Cartesian Natural Tensor Networks (CarNet)

• Cartesian Natural Tensor Networks (CarNet) are a machine learning framework that uses irreducible, fully symmetric, and traceless Cartesian tensors to enforce exact 3D symmetries.
• The framework employs an E(3)-equivariant graph neural network with angular and radial features to accurately learn interatomic potentials and tensorial property maps.
• CarNet enables systematic decomposition and reconstruction of various physical tensors, ensuring accurate predictions for properties like dipoles, polarizabilities, and elastic constants.

Cartesian Natural Tensor Networks (CarNet) are an atomistic machine learning framework that works directly with Cartesian tensors while enforcing the exact symmetries of 3D space. Their central representation is the Cartesian natural tensor: a fully symmetric, traceless Cartesian tensor whose rank $n$ furnishes the $2n+1$-dimensional irreducible representation of $SO(3)$. On this basis, CarNet provides an $E(3)$-equivariant graph neural network that can learn interatomic potentials and general tensorial structure–property maps in a single Cartesian formalism, including rank‑1 dipole moments, rank‑2 polarizabilities and shielding tensors, and rank‑4 elastic constant tensors (Chen et al., 5 Oct 2025).

1. Emergence within Cartesian equivariant atomistic learning

CarNet arose within a broader shift from scalar-only atomistic graph networks toward explicitly equivariant Cartesian models. TensorNet introduced an $\mathrm{O}(3)$-equivariant message-passing architecture whose hidden atomic features are Cartesian rank‑2 tensors $X^{(i)}\in\mathbb{R}^{3\times 3}$, with cheap decomposition into scalar, vector, and symmetric traceless tensor components and feature mixing implemented by matrix products rather than higher-rank Clebsch–Gordan couplings (Simeon et al., 2023). In parallel, HotPP generalized Cartesian message passing to arbitrary tensor order, allowing node and message features to be tensors of rank $l=0,1,2,\dots$ and enabling direct prediction of dipoles and polarizabilities without changing the backbone, albeit with the redundancy of full Cartesian tensors rather than irreducible ones (Wang et al., 2024).

CarNet differs from those precursors by making the irreducible Cartesian tensor itself the primitive object. The framework was proposed to address what earlier Cartesian models lacked: a systematic symmetry-preserving theory for arbitrary physical tensors in Cartesian coordinates, including both decomposition into irreducible pieces and reconstruction back to the physically measured tensor with its required index symmetries. This gives CarNet a broader target class than low-rank Cartesian potentials and a more structured internal representation than arbitrary-order Cartesian tensor passing (Chen et al., 5 Oct 2025).

A plausible implication is that CarNet should be read less as a single architecture than as a Cartesian representation-theoretic program: it keeps the implementation and interpretability advantages of Cartesian coordinates while importing the irreducible-operator discipline that spherical-harmonic equivariant models had previously monopolized.

2. Cartesian natural tensors and irreducible Cartesian representation theory

A Cartesian natural tensor of rank $n$ is defined as a Cartesian tensor $X_{i_1\dots i_n}$ that is fully symmetric and traceless on any index pair,

$$X_{i_{\pi(1)}\dots i_{\pi(n)}}=X_{i_1\dots i_n},\qquad \delta_{i_a i_b}X_{i_1\dots i_a\dots i_b\dots i_n}=0.$$

In three dimensions such a tensor has $2n+1$0 independent components, so rank is simultaneously the Cartesian tensor order and the $2n+1$1 irrep label. Rank‑0 and rank‑1 cases are the usual scalar and vector; rank‑2 is the symmetric traceless $2n+1$2 tensor with five degrees of freedom (Chen et al., 5 Oct 2025).

These objects are the Cartesian counterpart of spherical harmonics. Given a unit vector $2n+1$3, one first forms the symmetric polyadic tensor $2n+1$4, then projects to the symmetric traceless subspace. For $2n+1$5,

$2n+1$6

The same projection logic appears in explicit algebraic constructions of Cartesian harmonic tensors, where symmetrization and trace subtraction are written in closed form with Kronecker deltas; this supplies a practical low-rank backend for Cartesian irreducible operations (Parke, 2023).

The natural tensor product $2n+1$7 is the Cartesian analogue of Clebsch–Gordan coupling. For tensors of ranks $2n+1$8 and $2n+1$9,

$SO(3)$0

This permits the same angular-momentum selection rules as spherical networks, but expressed through Cartesian contractions, symmetrizations, and trace removal rather than Wigner coefficients (Chen et al., 5 Oct 2025).

