MACE-Field: Field-Aware Equivariant Potential

• MACE-Field is a field-aware O(3)-equivariant interatomic potential that models dielectric response via a learned scalar electric enthalpy functional.
• It injects a static electric field in every message-passing layer using Clebsch–Gordan tensor products, ensuring rotational equivariance and derivative consistency.
• The model supports cross-chemistry transfer and finite-field molecular dynamics, accurately reproducing dielectric observables for inorganic insulating solids.

MACE-Field is a field-aware $O(3)$-equivariant interatomic potential for inorganic insulating solids that extends the MACE architecture to a uniform static electric field while preserving rotational equivariance and derivative consistency. Rather than predicting polarization or dielectric tensors as independent targets, it learns a single scalar electric enthalpy functional $\mathcal F(\{\mathbf R\},\mathbf E)$, from which polarization $\mathbf P$, Born effective charges $Z^*$, polarisability $\boldsymbol\alpha$, and $\varepsilon_\infty$ are obtained by exact differentiation. The design is a plug-in modification of standard MACE: the field is injected in each message-passing layer through Clebsch–Gordan tensor products and equivariant residual mixing, while the scalar readout is retained, so existing MACE foundation weights can be inherited with minimal change (Martin et al., 25 Aug 2025).

1. Origins and scientific setting

MACE-Field was introduced to address a specific limitation in machine-learned interatomic potentials for periodic solids: dielectric response requires models that respect the modern theory of polarization, yield derivative-consistent observables from a single thermodynamic scalar, and preserve $O(3)$ symmetry. Density-functional perturbation theory and finite-field electronic-structure methods provide that rigor, but their computational cost restricts accessible system sizes, timescales, and chemical diversity. MACE-Field is intended as a compact alternative that keeps the symmetry structure of MACE while making uniform-field simulations and dielectric-property prediction available across chemistries (Martin et al., 25 Aug 2025).

Its parent architecture, MACE, is a higher-order equivariant message-passing force field that builds local features from radial bases and spherical harmonics, then forms higher-body correlations via equivariant tensor products. In the original formulation, total energy is expressed as a sum of site energies and forces are obtained by differentiation, with explicit $E(3)$-equivariance and efficient construction of many-body features (Batatia et al., 2022). MACE-Field preserves that backbone and changes the latent dynamics rather than the readout. This is significant because prior “unified enthalpy” approaches had largely focused on single materials and bespoke architectures, whereas MACE-Field was designed to support cross-chemistry transfer and reuse of pretrained MACE models (Martin et al., 25 Aug 2025).

A common misconception is to treat MACE-Field as a conventional multi-head regressor for $E$, $\mathbf P$, $\mathcal F(\{\mathbf R\},\mathbf E)$0, and $\mathcal F(\{\mathbf R\},\mathbf E)$1. Its defining feature is the opposite: those quantities are constrained to arise from one scalar enthalpy, so reciprocity and sum rules are structural rather than penalty-enforced (Martin et al., 25 Aug 2025).

2. Electric enthalpy formalism and derivative consistency

The central object in MACE-Field is the fixed-field electric enthalpy

$\mathcal F(\{\mathbf R\},\mathbf E)$2

where $\mathcal F(\{\mathbf R\},\mathbf E)$3 is the zero-field energy and $\mathcal F(\{\mathbf R\},\mathbf E)$4 is the cell volume. By design, the network output remains a scalar invariant under joint rotation of the atomic configuration and the applied field (Martin et al., 25 Aug 2025).

All dielectric observables are exact derivatives of $\mathcal F(\{\mathbf R\},\mathbf E)$5:

$\mathcal F(\{\mathbf R\},\mathbf E)$6

$\mathcal F(\{\mathbf R\},\mathbf E)$7

$\mathcal F(\{\mathbf R\},\mathbf E)$8

and

$\mathcal F(\{\mathbf R\},\mathbf E)$9

Because these are derivatives of a single scalar, Maxwell symmetry and the acoustic sum rule follow by construction:

$\mathbf P$0

The same construction also implies the correct rotation law for the dielectric tensor under co-rotation of lattice and field (Martin et al., 25 Aug 2025).

