Korteweg–de Vries

Soliton dynamics with third-order dispersion:

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+{\delta }^2\frac{{\partial }^{3}u}{\partial x^3}=0,\delta =0.022,$$

with periodic boundaries and $u(0,x)=\cos (\pi x)$. The initial profile steepens and breaks into a train of interacting solitons.

Results

relative L2 error 1.5e-04recipe adaptive pseudo-time50k steps, single GPU

Adaptive pseudo-time stepping gives the best result on KdV: 1.5e-04 relative L2 error, about 2× better than the plain baseline (2.9e-04). The soliton train's fine dispersive structure is where the damping term helps — it suppresses the transient oscillations the optimizer otherwise chases during soliton fission. This example is also our cleanest optimizer comparison: swapping SOAP for Adam at identical settings costs 8× (2.2e-03) — see the SOAP deep-dive.

Reference, PINN prediction, and absolute error across soliton fission and interaction.

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

Run

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

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

Notes

• Third derivatives are computed with Taylor-mode automatic differentiation (jax.experimental.jet), which is much cheaper than nesting grad three times.