eSEN-30M-OAM: Universal MLIP Benchmark

• eSEN-30M-OAM is a pretrained universal machine-learning interatomic potential leveraging equivariant message-passing and smooth, conservative energy predictions.
• It integrates advanced techniques such as partial charge prediction, Mixture-of-Experts specialization, and long-range Coulomb treatment to enhance predictive accuracy.
• Benchmarked on diverse systems like Heusler alloys and zeolites, it consistently achieves high DFT correlation for lattice constants, formation energies, and stability screening.

eSEN-30M-OAM is a pretrained universal machine-learning interatomic potential (MLIP) used for atomistic structure optimization and energy evaluation in high-throughput materials workflows, and benchmarked as a broad-coverage potential for chemically diverse inorganic systems such as Heusler alloys and zeolites (Xiao et al., 28 Aug 2025, Ito et al., 9 Sep 2025). It belongs to the eSEN family of equivariant graph-based interatomic potentials. The acronym has been expanded in the literature as both “equivariant Smooth Energy Network” and, in the mlip v2 description, “Equivariant Spherical Embedding Network,” reflecting a family lineage that combines conservative energy-based prediction, equivariant message passing, and large-scale pretrained deployment (Fu et al., 17 Feb 2025, Brunken et al., 21 May 2026).

1. Model identity and nomenclature

The designation eSEN-30M-OAM appears explicitly in application and benchmarking studies as a universal pretrained MLIP. In the Heusler benchmark, it is the interatomic potential used for structure optimization, formation-energy evaluation, and convex-hull screening within an ML-accelerated high-throughput workflow (Xiao et al., 28 Aug 2025). In the zeolite benchmark, it is listed among pretrained universal MLIPs and is described as being provided in the OMat24 model suite, trained on the OMat24 dataset and fine-tuned on MPtrj and Alexandria datasets (Ito et al., 9 Sep 2025).

The broader eSEN family was introduced as a state-of-the-art atom-graph-based equivariant MLIP designed to improve physical property prediction through smooth, conservative potential energy surfaces (Fu et al., 17 Feb 2025). The later mlip v2 paper presents eSEN as a modern architecture integrated into a unified software framework, emphasizing modularity, efficiency, Mixture-of-Experts (MoE), electrostatics, and scalable multi-dataset training (Brunken et al., 21 May 2026).

A notable point in the literature is that the exact label “eSEN-30M-OAM” is not explicitly described in the mlip v2 paper. That source instead states that no explicit 30M or “OAM” specialized variant is present in that version of the paper, while also describing released eSEN models of approximately 3.2 million parameters (Brunken et al., 21 May 2026). By contrast, other papers refer directly to eSEN-30M, eSEN-30M-MP, or eSEN-30M-OAM as 30M-scale pretrained variants used in benchmarking and screening (Fu et al., 17 Feb 2025, Li et al., 1 Jun 2026). The literature therefore distinguishes between the general architecture and particular pretrained checkpoints.

2. Architectural foundations

Within the eSEN family, the core architectural commitments are equivariance, smoothness, and energy-based conservative prediction. The earlier eSEN paper defines the model as a message-passing neural network for atomistic graphs with built-in SO(3) symmetry via spherical harmonics, direct energy prediction, and forces and stresses obtained from derivatives of the predicted energy (Fu et al., 17 Feb 2025). The force and stress relations are stated as

$$\mathbf{F} = -\nabla_{\mathbf{r}} E(\mathbf{r}, \mathbf{a}),$$

$$\bm{\sigma} = \frac{\partial E(\mathbf{r}, \mathbf{a}, \mathbf{l})}{\partial \bm{l}},$$

which enforce a conservative force field by construction (Fu et al., 17 Feb 2025).