Periodic polarization introduces Berry-phase multivaluedness. MACE-Field handles this through a branch-invariant folded difference. If $\mathbf P$1 and $\mathbf P$2 are the polarization-lattice vectors assembled as $\mathbf P$3, then

$\mathbf P$4

Training compares folded differences rather than raw Berry-phase values, which avoids spurious penalties from branch choice while leaving the model strictly derivative-based (Martin et al., 25 Aug 2025).

This formulation suggests a precise conceptual distinction: MACE-Field is not merely “field-conditioned MACE,” but an enthalpy model in which electromechanical response tensors are secondary derivatives of a symmetry-constrained scalar.

3. Equivariant architecture and field injection

MACE-Field retains the latent spherical-tensor representation of MACE. Atom-centered features are indexed as $\mathbf P$5, with atom index $\mathbf P$6, channel $\mathbf P$7, angular momentum $\mathbf P$8, and component $\mathbf P$9. A uniform electric field is represented as a rank-1 irrep $Z^*$0, and at each layer $Z^*$1 the field is coupled to latent features by a Clebsch–Gordan tensor product followed by equivariant residual mixing:

$Z^*$2

$Z^*$3

In angular-momentum notation this is the decomposition

$Z^*$4

If $Z^*$5, the update vanishes and the architecture reduces to standard MACE (Martin et al., 25 Aug 2025).

The design keeps the scalar $Z^*$6 readout unchanged. That is why pretrained MACE foundation weights can be loaded and fine-tuned rather than retraining the full network from scratch. The field perturbation is inserted after the ACE-style local feature construction and before the scalar energy readout, so latent multipoles are modified in a symmetry-allowed way without altering the underlying MACE output structure (Martin et al., 25 Aug 2025).

The implementation reported for MACE-Field uses the ScaleShiftFieldMACE backbone with field injection via cuEquivariance tensor products, with an e3nn option also available. The stated architecture uses 2 interaction layers, correlation order 3, cutoff $Z^*$7 Å, hidden-layer $Z^*$8, spherical-harmonic $Z^*$9, and 10 radial basis functions. Channel counts are 128 for cross-chemistry models and 64 for single-material molecular-dynamics models. Readout is scalar $\boldsymbol\alpha$0 per layer, summed, with an MLP readout 16×0e. The field representation is a uniform rank-1 irrep of odd parity shared across atoms. Inference cost scales linearly with system size as in standard MACE, and the field injection is reported to add minimal overhead (Martin et al., 25 Aug 2025).

4. Training regimes, datasets, and implementation practice

MACE-Field was trained in four principal settings that separate cross-chemistry learning from single-material finite-field dynamics.

Task	Dataset	Main supervision
Ferroelectric branches	$\boldsymbol\alpha$1 frames from $\boldsymbol\alpha$2 materials	$\boldsymbol\alpha$3, $\boldsymbol\alpha$4, Berry-phase $\boldsymbol\alpha$5
MP dielectrics	$\boldsymbol\alpha$6 insulating Materials Project entries	sitewise $\boldsymbol\alpha$7, tensor $\boldsymbol\alpha$8
BaTiO$\boldsymbol\alpha$9 finite-field MD	single material, $\varepsilon_\infty$0 supercell	$\varepsilon_\infty$1, $\varepsilon_\infty$2, stress, $\varepsilon_\infty$3, $\varepsilon_\infty$4, $\varepsilon_\infty$5
$\varepsilon_\infty$6-quartz spectroscopy	single material, $\varepsilon_\infty$7 supercell	trajectory-based IR, Raman, $\varepsilon_\infty$8 analysis

For the ferroelectric task, the dataset comprises about 2,500 path frames from about 250 ferroelectric materials spanning 61 elements. Each material contributes 10 frames along a nonpolar-to-polar distortion path, with Berry-phase polarization labels folded to the appropriate branch. Splits are 80/10/10 at material level. The loss combines energy, forces, and polarization with weights $\varepsilon_\infty$9, $O(3)$0, $O(3)$1, $O(3)$2, $O(3)$3, and $O(3)$4, using Huber losses and early stopping on validation polarization (Martin et al., 25 Aug 2025).