A major consequence is that arbitrary physical tensors can be decomposed into natural tensors and reconstructed. For a general rank‑2 tensor $SO(3)$1, the decomposition is the familiar split into scalar trace, antisymmetric part, and symmetric traceless part. For the rank‑4 elastic tensor with minor and major symmetries, the decomposition reduces to two scalar natural tensors, two rank‑2 natural tensors, and one rank‑4 natural tensor; rank‑3 contributions vanish under the elastic symmetries. CarNet formalizes this through a constructive algorithm that generates candidate natural tensors consistent with rank and index pattern, enforces linear independence via QR factorization and symmetry-informed elimination, and derives projector matrices for decomposition and reconstruction (Chen et al., 5 Oct 2025).

A complementary invariant-theoretic formulation shows that, for Cartesian tensor inputs, invariant polynomials are generated by connected tensor-network contractions built from the inputs, $SO(3)$2, and at most one $SO(3)$3; equivariant maps are then obtained as invariant-conditioned linear combinations of basis equivariant tensor networks. This places CarNet within a general symmetric-tensor-network formalism rather than a single hand-crafted architecture (Zhang et al., 18 Aug 2025).

3. Graph architecture and equivariant computation in CarNet

CarNet represents an atomistic structure as a graph $SO(3)$4 with atomic numbers $SO(3)$5, positions $SO(3)$6, edge vectors $SO(3)$7, distances $SO(3)$8, and directions $SO(3)$9. Initial atomic features are scalar natural tensors,

$E(3)$0

with $E(3)$1 a channel index. Angular edge features are natural tensors $E(3)$2 constructed from $E(3)$3, while radial dependence is expanded by Chebyshev polynomials with a smooth cutoff,

$E(3)$4

These feed radial MLPs that parametrize equivariant tensor products (Chen et al., 5 Oct 2025).

The core local aggregation is the atomic moment

$E(3)$5

where $E(3)$6 is a tensor-product path. Multiple paths to the same output rank are mixed with factorized weights. To encode higher-body interactions, CarNet forms hyper moments

$E(3)$7

where the correlation degree $E(3)$8 controls effective body order up to $E(3)$9. Messages derived from hyper moments update rank-$\mathrm{O}(3)$0 atomic features through residual equivariant mixing (Chen et al., 5 Oct 2025).

Scalar outputs such as energies are read out from rank‑0 features. Tensorial outputs are produced by collecting rank‑$\mathrm{O}(3)$1 features across layers,

$\mathrm{O}(3)$2

then reconstructing the physical Cartesian tensor from the relevant natural-tensor blocks. Structural extensive quantities, such as dipoles and polarizabilities, are obtained by summing atomic natural tensors over atoms; intensive quantities, such as elastic constants, are formed by averaging before reconstruction. Forces are recovered as $\mathrm{O}(3)$3 (Chen et al., 5 Oct 2025).

The main architectural control parameters are the maximum tensor rank $\mathrm{O}(3)$4, correlation degree $\mathrm{O}(3)$5, number of GNN layers, and the set of tensor-product paths. To control cost, CarNet introduces sparsified path modes—“full”, “level”, and “lite”—with “lite” reported as the best accuracy–efficiency trade-off and a training time about 60% of full mode while often more accurate on small datasets (Chen et al., 5 Oct 2025).

4. State conditioning, degeneracy, and attribute-aware Cartesian extensions

A recurrent limitation of atomistic networks that use only atomic numbers and positions is input degeneracy: different electronic states can share the same nuclear geometry. A TensorNet-based extension relevant to CarNet design addresses this by directly injecting global and local molecular attributes—total charge $\mathrm{O}(3)$6, spin $\mathrm{O}(3)$7, and partial charges $\mathrm{O}(3)$8—into the equivariant update with only scalar rescalings of tensor interactions. The modified update takes the form

$\mathrm{O}(3)$9

with an analogous scalar modulation of the quadratic residual term. Because the modulating factor is a scalar, exact $X^{(i)}\in\mathbb{R}^{3\times 3}$0-equivariance is preserved (Simeon et al., 2024).

The design is arranged so that neutral singlets recover the original TensorNet limit: if $X^{(i)}\in\mathbb{R}^{3\times 3}$1, the multiplicative factor is 1 and the equations reduce to the base architecture. In practice, the work used simple fixed scalar weights such as $X^{(i)}\in\mathbb{R}^{3\times 3}$2, $X^{(i)}\in\mathbb{R}^{3\times 3}$3, and $X^{(i)}\in\mathbb{R}^{3\times 3}$4 (Simeon et al., 2024).

