Equivariant Neural Prediction of the Stokes Resistance Tensors for Arbitrary Microparticle Shapes

Abstract

The Stokes-flow hydrodynamics of an irregular microparticle is encoded by its grand resistance matrix, a 6x6 tensor whose translational and rotational blocks (A and C) govern settling, diffusion, and orientational transport. Empirical drag correlations compress these tensors to a single scalar, discarding drag's orientation dependence and the rotational response. We present an SO(3)-equivariant neural network that predicts the full symmetric positive-definite A and C blocks from a particle's spherical-harmonic surface representation, equivariant by construction to floating-point precision. Trained on 1.1x10^5 random shapes from near-spherical to very rough, with a sealed test set of 18,000, it achieves 1.4% and 2.7% mean relative error on the two blocks while evaluating each shape 4-5 orders of magnitude faster than the regularised-Stokeslet solver that generated its labels. A spectral-convergence study confirms the representation is faithful: truncating a shape at spherical-harmonic degree 15 changes its resistance by a median of 0.14% (translation) and 0.38% (rotation), while the training shapes, generated band-limited at degree 15, carry no truncation error. Across 1.16x10^6 orientation-sampled settling, rotation and diffusion events, the surrogate reveals lateral drift up to 11 deg, rotational misalignment up to 46 deg, and shape-induced diffusion spreads of 30% (translational) and 2.4x (rotational), all identically zero under any scalar or spheroid reduction. Even the orientation-averaged scalar friction the correlations target, accurate to 1.7-2.8% (median; 2.2-3.2% mean), carries no tensor orientation, whereas the surrogate reproduces that scalar to ~1% while supplying the full anisotropic tensors. A fast, equivariant tensor surrogate can replace both solver and scalar approximation in atmospheric dust transport, microplastic fate and colloidal Brownian dynamics.