MACE-OFF23: Advanced ML Interatomic Potentials

• MACE-OFF23 is a family of machine learning interatomic potentials that employs multilayer atomic cluster expansion and equivariant message-passing to predict energies with high precision.
• It combines symmetry-adapted local descriptors and nonlinear message-passing architectures to achieve state-of-the-art accuracy and efficiency across diverse chemical systems.
• The framework enables rapid, scalable simulations from molecular to condensed-phase regimes, setting new benchmarks for transferability and stability in ML force fields.

MACE-OFF23 denotes a family of state-of-the-art machine learning interatomic potentials and force fields built on the Multilayer Atomic Cluster Expansion (MACE) framework. This suite of models exemplifies the integration of symmetry-adapted local descriptors with modern equivariant message-passing neural architectures, trained on first-principles data for broad chemical and physical transferability. MACE-OFF23 has become a canonical reference for molecule-level force fields in computational chemistry and materials science, supporting applications ranging from molecular simulation and condensed-phase property prediction to lattice dynamics and biomolecular modeling.

1. Theoretical Foundations: From GAP to ACE to MACE

MACE-OFF23 is grounded in the evolution of machine-learned interatomic potentials, building upon the conceptual progression from SOAP-GAP (Gaussian Approximation Potential) through the linear Atomic Cluster Expansion (ACE) to multilayer, equivariant neural models.

• SOAP-GAP: Utilizes Gaussian process regression on smooth overlap of atomic positions (SOAP) descriptors, achieving strict invariance and tunable smoothness but incurring high computational cost and limited nonlinearity due to fixed-kernel GPR (Bernstein, 2024).
• Linear ACE: Expresses local atomic neighbor densities in an orthonormal radial basis with spherical harmonics, enabling efficient high-order body term expansions but remaining restricted by linear model expressivity absent very large basis size (Bernstein, 2024).
• MACE: Lifts ACE descriptors into an equivariant message-passing graph neural network. Each atom’s feature vector is iteratively updated via neighbor-wise tensor contractions and learnable nonlinear functions over basis sets up to a selected angular momentum and body order. Final atomic contributions are read out via MLPs, yielding a strictly local, symmetrically consistent total energy (Bernstein, 2024, Kovács et al., 2023).

MACE-OFF23 models maintain the systematic improvability and high symmetry fidelity of ACE while achieving the many-body nonlinear expressivity, data efficiency, and transferability associated with neural message-passing frameworks.

2. Model Architecture, Mathematical Formulation, and Training

MACE-OFF23 employs a hierarchical, local energy decomposition wherein atomic energies $E_i$ depend on rotationally and permutationally equivariant descriptors extracted within a radial cutoff, updated over two message-passing layers.

Key mathematical elements (Bernstein, 2024, Kovács et al., 2023):

• Local neighbor density:

$$\rho_i(r) = \sum_{j \in N(i)} f_c(r_{ij}) \sum_{n,\ell,m} c_{ijn\ell} R_n(r_{ij}) Y_{\ell m}(\hat{r}_{ij})$$

• Layerwise updates:

$$a_i^{(t+1)} = \sum_{j \in N(i)} M(a_i^{(t)}, a_j^{(t)}, \{R_n, Y_{\ell m}\})$$

with $M$ a symmetry-adapted MLP.

• Atomic energy and total readout:

$E = \sum_i F(a_i^{(T)})$

where $F$ is a final-layer MLP.

MACE-OFF23 typically uses a body-order truncation of 4, two message-passing layers, maximum angular momentum $\ell_{max}=2$, and radial basis size of 32–64. Training minimizes a composite loss of energy and force mean-square errors, with force weighting optimized to balance O(3N) constraints from each configuration. Optimization is performed via Adam/AdamW with regularization, cosine scheduling, and early stopping (Bernstein, 2024, Kovács et al., 2023).

3. Benchmarks: Accuracy, Transferability, and Stability

MACE-OFF23 has demonstrated state-of-the-art accuracy and transferability on a range of benchmarks, notably:

Organic and Condensed-Phase Systems (Kovács et al., 2023, Bernstein, 2024)

Dataset/Task	Metric	MACE-OFF23 Performance
SPICE (10 elements, organic)	RMSE (energy)	1–2 meV/atom
SPICE (organic)	RMSE (force)	20–30 meV/Å
TorsionNet-500 (drug-like torsions)	Barrier error	0.5/0.25/0.15 kcal/mol (S/M/L)
Paracetamol Form II (Raman spec.)	Relative err.	2–3% (THz/high-freq. bands)
X23 sublimation enthalpies	MAE	1.7 kcal/mol (L)
Water IR/Raman (PIGS, quantum)	Rel. error	<3% (all bands)
109 organic liquids	Density MAE	0.09 g/cm³ (vs 0.21, ANI-2x)

