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

Abstract

Computing the viscous Stokes flow around a shape requires solving a boundary-integral equation, and for a new shape the solve must begin from scratch. Learned operators promise to spread this cost across shapes, but it is not clear what such an operator retains of the solver it replaces, or what determines whether it transfers to shapes it was not trained on. We make both questions answerable by choosing a problem which is exactly solvable except for a single term: a second-kind boundary-integral problem is solved exactly using a kernel-independent fast summation, while the boundary correction, which has no closed form, is learned. The solver's guarantees carry over unchanged: exactness on the closed-form part, $O(N)$ scaling, $SO(3)$-equivariance to machine precision, and an $O(N)$ differentiable adjoint. We then make precise what the learning contributes. It does not contribute to accuracy, differentiability, or scaling $O(N)$, all of which are provided by the solver. Learning contributes a one-time cost in that the forward map is trained once and then evaluated on a new shape in a single pass rather than resolved. Measured against baselines, the learned map is $5$ to $16\times$ more data-efficient than a black-box DeepONet and maintains a $\sim 2.5\times$ lower in-distribution error than a geometry-aware operator, although that operator is stronger in the low-data limit and the learned map is less reliable out of distribution. We trace this fragility to the global parameterization and reduce it with a local equivariant kernel. The exactly-solvable setting yields clarity about the mechanism: geometric generalization is governed by invariance and coverage, not by conditioning or by capacity.

24 pages, 6 figures, 10 tables