The effect on degeneracy is direct. In a toy merged dataset $X^{(i)}\in\mathbb{R}^{3\times 3}$5 containing charge-distinct states with identical geometries, original TensorNet gives $X^{(i)}\in\mathbb{R}^{3\times 3}$6 meV and $X^{(i)}\in\mathbb{R}^{3\times 3}$7 meV/Å, whereas TensorNet+$X^{(i)}\in\mathbb{R}^{3\times 3}$8 on the same merged set gives $X^{(i)}\in\mathbb{R}^{3\times 3}$9 meV and $l=0,1,2,\dots$0 meV/Å. On QMspin, where singlet and triplet carbene states create a spin degeneracy, baseline TensorNet gives an energy MAE of 432 meV, while TensorNet+spin $l=0,1,2,\dots$1 gives 43 meV, about $l=0,1,2,\dots$2 kcal/mol (Simeon et al., 2024).

This suggests that, within a CarNet program, state variables should be treated as first-class equivariant conditioners rather than appended only at invariant readout. The same paper explicitly proposes that future variants make these attribute weights learnable single or channel-wise scalars, potentially depending on atomic number (Simeon et al., 2024).

5. Empirical performance across potentials and tensorial properties

CarNet was evaluated on interatomic potentials, multitask molecular property prediction, and crystal elasticity. The reported results show low scalar errors, accurate tensor reconstruction, and stable downstream simulation behavior (Chen et al., 5 Oct 2025).

Benchmark	CarNet result	Comparison
LiPS potential	3L: 0.09 meV/atom, 5.6 meV/Å	NequIP: 0.12, 7.7; CAMP: 0.12, 7.4
Water potential	3L: 0.54 meV/molecule, 31 meV/Å	MACE: 0.63, 36; CAMP: 0.59, 34
Ethanol multitask	$l=0,1,2,\dots$3 kcal/mol; $l=0,1,2,\dots$4 kcal/mol/Å; $l=0,1,2,\dots$5 D; $l=0,1,2,\dots$6 Bohr$l=0,1,2,\dots$7; $l=0,1,2,\dots$8 ppm; $l=0,1,2,\dots$9 ppm	TensorNet: $n$0; $n$1; $n$2; $n$3; $n$4
Elastic tensor	$n$5 GPa; $n$6 GPa; $n$7 GPa; $n$8 GPa	MatTen: $n$9; $X_{i_1\dots i_n}$0; $X_{i_1\dots i_n}$1; $X_{i_1\dots i_n}$2

On LiPS, a dataset of $X_{i_1\dots i_n}$3 structures from ab initio MD, CarNet(3L) improves both energy and force MAE relative to NequIP and CAMP. On water, using $X_{i_1\dots i_n}$4 configurations of 192-molecule snapshots, CarNet(3L) matches the best reported energy RMSE and improves force RMSE to 31 meV/Å. In 50 ps NVT MD validations, no instabilities were reported; diffusion coefficients were $X_{i_1\dots i_n}$5 for $X_{i_1\dots i_n}$6 in LiPS versus AIMD $X_{i_1\dots i_n}$7, and $X_{i_1\dots i_n}$8 for oxygen in water versus AIMD $X_{i_1\dots i_n}$9 (Chen et al., 5 Oct 2025).

The ethanol multitask benchmark is especially important because it mixes scalar, vector, rank‑2 structural, and rank‑2 atomic targets in one model: energy $$X_{i_{\pi(1)}\dots i_{\pi(n)}}=X_{i_1\dots i_n},\qquad \delta_{i_a i_b}X_{i_1\dots i_a\dots i_b\dots i_n}=0.$$0, forces $$X_{i_{\pi(1)}\dots i_{\pi(n)}}=X_{i_1\dots i_n},\qquad \delta_{i_a i_b}X_{i_1\dots i_a\dots i_b\dots i_n}=0.$$1, dipole $$X_{i_{\pi(1)}\dots i_{\pi(n)}}=X_{i_1\dots i_n},\qquad \delta_{i_a i_b}X_{i_1\dots i_a\dots i_b\dots i_n}=0.$$2, polarizability $$X_{i_{\pi(1)}\dots i_{\pi(n)}}=X_{i_1\dots i_n},\qquad \delta_{i_a i_b}X_{i_1\dots i_a\dots i_b\dots i_n}=0.$$3, chemical shift $$X_{i_{\pi(1)}\dots i_{\pi(n)}}=X_{i_1\dots i_n},\qquad \delta_{i_a i_b}X_{i_1\dots i_a\dots i_b\dots i_n}=0.$$4, and nuclear shielding tensor $$X_{i_{\pi(1)}\dots i_{\pi(n)}}=X_{i_1\dots i_n},\qquad \delta_{i_a i_b}X_{i_1\dots i_a\dots i_b\dots i_n}=0.$$5. CarNet improves the chemical-shift and dipole errors by about a factor of 3 versus TensorNet, and single-task experiments show that increasing the maximum natural-tensor rank from $$X_{i_{\pi(1)}\dots i_{\pi(n)}}=X_{i_1\dots i_n},\qquad \delta_{i_a i_b}X_{i_1\dots i_a\dots i_b\dots i_n}=0.$$6 to $$X_{i_{\pi(1)}\dots i_{\pi(n)}}=X_{i_1\dots i_n},\qquad \delta_{i_a i_b}X_{i_1\dots i_a\dots i_b\dots i_n}=0.$$7 further improves $$X_{i_{\pi(1)}\dots i_{\pi(n)}}=X_{i_1\dots i_n},\qquad \delta_{i_a i_b}X_{i_1\dots i_a\dots i_b\dots i_n}=0.$$8 and $$X_{i_{\pi(1)}\dots i_{\pi(n)}}=X_{i_1\dots i_n},\qquad \delta_{i_a i_b}X_{i_1\dots i_a\dots i_b\dots i_n}=0.$$9, indicating that rank‑3 irreps capture additional many-body anisotropy (Chen et al., 5 Oct 2025).

