Solver Exactness, Learned Flexibility: Equivariant Boundary-Correction Operators for Stokes Flow

Abstract

The drag and mobility of bodies in viscous (Stokes) flow govern problems in shape design, suspensions, or microorganism swimming. Classical solvers compute them accurately but expensively; purely learned surrogates are fast but unreliable off their training data. We combine both: a solver's exactness for the part of the solution operator known in closed form, and learning for the part that is not. For incompressible Stokes flow the elliptic-core kernel is already known: in free space the Leray projector is a single rotation-equivariant Stokeslet with no free parameters, and the boundary-integral solver built on it is exact to machine precision at $O(N)$. The one object with no closed form is the boundary correction. We split the operator: fix the core exactly and equivariantly, and learn only that correction, as a well-conditioned second-kind operator. On a Stokes testbed where the exact solve is ground truth, the split gives a working solver ($2 \times 10^{-3}$ end-to-end, $5$-$16\times$ more data-efficient than a black-box DeepONet) and overturns three expectations. (i) Conditioning is not the bottleneck: a $10^{16}$-conditioned first-kind and a bounded second-kind operator give the same error. (ii) Cross-shape generalization is governed by the descriptor's equivariance, not capacity: a noninvariant descriptor degrades by $>10^5\times$ under rotation, while canonicalization restores near-machine transfer. (iii) Coverage, not expressivity, is the lever; a local equivariant kernel removes the heavy out-of-distribution tail, cutting worst-case interior error from $O(10)$ to $\sim 10^{-7}$. We then open the central exterior problem in 3D: a completed double layer, made exact by quadrature by expansion, is second-kind well-conditioned and $SO(3)$-equivariant, reproduces the analytic drag of spheres and ellipsoids, and composes across bodies.