Kuramoto–Sivashinsky

Spatiotemporal chaos with fourth-order hyper-diffusion:

$$\frac{\partial u}{\partial t}+\alpha u\frac{\partial u}{\partial x}+\beta \frac{{\partial }^{2}u}{\partial x^2}+\gamma \frac{{\partial }^{4}u}{\partial x^4}=0,\alpha =\frac{100}{16},\beta =\frac{100}{{16}^2},\gamma =\frac{100}{{16}^4},$$

with periodic boundaries, trained through the transition to chaos.

Results

relative L2 error 4.3e-03recipe adaptive pseudo-time100k steps, single GPU

The chaotic trajectory is tracked to a relative L2 error of 4.3e-03 — and here pseudo-time stepping is transformative, cutting the baseline's 9.4e-02 by 22×. As the dynamics transition to chaos, the plain residual loss develops transient minima that pull training off the trajectory; the damping term suppresses exactly those excursions. Two further observations sharpen the picture: with pseudo-time active, causal weighting becomes redundant (turning it off even edges slightly ahead at 3.6e-03), and a parameter-matched comparison of PirateNet, ModifiedMlp, and a plain MLP shows all three tie at the baseline level — the method, not the network, is what moves this benchmark.

Reference, PINN prediction, and absolute error through the transition to chaos.

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

Run

cd examples/ks
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

• Longer horizons use time-window training (--config.training.num_time_windows=4), propagating the network's own prediction as the next initial condition.
• Fourth derivatives via Taylor-mode AD (jax.experimental.jet).