Preserving Geometric Symmetry in Uncertainty Estimation for Molecular Forces

Accurate and reliable uncertainty estimation (UE) for vector-valued physical properties is crucial for scientific discovery in fields like drug and materials discovery. For example, atomic forces are central for finding optimal structures and identifying equilibrium systems, and a fundamental requirement of these vectors is that they must be equivariant to 3D rotations. However, existing UQ methods often fail to respect these geometric constraints, leading to poorly calibrated uncertainty and degraded predictive performance. To address the problem, we introduce a novel framework for equivariant multivariate evidential regression. Our primary contribution is a theoretically grounded parameterization of the evidential prior's scale matrix, constructed via a decomposition into an invariant scalar component and an equivariant low-rank term. Furthermore, we address the lack of suitable evaluation standards for vector fields by introducing a rigorous calibration metric based on the Euclidean distance. Extensive experiments on molecular property prediction benchmarks, including MD17, QM7-X and OC20, show that our framework consistently outperforms established baselines, achieving state-of-the-art predictive accuracy with well-calibrated uncertainty. This work provides a principled and practical approach to uncertainty estimation for equivariant vector-valued properties, paving the way for more trustworthy machine learning applications in scientific discovery.