MACE Potential: Equivariant Many-Body Modeling

• MACE potential is a machine-learned framework that combines higher-body-order message passing with Atomic Cluster Expansion in an E(3)-equivariant graph neural network.
• It constructs many-body features directly via tensor products, enabling efficient four-body interactions with only two message-passing layers and lowering computational latency.
• The model family spans diverse applications from materials screening to electrochemical interfaces while rigorously preserving symmetry and local scaling.

The MACE potential is a family of machine-learned interatomic potentials built around higher-order equivariant message passing and the Atomic Cluster Expansion (ACE), designed to predict potential energies, forces, and, in many implementations, stresses or virials for atomistic systems at near-DFT fidelity while retaining strict symmetry constraints and local scaling (Batatia et al., 2022). In the literature, MACE is described both as “Message Passing with Atomic Cluster Expansion” and as “Multilayer Atomic Cluster Expansion,” reflecting the same underlying idea: ACE-like many-body basis construction embedded in an E(3)-equivariant graph neural architecture (Park et al., 24 Mar 2025, Bernstein, 2024). The term therefore refers not to a single model, but to an extensible architectural class spanning foundation models, domain-specific fine-tunes, polarizable and potential-conditioned variants, excited-state extensions, and hybrid electronic-structure/ML schemes.

1. Architectural foundations

MACE is an E(3)-equivariant or SO(3)-equivariant graph neural network for atomistic modeling. Atomic environments are represented on a neighbor graph with a finite cutoff radius, and the model is constructed so that scalar predictions such as total energy are invariant under translations, rotations, and permutations of like atoms, while vector and tensor quantities transform equivariantly. This is achieved by expressing features in irreducible representations, using spherical harmonics for angular structure, radial basis functions for interatomic distances, and Clebsch–Gordan couplings for symmetry-preserving tensor products (Batatia et al., 2022, Bernstein, 2024).

What distinguishes MACE from earlier equivariant MPNNs is its explicit higher-body-order message construction. Rather than relying primarily on repeated two-body message passing to increase effective body order, MACE constructs many-body features directly at nodes by taking tensor products of pooled neighbor features. In the original architecture paper, the key practical consequence is that four-body messages can be realized with only two message-passing layers, reducing depth, receptive-field growth, and latency while retaining expressivity (Batatia et al., 2022). This feature is central to the architecture’s data efficiency and to its recurring use as a foundation-model backbone across chemistry and materials.

The model is also strictly local in its baseline form. For an atom $i$, the neighborhood is defined by a radial cutoff, and the total energy is decomposed into atomic site energies,

$E = \sum_i E_i.$

This locality yields linear scaling with atom count for fixed neighbor density and permits deployment in MD, geometry optimization, phonons, and high-throughput screening (Kovács et al., 2023). A recurring consequence, however, is that long-range electrostatics, polarization, and dispersion are not automatically represented unless they are either learned effectively within the cutoff or added explicitly.

The ACE lineage is not merely historical. MACE inherits the ACE view that many-body atom-centered basis functions can be built systematically from radial–angular projections of the neighbor density and their rotational couplings. The review literature presents MACE as a nonlinear neural extension of ACE: ACE supplies the symmetry-adapted basis logic, while the neural layers supply adaptive representation learning and nonlinear readout (Bernstein, 2024). This suggests why MACE often occupies a middle ground between hand-crafted body-ordered models and generic GNNs: it keeps rigorous geometric structure while allowing flexible fitting.

2. Mathematical formulation and training objectives

At the core of MACE is the atom-centered energy decomposition together with analytic differentiation for forces,

$$E(\{\mathbf{R}_i\}) = \sum_i E_i(\mathcal{N}_i), \qquad \mathbf{F}_i = -\frac{\partial E}{\partial \mathbf{R}_i}.$$

Here $\mathcal{N}_i$ denotes the local environment of atom $i$ inside the cutoff (Park et al., 24 Mar 2025, Kovács et al., 2023). In MACE-OFF and related formulations, neighbor information is first projected into equivariant one-particle bases, then pooled into atomic bases, then lifted to higher-order many-body features by tensor products and generalized Clebsch–Gordan coupling. Only the scalar $L=0$ channels enter the energy readout, ensuring rotational invariance of $E$ (Kovács et al., 2023).

