Lid-driven Cavity

Steady incompressible flow in a square cavity driven by a moving lid — a boundary-value problem (no time variable):

$$(\mathbf{u}\cdot \mathrm{\nabla })\mathbf{u}+\mathrm{\nabla }p-\nu {\mathrm{\nabla }}^2\mathbf{u}=0,\mathrm{\nabla }\cdot \mathbf{u}=0\text{in }[0,1{]}^2.$$

Problem setup

The domain is the unit square $(x,y)\in [0,1{]}^2$. The top lid slides tangentially with $u=1,v=0$, while the left, right, and bottom walls are no-slip ($u=v=0$). The Reynolds number is $\mathrm{Re}=UL/\nu$ with lid speed $U=1$ and cavity size $L=1$; the showcase runs at $\mathrm{Re}=5000$ ($\nu =1/5000$), where a strong primary vortex fills the cavity and counter-rotating eddies develop in the bottom corners.

Geometry and boundary conditions. The lid velocity is discontinuous at the two top corners, which concentrates error and makes high-Re training stiff.

Both boundary conditions enter the loss as sampled point sets: a u_bc term fitting the lid profile (including the discontinuous corners) and a v_bc term enforcing zero normal flow, alongside the three momentum/continuity residuals — five loss terms balanced by grad-norm weighting.

Results

relative L2 error 0.043recipe adaptive pseudo-timeRe 5000, 50k steps, single GPU

Direct training at Re 5000 stalls: the baseline plateaus at a relative L2 error of 0.64 in the velocity magnitude, trapped in a low-circulation spurious solution (swapping SOAP for Adam changes nothing — 0.68). Adaptive pseudo-time stepping acts as an artificial-time continuation toward the steady state and reaches 0.043 — a 15× improvement and the difference between a wrong flow field and a quantitatively correct one. The comparison is sharp: constant pseudo-time weights do not help here (0.84); the adaptive step-size estimate is what tracks the slowly developing circulation.

Reference and predicted velocity magnitude at Re 5000, and the absolute error.

Training losses and relative L2 error of the showcase (adaptive pseudo-time) run.

Run

cd examples/lid_driven_cavity
python3 main.py --config=configs/pseudo_time.py

Config	Description
configs/baseline.py	Modified MLP baseline
configs/pseudo_time.py	Adaptive pseudo-time stepping (showcase)
configs/fixed_pseudo_time.py	Constant pseudo-time weights

Notes

• Steady problems subclass ForwardBVP — same trainer, no causal weighting.
• Reference data ships for Re ∈ {100, 400, 1000, 1600, 3200, 5000} (--config.Re=...).