The mlip v2 description recasts eSEN as an E(3)-equivariant architecture embedded in a modular Graph \rightarrow Graph framework, with embeddings, message-passing layers, and readout heads exposed as interchangeable components (Brunken et al., 21 May 2026). A central efficiency mechanism is the SO(3) $\rightarrow$ SO(2) reduction, attributed to Passaro and Zitnick, which reduces the complexity of equivariant tensor products from $O(L^6)$ to $O(L^3)$, where $L$ is the maximum angular-momentum order (Brunken et al., 21 May 2026). The same source states that the architecture uses spherical harmonics for geometric encoding and the e3j backend for fast equivariant tensor products via CUDA/Pallas kernels.

The earlier eSEN work places particular emphasis on smoothness of the potential energy surface. The design choices include polynomial envelope functions at the cutoff, no maximum-neighbor cap, a low-pass radial basis with default $\sim 10$ functions, and avoidance of discretization onto spatial grids for nonlinearities (Fu et al., 17 Feb 2025). These choices are motivated by the requirement that lower held-out error alone is not sufficient for physically reliable downstream behavior; instead, stable energy-conserving molecular dynamics is treated as a prerequisite for accurate property prediction (Fu et al., 17 Feb 2025).

3. Physical modeling, electrostatics, and expert specialization

The mlip v2 implementation extends eSEN beyond short-range local environments by incorporating partial charge prediction, charge correction, charge conditioning, and long-range Coulomb treatment (Brunken et al., 21 May 2026). Models predict atomic partial charges $q_i$, then apply a correction to enforce total-charge conservation. The framework also embeds the total system charge and concatenates it with atomic embeddings, supporting neutral and charged systems. Long-range electrostatics is included through a Coulomb term with a softened or regularized distance function in a PhysNet-style formulation (Brunken et al., 21 May 2026).

A second major feature is the optional Mixture-of-Experts formulation. In this setup, experts are subsets of network parameters activated according to input properties such as chemical composition, dataset origin, or charge state. Routing can be based on dataset name or charge/multiplicity, and after training the MoE can be specialized (“contracted”) to one expert for inference, recovering runtime efficiency comparable to a standard single-head model (Brunken et al., 21 May 2026). The corresponding multi-dataset readout is written as

$$\tilde{E}_g = \sum_{i \in g} \left[ \mathcal{R}_{d_g}(h_i) + E_0^{(d_g)}(z_i) \right],$$

where $\mathcal{R}_{d_g}$ is the expert readout head for dataset $$\bm{\sigma} = \frac{\partial E(\mathbf{r}, \mathbf{a}, \mathbf{l})}{\partial \bm{l}},$$0, and $$\bm{\sigma} = \frac{\partial E(\mathbf{r}, \mathbf{a}, \mathbf{l})}{\partial \bm{l}},$$1 is the per-dataset atomic reference energy (Brunken et al., 21 May 2026).

These mechanisms are directly relevant to the large pretrained 30M-scale checkpoints. The Heusler and zeolite studies both treat eSEN-30M-OAM as a general-purpose pretrained MLIP not fine-tuned to the target systems, and both report strong transferability to previously unseen materials classes (Xiao et al., 28 Aug 2025, Ito et al., 9 Sep 2025). A plausible implication is that the combination of equivariant representation learning, conservative energy prediction, and broad pretraining is central to its cross-domain performance, although the application papers do not decompose that contribution experimentally.

4. Training scale and family variants

The eSEN literature includes both smaller and larger model instantiations. The mlip v2 appendix summary reports a released eSEN with approximately 3.2 million parameters and the hyperparameters num_layers = 2, sphere_channels = 128, hidden_channels = 128, edge_channels = 128, l_max, m_max = 2, 2, batch_size = 128, learning_rate = 1.5e-3, radial envelope = poly., charge prediction = True, and total charge embed = True (Brunken et al., 21 May 2026). The same source also states that the largest eSEN model discussed there was trained on a curated SPICE2 subset of OMOL25 containing 1.76 million structures, 17 elements, and charged systems (Brunken et al., 21 May 2026).