For the cross-chemistry dielectric-response task, the dataset contains about 6,000 insulating Materials Project entries with DFPT Born effective charges and electronic polarisabilities, spanning 81 elements. Splits are again 80/10/10, here at Materials Project identifier level. Labels are component-wise $O(3)$5 and full anisotropic $O(3)$6 $O(3)$7 tensors. Forces and stresses were not used for this dataset, and the reported loss weights are $O(3)$8, $O(3)$9, $E(3)$0, $E(3)$1, with early stopping on validation $E(3)$2 (Martin et al., 25 Aug 2025).

The single-material BaTiO$E(3)$3 and $E(3)$4-quartz models target dynamical observables. For tetragonal BaTiO$E(3)$5 (mp-5986; $E(3)$6), the model is trained on ab initio molecular-dynamics frames with energy, forces, stress, Berry-phase polarization, Born effective charges, and polarisability, then used in finite-field MD with sinusoidal driving field $E(3)$7, amplitude $E(3)$8 eV/Å, and period $E(3)$9 steps. For $E$0-SiO$E$1 (mp-7000; $E$2), the model is run at zero external field in a 300 K NVT trajectory of about 200 ps, from which IR absorption, Raman response, and dielectric spectra are obtained by time-correlation analysis, with branch-invariant wrapping of $E$3 during analysis (Martin et al., 25 Aug 2025).

Training uses Adam with AMSGrad and decoupled weight decay $E$4, plus gradient EMA with decay about 0.995. The scheduler is cosine learning rate with warmup. Reported learning rates are $E$5 for most experiments and $E$6 for the $E$7-quartz MD model. Batch sizes of 2–8 were chosen to saturate GPU throughput under AMP, with DDP across GPUs, mixed precision, gradient clipping, and retention of higher-order autograd graphs only when $E$8 or $E$9 labels are present in the batch (Martin et al., 25 Aug 2025).

5. Empirical results and demonstrated physical behavior

On ferroelectric distortion paths, MACE-Field reproduces Berry-phase polarization branches with tight parity between predicted polarization, computed as $\mathbf P$0, and reference labels across frames, materials, and splits. Spontaneous polarization extracted from folded endpoint differences correlates strongly with DFPT references, and example branch families are reproduced without spurious lattice hopping. The reported “fan” plots of polarization versus distortion, normalized by the polarization quantum, match DFPT distributions across the dataset (Martin et al., 25 Aug 2025).

In finite-field BaTiO$\mathbf P$1 molecular dynamics, the single-material model reproduces a hysteretic polarization–field loop at 0 K. The reported remanent polarizations are $\mathbf P$2 and $\mathbf P$3, while the coercive fields are $\mathbf P$4 and $\mathbf P$5. The switching process involves titanium and oxygen sublattices with damped oscillation after reversal, consistent with tetragonal ferroelectric behavior. These values are described as comparable to prior unified-enthalpy ML baselines and realistic relative to DFT and experiment (Martin et al., 25 Aug 2025).

For $\mathbf P$6-quartz at 300 K, MACE-Field reproduces IR and Raman band structure and intensities from time-correlation functions. Reported IR peaks are 1036, 428, and 759 cm$\mathbf P$7, compared against 1041, 420, and 765 cm$\mathbf P$8 for Allegro-pol. The relative high-frequency dielectric constants are $\mathbf P$9 and $\mathcal F(\{\mathbf R\},\mathbf E)$00, while the static values are $\mathcal F(\{\mathbf R\},\mathbf E)$01 and $\mathcal F(\{\mathbf R\},\mathbf E)$02. The frequency-dependent $\mathcal F(\{\mathbf R\},\mathbf E)$03 exhibits the expected high-frequency plateau at $\mathcal F(\{\mathbf R\},\mathbf E)$04 and low-frequency rise toward $\mathcal F(\{\mathbf R\},\mathbf E)$05; the reported underestimation of $\mathcal F(\{\mathbf R\},\mathbf E)$06 is explicitly attributed to a combination of architecture, data, and analysis choices (Martin et al., 25 Aug 2025).