The original MACE formulation writes a general message-passing stack in which features $h_{i,kLM}^{(t)}$ are updated layer by layer, and the total energy is accumulated through hierarchical readout,

$$E = \sum_i \sum_{t=1}^{T} E_i^{(t)}, \qquad \mathbf{F}_i = -\nabla_{\mathbf{R}_i} E.$$

Training is typically carried out with weighted losses over energies and forces, and often stresses or virials when those targets are available. In the original NeurIPS paper, the loss was a combined MSE with $\lambda_E = 1$ and $E = \sum_i E_i.$0 (Batatia et al., 2022). Later application papers often used two-stage schedules that emphasize force fitting early and energy refinement later, or Huber-based robust losses for energies, forces, and stresses (Kovacs et al., 2023, Alghamdi et al., 6 Oct 2025).

A representative supervised objective, used in several application papers, has the form

$E = \sum_i E_i.$1

with specific weight schedules depending on domain and data regime (Park et al., 24 Mar 2025). For example, ionic-liquid MACE training employed a two-stage schedule with $E = \sum_i E_i.$2 for epochs 1–75 and $E = \sum_i E_i.$3 for epochs 76–100, while stresses were excluded (Park et al., 24 Mar 2025). Fine-tuning of MACE-MPA-0 for Li diffusion in LiF used weighted Huber losses with explicit deltas for energies, forces, and stresses and committee-based uncertainty assessment when needed (Alghamdi et al., 6 Oct 2025).

Pretraining has become an important part of the MACE ecosystem. Foundation models are trained on large DFT-derived datasets such as MPtrj, Alexandria, and OMol25, then reused out of the box or fine-tuned. This changes the role of the training objective: instead of learning an entire PES from scratch, fine-tuning may correct a broadly competent prior toward a narrower target distribution. A plausible implication is that MACE’s practical success now depends as much on dataset design and pretraining scope as on the bare architecture.

3. Model families and extensions

MACE now denotes a model family rather than a single instantiation. Different variants change the data domain, the physical augmentation, or the prediction target.

Model family	Distinguishing feature	Domain
MACE-MP-0 / MACE-MPA-0	Foundation models pretrained on broad materials data	Inorganic crystals, transfer learning
MACE-OFF23	Short-range transferable force fields for organics	Molecules, liquids, biomolecules
MACE-MP-MOF0	Fine-tuned phonon-oriented MOF model	MOF phonons and QHA
MACE-POLAR / CSP-MACE-Å	Polarization plus explicit intermolecular corrections	Molecular crystals, CSP
PE-MACE	Explicit electric-potential embedding	Electrochemical interfaces
X-MACE	DeepSets-based excited-state extension	Excited-state PES and conical intersections
DFTB+MACE	MACE residual on top of DFTB electronic terms	Mg compounds, charge-transfer materials

The materials foundation line includes MACE-MP-0 and MACE-MPA-0. MACE-MP-0 was used as a general-purpose inorganic crystal potential in silica, molten salts, and MOFs; MACE-MPA-0 extends the pretraining set by combining MPtrj with part of Alexandria, giving about 3.5 million configurations and improving chemical breadth for fine-tuning tasks such as Li diffusion in LiF (Nasir et al., 2024, Alghamdi et al., 6 Oct 2025). The organic line, MACE-OFF23, is purely short-range and trained on $E = \sum_i E_i.$4B97M-D3(BJ)/def2-TZVPPD data for neutral organic systems, with small, medium, and large variants specialized to speed–accuracy tradeoffs (Kovács et al., 2023).

Several extensions make the baseline architecture less purely local or less purely ground-state. MACE-POLAR adds a principled treatment of long-range electrostatics via polarization and forms the intramolecular/intermolecular backbone of CSP-MACE-Å, where it is combined with an XDM-form dispersion term and a learned delta correction to reproduce B86bPBE-XDM intermolecular energies in molecular crystals (Midgley et al., 27 May 2026). PE-MACE embeds a scalar electric potential as a $E = \sum_i E_i.$5 feature from the first layer, thereby learning a conditional PES $E = \sum_i E_i.$6 for constant-potential electrochemical simulations (Zhou et al., 8 Apr 2026). X-MACE integrates DeepSets and a Hermitian decoder to represent unordered sets of adiabatic excited-state energies smoothly across conical intersections (Barrett et al., 18 Feb 2025).

