Title: On the Geometry of Uncertainty: Equivariant Uncertainty Estimation for Molecular Vector Properties
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
Keywords: Uncertainty Quantification, Equivariant Graph Neural Networks, Evidential Deep Learning, Molecular Modeling

Abstract: 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 key contribution is a new parameterization for the covariance matrix, which is provably equivariant, identifiable, and correctly parameterized. We further introduce a calibration metric based on the Mahalanobis distance to rigorously assess equivariant multivariate uncertainty. Extensive experiments on molecular property prediction benchmarks, including MD17, OC20, and QM7-X, show that our framework consistently outperforms established baselines, achieving state-of-the-art predictive accuracy with well-calibrated uncertainty estimates. 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.

TL;DR: This paper introduces a novel uncertainty estimation method that accurately predicts vector-valued properties, such as molecular forces, by explicitly accounting for underlying 3D rotational symmetries.