Across the Materials Project dielectric set, parity plots show strong agreement for diagonal components of $\mathcal F(\{\mathbf R\},\mathbf E)$07 and $\mathcal F(\{\mathbf R\},\mathbf E)$08, with broader scatter for off-diagonals because their magnitudes are smaller. The acoustic sum rule $\mathcal F(\{\mathbf R\},\mathbf E)$09 holds numerically. By construction, $\mathcal F(\{\mathbf R\},\mathbf E)$10, and converted $\mathcal F(\{\mathbf R\},\mathbf E)$11 exhibits anisotropy consistent with point-group constraints. Larger relative errors are concentrated near near-zero targets, especially weakly polar sites and highly anisotropic or small-gap systems, while the paper states that absolute errors remain small (Martin et al., 25 Aug 2025).

The architecture was also benchmarked comparatively with Allegro-pol. The distinction is structural: Allegro-pol embeds the field at the input and treats it alongside geometric vectors, whereas MACE-Field injects the field in every message-passing layer through Clebsch–Gordan products and equivariant residual mixing while preserving the standard MACE readout. Reported quartz spectra and dielectric quantities are comparable, BaTiO$\mathcal F(\{\mathbf R\},\mathbf E)$12 hysteresis remains realistic, and inference cost is described as similar to standard MACE because the added field injection incurs negligible overhead (Martin et al., 25 Aug 2025).

6. Scope, limitations, and relation to adjacent MACE variants

MACE-Field is explicitly scoped to inorganic insulating solids. Metals are excluded because Berry-phase polarization is ill-defined there and the associated finite-field thermodynamics differ. The model assumes a spatially homogeneous external field, so field gradients and explicit quadrupolar couplings are outside its stated treatment. Strong fields may produce nonlinear effects not fully captured by the latent linear field injection, although higher-order dependencies can still emerge through stacked layers and nonlinear readout. The paper also notes sensitivity near Berry-phase branch crossings or symmetry changes, possible quality variation in DFPT labels, challenges in highly anharmonic or disordered phases, and occasional tiny negative $\mathcal F(\{\mathbf R\},\mathbf E)$13 eigenvalues before enthalpy convergence (Martin et al., 25 Aug 2025).

These caveats help delimit the method’s claim. MACE-Field is not a universal long-range electrostatics model, nor is it intended as a substitute for all finite-field electronic-structure calculations. It is instead a derivative-consistent enthalpy surrogate over the regime defined by uniform fields, insulating periodic solids, and the training distributions described above.

Within the broader MACE ecosystem, MACE-Field is distinct from “FieldMACE,” which augments MACE with an attention-weighted multipole expansion to model environmental and long-range electrostatic effects in ground- and excited-state QM/MM simulations. FieldMACE targets long-range environment coupling and nonadiabatic dynamics, whereas MACE-Field targets uniform-field dielectric response and finite-field molecular dynamics in periodic inorganic solids (Barrett et al., 28 Feb 2025). Both are field-aware extensions of MACE, but they encode different physical problems.

A practical implication of the plug-in architecture is that MACE-Field can plausibly inherit backend and precision optimizations developed for MACE more generally. Separate work on low-precision execution for SO(3)-equivariant MACE reports that cuEquivariance reduces inference latency by about $\mathcal F(\{\mathbf R\},\mathbf E)$14, and that casting only linear layers to BF16 or FP16 with FP32 accumulation yields roughly $\mathcal F(\{\mathbf R\},\mathbf E)$15 total speedups versus FP64 while preserving downstream molecular-dynamics observables within run-to-run variability (Benoit, 23 Oct 2025). This does not constitute a direct benchmark of MACE-Field, but it suggests an implementation path for deploying field-aware variants on the same backend.

In that sense, MACE-Field occupies a specific position in the evolution of MACE-based models: it preserves the higher-order equivariant message-passing core of MACE (Batatia et al., 2022), adds a symmetry-compatible mechanism for coupling uniform electric fields to latent spherical tensors, and reinterprets dielectric response as differentiation of a learned scalar enthalpy. Its significance lies less in adding a new output head than in importing the thermodynamic structure of finite-field response into an equivariant MLIP with cross-chemistry transfer.