Equivariant Evidential Deep Learning for Interatomic Potentials

Uncertainty quantification (UQ) is critical for assessing the reliability of machine learning interatomic potentials (MLIPs) in molecular dynamics (MD) simulations, identifying extrapolation regimes and enabling uncertainty-aware workflows such as active learning for training dataset construction. Existing UQ approaches for MLIPs are often limited by high computational cost or suboptimal performance. Evidential deep learning (EDL) provides a theoretically grounded single-model alternative that determines both aleatoric and epistemic uncertainty in a single forward pass. However, extending evidential formulations from scalar targets to vector-valued quantities such as atomic forces introduces substantial challenges, particularly in maintaining statistical self-consistency under rotational transformations. To address this, we propose \textit{Equivariant Evidential Deep Learning for Interatomic Potentials} ($\text{e}^2$IP), a backbone-agnostic framework that models atomic forces and their uncertainty jointly by representing uncertainty as a full $3\times3$ symmetric positive definite covariance tensor that transforms equivariantly under rotations. Experiments on diverse molecular benchmarks show that $\text{e}^2$IP provides a stronger accuracy-efficiency-reliability balance than the non-equivariant evidential baseline and the widely used ensemble method. It also achieves better data efficiency through the fully equivariant architecture while retaining single-model inference efficiency.

💡 Research Summary

The paper introduces e²IP (Equivariant Evidential Interatomic Potentials), a single‑model framework that simultaneously predicts atomic forces and quantifies their uncertainty in a rotation‑consistent manner. Traditional uncertainty quantification for machine‑learning interatomic potentials (MLIPs) relies on ensembles, which are accurate but computationally expensive, or on evidential deep learning (EDL), which so far has been limited to scalar targets. Extending EDL to vector‑valued forces is non‑trivial because the predicted uncertainty must transform as a rank‑2 tensor under SE(3) rotations: Σ′ = R Σ Rᵀ.

To satisfy this requirement, e²IP represents each atom’s force uncertainty with a full 3 × 3 symmetric positive‑definite (SPD) covariance matrix Σ. The covariance is constructed via a Lie‑algebra parameterization: a symmetric matrix S is predicted in the (0ᵉ ⊕ 2ᵉ) irreducible representation space (scalar trace component plus traceless symmetric component), then mapped to Cartesian form and exponentiated, Σ = exp(S). The matrix exponential guarantees SPD and is equivariant because exp(RSRᵀ) = R exp(S) Rᵀ.

Uncertainty modeling follows a Normal‑Inverse‑Wishart (NIW) evidential prior. The scale matrix Ψ is re‑parameterized as ν Σ₀, where Σ₀ is the SPD matrix built above, and ν, κ are evidence parameters controlling the strength of the prior. Marginalizing the NIW yields a multivariate Student‑t predictive distribution, from which aleatoric uncertainty U_ale = E