MACE Foundation Models

• MACE foundation models are advanced machine learning architectures that integrate higher-order equivariant message passing and atomic cluster expansions to predict atomistic properties.
• They achieve state-of-the-art accuracy in energies, forces, and derived properties across diverse applications by leveraging scalable training and unified data integration techniques.
• Recent innovations include uncertainty quantification, inverse design integration, and safety-driven concept removal in generative models, broadening their practical applications.

MACE foundation models constitute a family of advanced machine learning architectures that unify higher-order equivariant message passing, body-order atomic cluster expansion, and scalable training strategies to enable accurate predictions of atomistic properties across chemistry, materials science, and related domains. While their core application is as universal, transferable interatomic potentials for molecular and crystalline systems, recent variants also address explainability in image classification and safety-driven concept removal in generative models.

1. Architectural Foundations: Higher-order Message Passing and Equivariance

The defining principle behind MACE models is the hierarchical many-body expansion embedded in the message passing neural network (MPNN) paradigm. In contrast to traditional MPNNs that propagate two-body messages and require deep networks for expressivity, MACE constructs messages for each node as a sum over contributions with systematically increasing body order: $$m_i^{(t)} = \sum_j u_1(\sigma_i^{(t)}, \sigma_j^{(t)}) + \sum_{j_1,j_2} u_2(\sigma_i^{(t)}, \sigma_{j_1}^{(t)}, \sigma_{j_2}^{(t)}) + \dots + \sum_{j_1...j_\nu} u_\nu(\sigma_i^{(t)}, \sigma_{j_1}^{(t)}, \dots, \sigma_{j_\nu}^{(t)}).$$ Messages and update functions are equivariant under rotation (O(3)), translation, and permutation, ensuring that features such as atomic energies, forces, and dipoles transform correctly under symmetry operations. The use of tensor product contractions with generalized Clebsch–Gordan coefficients allows scalar (L=0) and directional (L>0) atomic features to be coupled, preserving physical invariances and enabling the prediction of vectorial and tensorial properties such as forces, dipole moments, and stress tensors (Batatia et al., 2022, Kovacs et al., 2023, Batatia et al., 2023, Bhatia et al., 26 Aug 2025).

MACE’s architectural choices—bounded nonlinearity (limited to radial expansion and final readout) and efficient many-body explicit encoding—allow high expressivity at reduced computational cost and enable stable, transferable representations across diverse chemistries and phases.

2. Model Training, Data Integration, and Scalability

MACE foundation models are typically trained via minimization of composite loss functions over energies, forces, and stress tensors, often formulated using weighted Huber losses to balance contributions appropriate to atomic and system scales. Training data spans large curated datasets such as the Materials Project (MPtrj, >150k crystal structures), QCML (millions of molecular conformers), and custom DFT data for specialized systems (ionic liquids, double halide perovskites).

Scalability in training is achieved by addressing two key challenges:

• Load balancing across GPUs: Efficient multi-objective bin-packing algorithms ensure balanced mini-batches of graphs of varying size and sparsity, reducing idle time and memory padding overhead (Firoz et al., 14 Apr 2025).
• Kernel optimization: Fusion of computationally intensive tensor contraction kernels, leveraging sparsity in angular momentum coupling (Clebsch–Gordan coefficients), maximizes memory and compute bandwidth, raising per-epoch throughput on large clusters (e.g., training time reduced from 12 to 2 minutes for 2.6M samples on 740 GPUs) (Firoz et al., 14 Apr 2025).

Integration of heterogeneous datasets—merging molecular and crystalline benchmarks, reconciling energy scales, and force alignment—is made possible by systematic Total Energy Alignment (TEA): inner-core energy matching and atomization-energy scaling corrections bring disparate data onto a unified potential energy surface (Shiota et al., 17 Dec 2024).

3. Atomic Property Prediction: Accuracy and Generalization

Across benchmarks, MACE models deliver state-of-the-art accuracy in predicting:

• Energies: Mean absolute errors (MAE) as low as 2.2 meV/atom (rMD17), 0.404 eV (organic reaction energies, Osaka24), and robust lattice energies for crystals (Batatia et al., 2022, Shiota et al., 17 Dec 2024).
• Forces: RMSE as low as 6.6 meV/Å for molecules, 36–37 meV/Å for liquid water, and accurate force fields for amorphous carbons and biaryl torsions (Kovacs et al., 2023).
• Derived properties: Molecular vibrational spectra, lattice constants (MAE ~0.018 Å), radial distribution functions, density, diffusion coefficients for ionic liquids, and dynamic stability for inorganic solids (Kovacs et al., 2023, Batatia et al., 2023, Park et al., 24 Mar 2025, Yang et al., 21 Oct 2025).

Transferability is demonstrated by the ability to perform “out-of-the-box” MD simulations for systems and domains unseen during training (liquids, biomolecules, amorphous materials, interfaces), attributed to the model’s construction of robust high-body-order, equivariant descriptors.

