Cheng, Huang, Wang, and Zhou proposed a neural network method for solving incompressible Navier-Stokes equations that enforces the divergence-free constraint by construction rather than by penalty. Velocity is represented through stream functions in two dimensions and vector potentials in three, guaranteeing that any output is automatically incompressible. The system then decouples velocity from pressure, solving them sequentially rather than simultaneously, and handles nonlinearity through Gauss-Newton linearization that converts the problem into a sequence of linear subproblems.
The structural insight is that embedding a physical law into the architecture eliminates an entire class of error that no amount of training can remove. Standard neural approaches approximate the divergence-free condition by adding a penalty term — the network is punished for compressibility but never prevented from producing it. The residual compressibility is small but nonzero, and it accumulates over time, contaminating long-horizon predictions. The stream-function representation makes this category of error structurally impossible. The constraint is not learned; it is inherited.
This principle — that architectural enforcement outperforms learned compliance — appears across engineering. Type systems in programming prevent categories of bugs that testing can only detect. Physical guardrails on roads prevent departure, while lane markings only discourage it. In any system where violation of a constraint produces cascading downstream errors, building the constraint into the structure is not conservative design — it is the only design that remains correct under distribution shift.
(arXiv:2603.17906)