The earlier eSEN paper explicitly describes an eSEN-30M large-model variant with 30 million parameters and maximum degree $$\bm{\sigma} = \frac{\partial E(\mathbf{r}, \mathbf{a}, \mathbf{l})}{\partial \bm{l}},$$2, trained for 60 epochs of direct-force pretraining followed by 40 epochs of conservative fine-tuning (Fu et al., 17 Feb 2025). In that work, direct-force training is used for rapid convergence, while the final physically faithful model is obtained by removing the direct-force head and optimizing conservative energy-gradient forces (Fu et al., 17 Feb 2025).

The DPA4 benchmark uses eSEN-30M-MP as a 30.1M-parameter baseline on Matbench Discovery, and the DistMLIP paper refers to public checkpoints of eSEN-30M-OAM with 30.2M parameters, describing them as too large for efficient parallelized simulation in that setting (Li et al., 1 Jun 2026, Han et al., 28 May 2025). These sources show that 30M-class eSEN models existed as benchmark-relevant checkpoints even where a full architectural exposition of the specific OAM-labeled variant is not provided.

5. Use in high-throughput materials discovery

In the Heusler-alloy workflow, eSEN-30M-OAM is used for the initial geometry construction and structure optimization of quaternary and all-$$\bm{\sigma} = \frac{\partial E(\mathbf{r}, \mathbf{a}, \mathbf{l})}{\partial \bm{l}},$$3 Heusler compounds, for the evaluation of formation energy $$\bm{\sigma} = \frac{\partial E(\mathbf{r}, \mathbf{a}, \mathbf{l})}{\partial \bm{l}},$$4, and for the calculation of convex-hull stability $$\bm{\sigma} = \frac{\partial E(\mathbf{r}, \mathbf{a}, \mathbf{l})}{\partial \bm{l}},$$5 (Xiao et al., 28 Aug 2025). Initial lattice constants are estimated from similar known Heusler compounds in the DxMag database, and a conventional cubic cell is then strained by $$\bm{\sigma} = \frac{\partial E(\mathbf{r}, \mathbf{a}, \mathbf{l})}{\partial \bm{l}},$$6 and $$\bm{\sigma} = \frac{\partial E(\mathbf{r}, \mathbf{a}, \mathbf{l})}{\partial \bm{l}},$$7 on lattice parameters, including both uniform and uniaxial strains, to generate starting configurations (Xiao et al., 28 Aug 2025). All strained structures are converted to primitive cells and relaxed with the Atomic Simulation Environment using the FIRE optimizer with symmetry constraints enforced, after which the optimized structures are ranked by total energy and the lowest-energy structure is retained as the candidate ground state (Xiao et al., 28 Aug 2025).

The workflow screens tetragonality using the threshold $$\bm{\sigma} = \frac{\partial E(\mathbf{r}, \mathbf{a}, \mathbf{l})}{\partial \bm{l}},$$8, and thermodynamic stability using $$\bm{\sigma} = \frac{\partial E(\mathbf{r}, \mathbf{a}, \mathbf{l})}{\partial \bm{l}},$$9 eV/atom and $\rightarrow$0 eV/atom (Xiao et al., 28 Aug 2025). On a benchmark of 10,000 ternary Heusler compounds, eSEN achieved the highest or among the highest fractions of compounds within tight error thresholds relative to DFT. On cubic Heuslers, it reached $\rightarrow$1 for lattice constant and $\rightarrow$2 for formation energy (Xiao et al., 28 Aug 2025). The same study reports that eSEN yielded fewer spurious local minima than other MLIPs and that subsequent DFT validation retained phase, $\rightarrow$3, and $\rightarrow$4 ordering for more than 96% of stability-screened outputs (Xiao et al., 28 Aug 2025).

