic distance, Laplacian smoothing, spectral filtering, and ARAP deformation directly on point clouds (Pang et al., 2024).

DeltaConv takes a more intrinsic differential-geometric route. It separates features into scalar and vector streams and connects them through discrete gradient, divergence, curl, co-gradient, and Hodge-Laplacian operators. The vector stream stores tangential vector coefficients at each point, allowing direction-dependent processing without a global tangent coordinate system. The core claim is that this enables anisotropic filtering directly on point clouds while remaining coordinate-independent and robust to irregular sampling (Wiersma et al., 2021).

Geometric Neural Operators push this direction toward reusable geometry models. They apply operator layers on local point-cloud neighborhoods $u(x_q) = M(x_q, F, G, S, p),$8, discretize integral operators by message passing,

$u(x_q) = M(x_q, F, G, S, p),$9

and train pretrained models to estimate Monge-gauge height functions, normals, inverse first fundamental forms, second fundamental forms, and Gaussian curvature. These pretrained GNPs are then reused in geometry estimation, mean-curvature flow, and Laplace-Beltrami PDE solves on noisy point clouds of arbitrary topology (Quackenbush et al., 6 Mar 2025).

For parametric PDEs on complex domains, Point Cloud Neural Operator provides perhaps the most explicit PDE-native point-cloud formulation. Its input augments field values with coordinates and a density term,

$x_q$0

and each layer combines a local linear map, a Fourier-kernel integral operator, and a local differential aggregation based on least-squares gradients: $x_q$1 The model is stated to be permutation invariant, to have layer complexity linear in the number of points when connectivity is sparse, and to admit a universal approximation result at the continuous level (Zeng et al., 24 Jan 2025).

6. Physics-informed training and symmetry-aware loss design

A distinct, but related, use of pointwise structure appears in physics-informed neural operators. In this literature, “point symmetry” refers not to point clouds but to Lie point symmetries acting on the independent and dependent variables of a PDE,

$x_q$2

The baseline PINO objective is the residual loss

$x_q$3

Earlier symmetry-aware loss augmentation added prolonged point-symmetry actions on the residual, whereas the generalized-symmetry formulation replaces standard generators by their evolutionary representatives,

$x_q$4

and trains with

$x_q$5

The underlying proposition is that a generalized vector field is a symmetry of a differential system if and only if its evolutionary representative is (Wang et al., 1 Feb 2025).

The motivation for this replacement is that standard point symmetry losses can be uninformative. For Burgers’ equation,

$x_q$6

the translation generator $x_q$7 yields

$x_q$8

while its evolutionary representative satisfies

$x_q$9

The former contributes no training signal; the latter contributes a derivative of the residual. Analogous behavior is given for Darcy flow, where finite-dimensional point symmetries can collapse to zero while evolutionary representatives produce derivative terms such as $F$0 and $F$1. The stated practical effect is improved data efficiency, accuracy during training, and zero-shot resolution transfer on Darcy flow, together with gains on $F$2D Burgers’ equation (Wang et al., 1 Feb 2025).

This line of work is conceptually important because it reframes symmetry information as operator regularization on sampled fields. It also clarifies that point-based inductive biases in neural operators are not limited to geometry representation; they also include pointwise physics priors and derivative-aware losses.

7. Applications, empirical behavior, and limitations

Empirically, point-based neural operators are especially prominent in settings where geometry is irregular, freeform, or naturally represented as a point cloud. The engineering benchmark on six industry-scale datasets concludes that point-based operators are the most natural family for freeform $F$3D geometry and irregular discretizations, with transformer-based variants such as GNOT and especially Transolver substantially stronger than plain PointNet. The same benchmark also states that point-based methods are not universally best: branch-trunk models are usually better on Heat sink and Bracket, grid-based models outperform point-based models on Heat sink and JEB, and time-dependent Bracket-time benefits from explicit temporal branch models. Parametric input fusion by direct concatenation or branch-enhanced fusion improves point-based models substantially on parameterized engineering problems (Zhong et al., 7 Oct 2025).

Beyond surrogate PDE prediction, learned point-based operators now support direct operator approximation and downstream numerical tasks. NeLo applies a learned Laplacian to point-cloud geometry processing, GNPs are used inside mean-curvature flow and Laplace-Beltrami solvers, PCNO is evaluated on boundary-layer problems, variable-domain Darcy flow, airfoil-plus-flap topology variation, vehicle pressure-load prediction, and parachute inflation dynamics, and RINO demonstrates operator learning from arbitrarily sampled inputs in antiderivative, nonlinear Darcy, and Burgers examples (Pang et al., 2024, Quackenbush et al., 6 Mar 2025, Zeng et al., 24 Jan 2025, Bahmani et al., 2024).

The limitations reported across these works are equally consistent. NeLo states that there is no theoretical convergence guarantee as point density grows, that performance may degrade if test point density differs greatly from training, and that the current method is not rotation-invariant (Pang et al., 2024). DSE assumes that only a small number of modes are needed, incurs per-sample transform-matrix construction when point clouds differ across samples, and notes orthogonality and inversion issues outside standard Fourier settings (Lingsch et al., 2023). PCNO states that topology generalization is not guaranteed, since training on one topology in the airfoil experiment did not generalize well to the other, and it leaves time-dependent PDE operators as an open direction (Zeng et al., 24 Jan 2025). RINO emphasizes that resolution independence still requires sufficiently rich sampling and therefore does not evade Nyquist–Shannon-type limits (Bahmani et al., 2024).

A plausible implication is that the field is converging on a shared design doctrine rather than a single dominant architecture. Coordinates, pointwise features, quadrature or density information, spectral bases, tokenized nonlocal interactions, and local differential estimators are increasingly combined rather than treated as mutually exclusive choices. Point-based neural operators, in this sense, are best understood not as one architecture but as a family of operator-learning strategies for irregularly sampled function data and geometry.