Scalar-pathway fidelity improves physical accuracy in short-range equivariant interatomic potentials

Technical Summary: Scalar-Pathway Fidelity in Short-Range Equivariant Interatomic Potentials

Problem Statement
Short-range equivariant interatomic potentials (MLIPs), such as MACE, Allegro, and NequIP, have advanced the field by representing local atomic geometry through $O(3)$-equivariant tensor features. These architectures successfully capture high-body-order geometric correlations. However, a critical bottleneck remains: the final deployment-relevant predictions (energy, forces, and stress) are derived by projecting these rich tensor representations onto invariant scalar ($\ell=0$) channels.

The authors identify that this "scalar pathway" is often a source of physical inaccuracy due to two systemic limitations:

1. Uniform Aggregation: Scalar messages are typically aggregated via unweighted, permutation-invariant sums. This lacks explicit adaptive amplitude gating conditioned on local coordination, density, or geometric distortion, leading to under-resolution in heterogeneous environments (e.g., surfaces, defects, strained regions).
2. Spectral Bias: Scalar nonlinearities (e.g., SiLU) exhibit a strong low-frequency spectral bias. This suppresses high-frequency structure in the potential energy surface (PES), resulting in physical errors such as softened short-range repulsion, red-shifted optical phonon modes, and energy drift during finite-timestep microcanonical integration.

Current approaches often treat the scalar readout as a fixed implementation detail rather than a controllable design dimension. The paper posits that residual physical errors in short-range MLIPs may arise significantly from this scalar compression stage rather than solely from the equivariant tensor backbone.

Methodology
To address these limitations, the authors introduce two lightweight, symmetry-preserving modules that act exclusively on the $\ell=0$ scalar channels, leaving the equivariant tensor-product backbone unchanged. These modules serve as controlled probes of the scalar interface:

1. Physics-Aware Neighborhood (PAN) Pooling:

   • Mechanism: Replaces uniform scalar aggregation with amplitude modulation using sigmoid gates.
   • Input: The gates are derived from invariant pairwise descriptors, including atomic numbers, a smooth coordination descriptor ($\rho_i$), and a cutoff-weighted cosine-angle variance ($\eta_i$).
   • Function: This allows the scalar aggregation to be sensitive to local geometric distortion, coordination loss, and density variations without breaking permutation invariance.

2. Physics-Guided Spectral (PGS) Mixers:

   • Mechanism: Augments scalar features with physically motivated spectral bases to improve short-range spectral resolution.
   • Implementation:
     • Edge Level: Injects a Fourier–Bessel expansion on pair distances.
     • Readout Level: Injects a tapered Exponential-of-Semicircle (ES) expansion on per-atom invariant latents.
   • Function: These bases are designed to better represent steep short-range curvature and high-frequency modes that standard nonlinearities smooth over.

Experimental Design
The authors primarily use MACE as a high-body-order mechanistic scaffold to test these interventions. The experimental protocol ensures that:

• The equivariant backbone, interaction depth, cutoff, and training budgets remain identical between baseline and modified models.
• Comparisons are made across five independent training seeds.
• The modules are tested on diverse systems: metallic Ag, covalent Si, a short-range ionic LiF/Li–F subset, and the MD17/rMD17 molecular benchmarks.
• Portability Tests: The PAN+PGS modules are inserted into Allegro and NequIP at architecturally analogous locations to verify if the benefits transfer across different short-range equivariant backbones.

Key Results

1. Accuracy Improvements:

   • Across Ag, Si, and LiF/Li–F, the combined MACE+PAN+PGS model reduced force Mean Absolute Errors (MAE) by 22–27% and energy MAE by 19–22% compared to baseline MACE.
   • On systems with stress labels, stress errors decreased by 27–28%.
   • In molecular benchmarks (rMD17), force errors were reduced by an average of 13.9%, with the most significant gains observed in flexible, anharmonic molecules.
   • Capacity-matched controls (widening/deepening the backbone without PAN/PGS) yielded only 4–7% improvement, demonstrating that the gains are not merely due to increased parameter count.

2. Portability:

   • Inserting PAN+PGS into Allegro and NequIP reduced force errors by 17–19%, confirming the correction is portable across distinct short-range equivariant backbones, though the magnitude of improvement is architecture-dependent (largest in MACE).

3. Physical Observables:
   The scalar-pathway correction specifically attenuated four distinct physical failure modes predicted by the authors:

   • Anharmonic Torsions: Reduced Kullback–Leibler divergence in aspirin torsional distributions by ~3x, correcting "barrier leakage."
   • Vibrational Modes: Reduced the red-shift of high-frequency optical phonons in Si, improving the vibrational density of states (VDOS) peak position error by a factor of ~4.
   • Integration Drift: Reduced secular energy drift in 500 ps NVE simulations of bulk Ag by a factor of ~7.
   • Liquid Structure: Attenuated spurious sub-shell peaks in the radial distribution function (RDF) of liquid Ag, correcting unphysical atomic over-approach.

4. Computational Cost:
   The improvements were achieved with approximately 5% additional inference FLOPs, placing the method in a low-error, low-overhead region of the cost–accuracy frontier.

Significance and Claims
The paper claims to identify scalar-pathway fidelity as a practical, orthogonal design dimension for short-range equivariant interatomic potentials. The authors argue that:

• Tensor expressivity and scalar fidelity are not substitutable; an expressive equivariant backbone can still lose physical detail if the downstream scalar operations are insufficiently adaptive or spectrally limited.
• The scalar pathway is not merely an implementation detail but a critical interface where geometric information is converted into an energy landscape.
• PAN and PGS address specific, distinct forms of information loss (geometric under-resolution and spectral bias) without requiring explicit physical terms (like long-range electrostatics) or replacing the equivariant backbone.
• The method is complementary to existing extensions (e.g., for long-range interactions or reactivity) rather than an alternative, specifically targeting the residual short-range error remaining after local representation is built.

The authors conclude that future short-range MLIPs should treat scalar aggregation and spectral mixing as first-class design components to ensure that the physical accuracy of the equivariant representation is preserved when it becomes a deployable energy surface.