Hybridization with electronic-structure models is also possible. In DFTB+MACE, the conventional DFTB repulsive term is replaced by a MACE many-body correction trained on the residual between DFT and the DFTB electronic contribution,

$E = \sum_i E_i.$7

so that the scheme retains explicit DFTB electronic structure while improving structural energetics and forces (Yu et al., 24 Jun 2026). This is conceptually distinct from a pure ML potential: MACE here corrects an approximate electronic Hamiltonian rather than directly replacing it.

4. Benchmark performance across molecules and materials

On canonical atomistic ML benchmarks, MACE established itself by combining strong accuracy with comparatively low latency. On 3BPA, the two-layer $E = \sum_i E_i.$8 model achieved energy/force RMSEs of 3.0 ± 0.2 meV and 8.8 ± 0.3 meV/Å at 300 K, and retained an advantage over NequIP and BOTNet under extrapolation to 600 K and 1200 K. On AcAc, the same family reached 0.9 ± 0.03 meV and 5.1 ± 0.10 meV/Å at 300 K. Reported inference latencies on an NVIDIA A100 were about 10.5 ms for invariant MACE, 17.5 ms for $E = \sum_i E_i.$9, and 24.3 ms for $$E(\{\mathbf{R}_i\}) = \sum_i E_i(\mathcal{N}_i), \qquad \mathbf{F}_i = -\frac{\partial E}{\partial \mathbf{R}_i}.$$0, compared with about 101–104 ms for BOTNet and NequIP on the same 3BPA setup (Batatia et al., 2022).

Subsequent evaluation broadened this picture from molecular benchmarks to large molecules, liquids, amorphous materials, and universal materials datasets. On MD22, a two-layer MACE with effective $$E(\{\mathbf{R}_i\}) = \sum_i E_i(\mathcal{N}_i), \qquad \mathbf{F}_i = -\frac{\partial E}{\partial \mathbf{R}_i}.$$1 Å receptive field outperformed sGDML and achieved lower force errors than VisNet-LSRM across systems such as tetrapeptides, nucleic-acid assemblies, and the buckyball catcher; on COMP6, the large model reported total MAEs of 0.48 kcal/mol in energy and 0.52 kcal/mol/Å in forces, improving substantially over ANI-1x and GM-NN; on HME21, it reached 16.5 ± 1.2 meV/atom energy MAE and 140.2 ± 3.8 meV/Å force MAE; and on the M3GNet universal materials dataset it gave 34.1 meV/atom energy MAE and 60.1 meV/Å force MAE, with stress RMSE 0.62 GPa (Kovacs et al., 2023).

The organic specialization MACE-OFF23 demonstrated that a purely short-range MACE can still reproduce condensed-phase observables when the training data encode the relevant intermolecular physics. The large model achieved per-atom energy RMSE of about 0.5–1.0 meV/atom and force RMSE about 15–20 meV/Å on held-out tests; for the X23 molecular-crystal set it reported mean sublimation-enthalpy error about 1.7 kcal/mol; for 109 organic liquids the medium model achieved density MAE 0.09 g/cm$$E(\{\mathbf{R}_i\}) = \sum_i E_i(\mathcal{N}_i), \qquad \mathbf{F}_i = -\frac{\partial E}{\partial \mathbf{R}_i}.$$2 with $$E(\{\mathbf{R}_i\}) = \sum_i E_i(\mathcal{N}_i), \qquad \mathbf{F}_i = -\frac{\partial E}{\partial \mathbf{R}_i}.$$3; and for water the RDFs and quantum-corrected vibrational spectra agreed well with experiment and MB-pol (Kovács et al., 2023).

These results do not imply uniform superiority in every regime. Several papers show that model ranking depends strongly on target domain, label quality, and whether long-range or non-equilibrium physics are explicitly encoded. A recurring pattern is that MACE excels when the relevant local many-body structure is well sampled, but may require augmentation or fine-tuning when the target regime differs qualitatively from pretraining.