Data efficiency is notable: chemical accuracy achieved when trained on as few as 50 configurations, including successful reproduction of vibrational spectra and MD observables (Kovacs et al., 2023).

4. Lattice Dynamics and Limitations

MACE models are widely benchmarked for lattice dynamics—phonon eigenfrequencies, group velocities, lattice thermal conductivity (LTC), and dynamic stability:

• Harmonic phonon properties: Accuracy improves with training set size; RMSE in ω² shows order-of-magnitude reduction for newer variants (Yang et al., 21 Oct 2025).
• LTC and IFC fitting: Despite achieving force RMSE comparable to leading models (e.g., EquiformerV2), MACE exhibits amplified errors in computing second-/third-order interatomic force constants (IFCs), resulting in overestimated LTC MAEs (e.g., 1.002 log(W·m⁻¹K⁻¹) vs 0.174 for EquiformerV2) and fewer dynamically stable structures in screening campaigns (Anam et al., 3 Sep 2025).

The amplification of force prediction errors in Hessian calculations is the primary error source, with further limitations observed when sampling configurational spaces inaccessible by the model or in highly anharmonic/dynamically unstable systems. The use of exclusively energy/force losses may be insufficient; inclusion of direct curvature (Hessian) information and adaptive sampling of high-gradient regions present clear avenues for improvement (Yang et al., 21 Oct 2025).

5. Extended Applications: Chemistry Emulation and Uncertainty Quantification

Recent MACE variants expand beyond force field prediction:

• Chemistry Emulation: An autoencoder and latent ODE structure enables MACE to emulate high-dimensional chemical kinetics in dynamic environments (e.g., AGB astrophysical outflows). Integrated training over full chemical pathways and latent-dynamics matching achieves speed-ups of ~24–28x over classical solvers and reproduces chemical evolution with high accuracy and sub-linear scaling versus simulation size (Maes et al., 6 May 2024).
• Uncertainty Quantification: Multi-head committee architectures, where only new output heads are trained (with shared AEDs fixed), provide reliable on-the-fly uncertainty estimation for energies and forces, closely correlating with true prediction errors. This supports active learning workflows to condense training datasets to 5% of their original size with negligible accuracy loss, facilitating robust screening and error monitoring (Beck et al., 13 Aug 2025).

6. Foundation Models in Practice: Integration, Active Learning, and Inverse Design

MACE-MP-0 and its derivatives serve as “foundation models” for atomic-scale simulations—a single model supports broad applications:

• Inverse Design: Integrated active learning frameworks combine generative crystal models (Con-CDVAE) with MACE-MP-0 for property-aligned generative materials design. Iterative feedback (GNN prediction, MACE-MP-0-based MD screening, DFT validation) refines models to match target properties (e.g., bulk modulus) with improved mean absolute percentage error (MAPE 0.4 → 0.14), supporting rapid discovery of novel materials (Li et al., 24 Feb 2025).
• Spectroscopy: MACE4IR enables direct, efficient prediction of infrared spectra, achieving accuracy comparable to DFT while reducing computational cost by orders of magnitude, even for complex and chemically diverse molecules (Bhatia et al., 26 Aug 2025).
• Safety and Explainability: In generative models, MACE supports large-scale concept erasure (up to 100 concepts) in text-to-image diffusion models through a modular process of cross-attention refinement and LoRA-based fine-tuning (Lu et al., 10 Mar 2024). For vision, MACE provides model-agnostic concept-based explanations, dissecting classifier behavior into human-interpretable “part” concepts and quantifying relevance (Kumar et al., 2020).

7. Future Directions and Outlook

Key forward-looking themes for MACE foundation models include:

• Physics-inspired enhancements: Training loss functions that directly incorporate Hessian curvature, higher-level quantum effects, and explicitly sampled high-gradient PES regions to improve lattice dynamics and anharmonic property reproduction (Yang et al., 21 Oct 2025).
• Adaptive and modular model design: Exploration of variable cutoff schemes, element-dependent radial bases, and multi-domain transfer learning—facilitated by Total Energy Alignment—to further unify and extend universality across chemical spaces (Shiota et al., 17 Dec 2024).
• Optimization and automation: Further kernel and memory optimizations, integration with high-performance MD engines, and workflow automation via active learning and uncertainty-driven sample selection to scale model deployment and accelerate iterations (Firoz et al., 14 Apr 2025, Beck et al., 13 Aug 2025).

MACE foundation models have become central to the development of universal, robust, and efficient machine-learned interatomic potentials, underpinning a broad suite of applications in computational chemistry, materials informatics, and molecular simulation. Ongoing research—especially at the interface of physics-inspired learning, uncertainty quantification, and data integration—is poised to yield new conceptual advances and practical tools for atomistic modeling at scale.