For crystal elasticity, CarNet predicts the full rank‑4 elastic constant tensor across $2n+1$00 inorganic crystals spanning all 7 crystal systems and 84 elements. Besides improving bulk, shear, and Young’s modulus errors relative to MatTen, it reproduces directional Young’s modulus for cubic rocksalt CaS with the expected cubic symmetry, indicating that the learned tensors preserve the correct anisotropic structure (Chen et al., 5 Oct 2025).

6. Relation to adjacent frameworks, common confusions, and open problems

CarNet is closely related to spherical-harmonic equivariant networks such as TFN, NequIP, e3nn, and MACE in one precise sense: both use irreducible representations and the same angular-momentum selection rule $2n+1$01. The difference is basis. Spherical models work with spherical tensors and Clebsch–Gordan coefficients; CarNet works with Cartesian natural tensors and implements the same representation theory through Cartesian contractions, symmetrization, and trace removal. For low to moderate tensor rank, this offers direct compatibility with physical Cartesian observables and avoids repeated basis conversion (Chen et al., 5 Oct 2025).

Subsequent work made this relation explicit by introducing Cartesian-3j and Cartesian-nj symbols, extending e3nn with irreducible Cartesian tensor products and contractions, and releasing the cartnn package. In that framework, Cartesian counterparts of MACE, NequIP, and Allegro achieved comparable but not better accuracy than the spherical originals, while suffering substantial memory growth at larger $2n+1$02 or higher correlation order. The same work concluded that, for irreducible Cartesian models, retaining only the highest-weight ICT product and contraction channels, as in TACE, appears to be the best current practice in speed and memory with competitive accuracy (Xu et al., 18 Dec 2025).

A related materials-specific development is CEITNet, which is described as essentially a concrete realization of the Cartesian-natural idea for order‑2 dielectric, order‑3 piezoelectric, and order‑4 elastic tensors. CEITNet performs most learning in channel space and assembles equivariant Cartesian outputs from local environment tensors. On GMTNet benchmarks it reports dielectric Fnorm $2n+1$03 versus GMTNet $2n+1$04, piezoelectric Fnorm $2n+1$05 versus $2n+1$06, and elastic EwT 10% $2n+1$07 versus $2n+1$08, while also improving inference throughput from $2n+1$09 to $2n+1$10 items/s for dielectric prediction (Jin et al., 4 Feb 2026).

A common confusion concerns the similar abbreviation used for Cartan Neural Networks. Those models are based on non-compact symmetric spaces $2n+1$11, solvable Lie group parameterizations, and Cartan geometry rather than irreducible Cartesian tensors. Their layers are homogeneous manifolds, not Cartesian tensor fields, so they are mathematically distinct from Cartesian Natural Tensor Networks despite the superficial acronym collision (Fré et al., 22 Jul 2025).

The main open problems reported for CarNet itself are computational rather than representational. Cost grows with maximum tensor rank $2n+1$12, correlation degree $2n+1$13, and the number of tensor-product paths; path selection remains partly empirical; natural-tensor GPU kernels lag behind the maturity of spherical libraries such as cuEquivariance and openequivariance; and the presented CarNet work does not include explicit long-range electrostatics. The authors identify extensions to time-reversal, spin, noncollinear magnetism, large-scale interatomic-potential frameworks, and non-atomistic 3D point-cloud tasks as natural directions (Chen et al., 5 Oct 2025).

A broader misconception is that Cartesian equivariance is necessarily limited to low-order outputs. CarNet’s decomposition and reconstruction machinery, together with later Cartesian-3j/nj developments, argues against that view. The more accurate statement is that arbitrary-rank Cartesian equivariance is available, but efficient path selection, redundancy control, and kernel optimization remain active design constraints.