arXiv subjects

Denis Gueyffier

Publications and source records attributed to Denis Gueyffier.

2 recordsLinked to original sources

arXiv2026

Guarantees by Construction for Learned Finite Volume Schemes on Steady Supersonic Flow

A second order finite volume scheme rests on two local quantities: a gradient reconstructed in each cell, and a limiter which scales it down where the reconstruction would overshoot. Both are set by fixed formulas, and on coarse unstructured meshes a small network can supply better values. But a network is free to output anything, and the usual safeguard is a penalty in the training loss, which discourages inadmissible states without preventing them. We replace the penalty by a hard constraint. The network still sets both quantities, and every value it can produce lies inside safe bounds: its stencil weights cannot cancel a neighbour, and its limiter is capped by the local flow. The flux, the wall treatment and the time step are not learned and carry their own guarantees. Admissibility therefore holds for every value of the weights rather than as an outcome of training, and no negative density or pressure occurred in any computation reported here. Because the scheme is safe whatever the network does, we could ask what the network contributes. We test it on supersonic channel flow over an obstacle, including the forward facing step of Woodward and Colella. Learning lowers the error by 38% on an unseen geometry and 29% on an unseen obstacle topology, measured against the same scheme with the network switched off. The method aims at the accuracy of a fine mesh for the cost of a coarse one, and refining once improves the error fourfold while multiplying the run time by eight. Learning secures half of this improvement for a sixth of this time. All of this comes from one of the two quantities the network sets. The gradient reconstruction reproduces the full effect on its own, and the limiter accounts for about a tenth as much. This also explains why the gain fades beyond the Mach numbers the weights were trained on.

physics.flu-dyn↗

arXiv2026

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

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.

physics.flu-dyn↗