When eSEN MLIP optimization and eSEN-based regressors were combined in the full screening workflow, more than 80% of ML-screened candidates satisfied DFT validation for stability and strong magnetocrystalline anisotropy energy, whereas workflows using ALIGNN-FF-based optimization retained only 25–54% of candidates under DFT validation (Xiao et al., 28 Aug 2025). In this context, the role of eSEN-30M-OAM is not merely energy evaluation; it is the structural backbone that conditions the fidelity of all downstream property predictions.

6. Benchmarking, deployment constraints, and comparative position

The zeolite benchmark provides the most direct cross-domain assessment of eSEN-30M-OAM as a universal pretrained potential. It compares pure-silica frameworks and aluminosilicates containing copper species, potassium, and organic cations against experimental data and DFT with PBE plus D3 dispersion correction (Ito et al., 9 Sep 2025). The study concludes that all MLIPs can well reproduce experimental or DFT-level geometries and energetics, and that eSEN-30M-OAM shows the most consistent performance across all zeolite structures studied (Ito et al., 9 Sep 2025).

Benchmark context	Metric	Reported result for eSEN-30M-OAM
Pure silica zeolites vs experiment	Relative-energy RMSE	$\rightarrow$5 kJ mol(Si)$\rightarrow$6
All $\rightarrow$7 zeolites vs DFT	Relative-energy RMSE	$\rightarrow$8 kJ mol(Si)$\rightarrow$9
Cu/CHA zeolites vs DFT	Relative-energy RMSE	$O(L^6)$0 kJ/mol(atom)
K-OSDA/ERI zeolites vs DFT	Relative-energy RMSE	$O(L^6)$1 kJ/mol(atom)

In the same study, eSEN preserved correct zeolite topology in cases where analytic potentials produced unphysical distortions, and it was reported to be free from the systematic overbinding bias observed for CHGNet on pure-silica energetics (Ito et al., 9 Sep 2025). The model was not retrained on zeolite-specific data, and none of the guest-containing zeolite structures appeared in OMat24, which the authors identify as evidence of strong out-of-distribution generalization (Ito et al., 9 Sep 2025).

Deployment at very large scale remains constrained by the computational footprint of equivariant 30M-class models. DistMLIP demonstrates distributed inference for eSEN via zero-redundancy graph-level parallelization, but its reported eSEN experiments use a 3.2M-parameter checkpoint with a 12 Å interaction range, not the 30M-OAM checkpoint (Han et al., 28 May 2025). On GaN, capacity grows from about 1,000 atoms on 1 GPU to 8,200 atoms on 8 GPUs, with 198.9 $O(L^6)$2s/(atom$O(L^6)$3step) on 8 GPUs; for $O(L^6)$4, the corresponding 8-GPU throughput is 241.2 $O(L^6)$5s/(atom$O(L^6)$6step) (Han et al., 28 May 2025). The same paper states that public checkpoints for eSEN-30M-OAM (30.2M parameters) are too large for efficient parallelized simulation in that environment, and even a 0.8M-parameter eSEN configuration ran out of memory in the reported distributed inference experiments because of extensive memory consumption (Han et al., 28 May 2025).

In comparative benchmarking, a distinct 30.1M-parameter eSEN-30M-MP baseline on Matbench Discovery was surpassed by DPA4-Air, which achieved CPS 0.804 versus 0.797, with 10.9$O(L^6)$7 fewer parameters and 42.9$O(L^6)$8 less training compute (Li et al., 1 Jun 2026). That comparison concerns the eSEN family rather than the OAM checkpoint specifically, but it establishes the current external reference point: eSEN-30M-class models remain highly competitive and widely used as baselines, while newer architectures are attempting to improve the accuracy-cost frontier.

Taken together, the literature positions eSEN-30M-OAM as a broadly transferable pretrained interatomic potential with demonstrated utility in high-throughput screening and universal-MLIP benchmarking, especially where reliable structural relaxation and DFT-compatible energetics are more important than minimal parameter count or maximal inference throughput (Xiao et al., 28 Aug 2025, Ito et al., 9 Sep 2025).