5. Representative scientific applications

In silica and zeolites, off-the-shelf MACE-MP-0 medium with D3 reproduced the established metastability ordering of siliceous zeolites relative to $$E(\{\mathbf{R}_i\}) = \sum_i E_i(\mathcal{N}_i), \qquad \mathbf{F}_i = -\frac{\partial E}{\partial \mathbf{R}_i}.$$4-quartz and tracked PBE+D3 closely. Representative framework energies per T-site relative to $$E(\{\mathbf{R}_i\}) = \sum_i E_i(\mathcal{N}_i), \qquad \mathbf{F}_i = -\frac{\partial E}{\partial \mathbf{R}_i}.$$5-quartz included AFI 10.5, MFI 8.5, and AST 13.8 kJ/mol per T, while dense polymorph energies included $$E(\{\mathbf{R}_i\}) = \sum_i E_i(\mathcal{N}_i), \qquad \mathbf{F}_i = -\frac{\partial E}{\partial \mathbf{R}_i}.$$6-cristobalite 2.5, coesite 2.2, and stishovite 37.9 kJ/mol per formula unit. Under pressure it predicted quartz $$E(\{\mathbf{R}_i\}) = \sum_i E_i(\mathcal{N}_i), \qquad \mathbf{F}_i = -\frac{\partial E}{\partial \mathbf{R}_i}.$$7 coesite transition near $$E(\{\mathbf{R}_i\}) = \sum_i E_i(\mathcal{N}_i), \qquad \mathbf{F}_i = -\frac{\partial E}{\partial \mathbf{R}_i}.$$8 GPa and coesite $$E(\{\mathbf{R}_i\}) = \sum_i E_i(\mathcal{N}_i), \qquad \mathbf{F}_i = -\frac{\partial E}{\partial \mathbf{R}_i}.$$9 stishovite near $\mathcal{N}_i$0 GPa, close to experiment, and captured the monoclinic-to-orthorhombic transition in ZSM-5 near $\mathcal{N}_i$1–$\mathcal{N}_i$2 GPa (Nasir et al., 2024).

In transport and molten-salt thermophysics, MACE has been used both out of the box and after light fine-tuning. For molten LiF, MACE-MP-0 small reproduced the experimental viscosity across the liquid range and predicted a melting temperature close to experiment, whereas Buckingham and BHM classical models underpredicted viscosity and underestimated melting by about 200–300 K (Devereux et al., 2024). For Li interstitial diffusion in crystalline LiF, foundational MACE-MPA-0 already gave activation energy $\mathcal{N}_i$3 eV, close to the DeePMD reference 0.24 eV, and fine-tuning with only 300 points produced $\mathcal{N}_i$4 eV and diffusivities close to the >40,000-structure DeePMD model (Alghamdi et al., 6 Oct 2025).

In ionic liquids, a newly trained MACE outperformed a DeePMD model trained on the same data in force accuracy and in MD observables. For PYR14BF4, MACE reached force MAE = 0.008 eV/Å and RMSE = 0.011 eV/Å, compared with DeePMD 0.039 and 0.053 eV/Å. It also reproduced more realistic density and diffusion behavior, whereas the DeePMD model produced over-dense packing and suppressed dynamics in some cases (Park et al., 24 Mar 2025).

In framework and lattice dynamics, fine-tuned MACE-MP-MOF0 corrected failures of base MACE-MP-0 in MOFs, notably symmetry distortions and spurious imaginary phonons. On curated test sets it achieved RMSD$\mathcal{N}_i$5 = 1.4 meV/atom, RMSD$\mathcal{N}_i$6 eV/Å, and RMSD$\mathcal{N}_i$7 meV/Å$\mathcal{N}_i$8, while frequency RMSDs relative to DFT for benchmark MOFs included 0.033 THz for MOF-5 and 0.082 THz for UiO-66 in the v2 split (Elena et al., 2024).