Materials and Lattice Dynamics (Yang et al., 21 Oct 2025)

System	Phonon RMSE (ω²)	Notes
Double halide perovskites	<0.05 THz²	Best model: omat-0-medium

Efficiency and Scaling (Bernstein, 2024, Firoz et al., 14 Apr 2025)

• MACE-OFF23: 0.04–0.12 ms/atom·step on GPU, 10–50× faster than SOAP-GAP (CPU)
• Efficient parallel scaling: epoch time reduced from 12 to 2 minutes on 740 GPUs (2.6M graphs) with optimized data distribution and kernel fusion (Firoz et al., 14 Apr 2025).

Limitations and Stability (Ranasinghe et al., 14 Mar 2025)

• Small models (OFF23 S): unstable beyond moderate distortion, fail long MD trajectories
• All MACE-OFF23 variants: risk of unphysical behavior at sub-Å clashes, lack explicit repulsive corrections
• Condensed-phase water: density overestimation and missing second hydration shell; best experimental agreement requires medium/large models and further correction

4. Applications and Domain Coverage

MACE-OFF23 supports diverse simulation modalities:

• Molecular sciences: Fast and accurate dihedral scans, reliable crystal structure prediction, explicit solvent biomolecular free energy computation, and peptide/protein folding with competitive backbone RMSD and vibrational spectra (Kovács et al., 2023).
• Condensed phases: Transferability validated on liquid water, organic solvents, and molecular crystals, reproducing spectroscopic observables and thermodynamic properties on par with high-level DFT (Kovács et al., 2023).
• Inorganic/solids: Foundation MACE models excel on lattice dynamics of complex perovskites; accurate reproduction of phonon spectra and stability classifications, except for systems with extreme anharmonicity or steep PES regions (Yang et al., 21 Oct 2025).
• Software and Hardware: Implemented in PyTorch/ACE, interfacing with LAMMPS and OpenMM, and heavily optimized for modern parallel GPU infrastructures (Firoz et al., 14 Apr 2025).

5. Methodological Advances Enabling MACE-OFF23

• Symmetry-adapted equivariant architecture: Neighborhood features and higher-body correlations rigorously encode spatial symmetry, improving data efficiency and physical fidelity.
• Locality and scalability: Purely local descriptors and message-passing ensure that computational cost grows linearly with system size, making the approach suitable for both small molecules and large periodic systems.
• Optimized training protocol: Advanced data distribution (multi-objective bin packing), kernel fusion, and exploitation of tensor contraction sparsity deliver dramatic training speedup, supporting foundation-scale datasets (Firoz et al., 14 Apr 2025).
• Layered expressivity: Nonlinear, learnable message-passing MLPs provide systematic improvability with respect to chemical/body-order and enable smooth chemical interpolation.

6. Limitations and Prospective Developments

MACE-OFF23, in its current iteration, is limited to purely short-range, local interactions and thus neutral, closed-shell, nonreactive systems. Key unresolved issues and avenues for further work include:

• Steric and condensed-phase artifacts: Need for explicit short-range repulsion to correct for spurious minima observed in steric clash testing and liquid structuring in water (Ranasinghe et al., 14 Mar 2025).
• Long-range interactions: Extensions to accommodate electrostatics, polarization, and dispersion are under development (e.g., multipolar or continuum modules) (Kovács et al., 2023).
• Anharmonicity and steep PES: Improved loss functions incorporating second-derivative (Hessian) information and adaptive dataset sampling strategies are indicated to enhance fitting in high-curvature/anharmonic regions (Yang et al., 21 Oct 2025).
• Active learning: Tailored, physics-driven active sampling strategies and inclusion of complex/charged species are prospective directions for broadening transferability.

7. Impact and Significance

MACE-OFF23 models articulate a general framework for foundation-level machine learning force fields, setting accuracy, scalability, and transferability benchmarks in molecular and materials modeling (Bernstein, 2024, Kovács et al., 2023). This approach has redefined practical standards for DFT-level accuracy at classical force-field cost, while underpinning rapid progress in automated, high-throughput simulation and discovery pipelines. Critical assessment highlights the necessity of model–system fit, stability benchmarking, and the inclusion of missing physical effects to realize the promise of these architectures in real-world scientific and industrial applications (Ranasinghe et al., 14 Mar 2025).