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Li et al. J. Mater. Inf. 2025, 5, 43 https://dx.doi.org/10.20517/jmi.2025.17 Page 3 of 23

Material representation
Early ML-IAPs relied on handcrafted invariant descriptors to encode the potential-energy surface by using
bond lengths and subsequently bond angles and dihedral angles [Figures 1 and 2A]. The advent of GNNs
has transformed this landscape by enabling end-to-end learning of atomic environments. In particular,
equivariant architectures preserve rotational and translational symmetries, while large language models
[12]
[13]
(LLMs) such as ChemBERTa and MolBERT have been repurposed to generate chemically informed
embeddings. Together, these advances have driven the development of a suite of state-of-the-art (SOTA)
ML-IAP frameworks that combine symmetry-aware message passing with data-driven feature
representations [Figure 1].
[10]

Embedding physical symmetries directly into network architectures, rather than applying symmetry
constraints only to final invariant outputs, has been instrumental in advancing ML-IAPs. Equivariant layers
maintain internal feature representations that transform under rotations and translations according to the
underlying symmetry group, guaranteeing that scalar predictions (for example, total energy) remain
invariant while vector and tensor targets (such as forces and dipole moments) exhibit the correct
equivariant behavior [Figure 2]. By unifying invariant and equivariant features throughout the model,
[14]
these architectures achieve both greater data efficiency and improved accuracy across downstream tasks, as
[14]
exemplified by NequIP exploration of higher-order tensor contributions to performance . Furthermore,
this approach parallels classical multipole theory in physics, encoding atomic properties as monopole,
dipole and quadrupole tensors and modeling their interactions via tensor products, integrating long-
standing theoretical formalisms into a modern deep-learning framework .
[15]

Equivariant models (also named geometrically equivariant models) explicitly embed the inherent
symmetries of physical systems, which is critical for accurately modeling tensorial quantities such as spin
Hall conductivity and piezoelectric coefficients . Many materials problems exhibit three-dimensional
[16]
translation, rotation and/or reflection invariances, corresponding respectively to the Euclidean groups
SO(3) (rotations), SE(3) (rotations and translations) and E(3) (including reflections). Unlike approaches
that rely on data augmentation to approximate symmetry, equivariant architectures integrate these group
actions directly into their internal feature transformations, ensuring that each layer preserves physical
consistency under the relevant symmetry operations. However, enforcing strict equivariance throughout the
network is not universally necessary: judicious relaxations of equivariance constraints have been shown to
enhance model generalization and computational efficiency in certain applications .
[17]

Furthermore, as the scale and complexity of materials datasets continue to grow, the computational burden
of fully E(3)-equivariant models increases steeply. An emerging strategy is to construct “lightweight”
architectures that disentangle rotational and translational symmetries by employing only SO(3)-equivariant
operations alongside translation-invariant scalar and vector features . By partitioning high-order tensor
[18]
products into rotationally equivariant and purely invariant components, these models markedly reduce the
cost of tensor contractions without sacrificing the symmetry-preserving properties essential for accurate
[11]
ML-IAP development .
Beyond atomic force vectors, spin degrees of freedom likewise transform as vectors under three-
dimensional Euclidean symmetries, motivating the extension of ML-IAPs to magnetic materials. MagNet
[19]
learns magnetic force vectors by mapping combined atomic and spin configurations to forces computed via
DFT, embedding E(3)-equivariance within its network layers to ensure physically consistent
transformations. In parallel, SpinGNN introduces two specialized architectures [the Heisenberg edge
[20]
graph neural network (HEGNN) and the spin distance edge graph neural network (SEGNN)] to represent