In magnetic alloys, a system-specific MACE-sqs model trained on spin-polarized PBE DFT for Fe–Ni alloys achieved validation RMSEs of about 2.0 meV/atom for energies and 24.3 meV/Å for forces, and was the most consistent MACE variant for equilibrium volumes, EOS, elastic constants, and thermal expansion. Yet all tested models predicted the wrong composition trend for the bcc-to-hcp transition pressure, indicating failure to capture pressure-induced magnetic collapse and composition-dependent magnetoelastic effects (Ramakrishna et al., 27 May 2026).

In electrochemical interfaces, PE-MACE at the Pt(111)/water interface learned a potential-conditioned force field with test energy RMSE 1.8 meV/atom, force RMSE 12.3 meV/Å, and relative force RMSE 1.93%. The companion PE-EDP electron-density model reached test NMAE 0.848%, enabling planar charge profiles and Bader analyses consistent with DFT and allowing 4 ns constant-potential MLMD, compared with 10 ps CP-AIMD validation trajectories (Zhou et al., 8 Apr 2026).

In crystal structure prediction, CSP-MACE-Å combined MACE-POLAR, XDM-style dispersion, and a learned delta intermolecular term. On a 19-compound AstraZeneca set, its energy-only ranking was comparable to PBE-NP DFT, with average rank 3.58 and $\mathcal{N}_i$9 kJ/mol, while harmonic free-energy reranking improved this to rank 2.11 and $i$0 kJ/mol. On a 28-compound blind-test set, energy-only ranking gave rank 3.86 and $i$1 kJ/mol, improving to rank 2.96 and $i$2 kJ/mol after harmonic free-energy reranking (Midgley et al., 27 May 2026).

6. Limitations, failure modes, and open directions

A central limitation of baseline MACE is locality. Several papers state explicitly that purely local MACE models do not include explicit electrostatics, polarization, or dispersion beyond the cutoff, even if some of these effects are partly learned from data (Kovács et al., 2023, Bernstein, 2024). This does not mean that MACE cannot handle such systems; rather, it means that success depends on whether the missing physics is effectively encoded within the finite receptive field or whether the architecture is augmented, as in MACE-POLAR, PE-MACE, or CSP-MACE-Å.

A second limitation is out-of-distribution behavior. Foundation models can be remarkably transferable, but multiple case studies show systematic failure modes. MACE-MP-0 was strong for equilibrium MOF structures yet struggled with MOF phonons until MOF-specific fine-tuning removed imaginary modes and symmetry distortions (Elena et al., 2024). Foundation models for ionic liquids did not reproduce some conformer distributions without targeted training on bulk ionic-liquid data (Park et al., 24 Mar 2025). In Fe–Ni alloys, even a well-trained system-specific model failed on the composition trend of the bcc-to-hcp transition because explicit spin degrees of freedom and magnetic collapse were outside the model class (Ramakrishna et al., 27 May 2026).

A third limitation concerns reactivity, defects, and extreme conditions. The silica study states that MACE-MP-0 is reliable for small-to-moderate distortions, pressure-driven coordination changes, and variable-cell relaxations, but that DFT remains the safer choice for defect formation, chemical reactions, and very extreme $i$3 conditions (Nasir et al., 2024). Similar caution appears in the LiF diffusion work, where foundational models exhibited PES “softening” and diffusivity underestimation that small fine-tunes corrected (Alghamdi et al., 6 Oct 2025). A plausible implication is that foundation MACE models should be viewed as strong priors, not universal guarantees.

Finally, several papers point to ongoing expansion of the architecture’s scope rather than closure of its limitations. X-MACE addresses non-smooth excited-state topologies near conical intersections by learning permutation-invariant latent invariants of multi-state energy sets (Barrett et al., 18 Feb 2025). PE-MACE and PE-EDP embed external electric potential explicitly, moving MACE toward operando electrochemistry (Zhou et al., 8 Apr 2026). DFTB+MACE preserves an electronic Hamiltonian and thus offers access to band structures, densities of states, and charge transfer beyond standard MLIP capability (Yu et al., 24 Jun 2026). This suggests that “MACE potential” now names a general equivariant many-body modeling framework whose frontier is defined less by one canonical form than by how its symmetry-adapted local representation is combined with missing physics: long-range fields, polarization, spin, excited-state structure, and electronic observables.