Burgers

Viscous Burgers equation developing a steep internal layer:

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\frac{0.01}{\pi }\frac{{\partial }^{2}u}{\partial x^2},(t,x)\in [0,1]\times [-1,1],$$

with $u(0,x)=-\sin (\pi x)$ and homogeneous Dirichlet boundaries. A viscous shock forms at $x=0$ and sharpens over time.

Results

relative L2 error 4.0e-05recipe baseline — architecture-insensitive100k steps, single GPU

The baseline recipe resolves the shock to a relative L2 error of 4.0e-05, with the residual error confined to the thin viscous layer. Burgers turns out to be a saturated benchmark for modern PINN training: in a parameter-matched comparison, a plain MLP, a modified MLP, and PirateNet all land within 1% of each other, and pseudo-time stepping changes nothing — with SOAP and grad-norm balancing in place, every reasonable configuration reaches the same answer. Treat it as a correctness check rather than a discriminating benchmark.

Reference, PINN prediction, and absolute error; the error is confined to the shock.

Training losses and relative L2 error of the showcase run.

Run

cd examples/burgers
python3 main.py --config=configs/baseline.py

Config	Description
configs/baseline.py	PirateNet + causal weighting
configs/pseudo_time.py	Adaptive pseudo-time stepping
configs/fixed_pseudo_time.py	Constant pseudo-time weights