Fluid Flow (FluidFlow-v0)

The Fluid Flow environment models a two-dimensional oscillatory shear flow. It is a canonical example of a system with nonlinear amplitude-frequency coupling that still admits a low-dimensional Koopman representation.

Dynamics

The continuous-time dynamics are

$$\begin{aligned} \dot{x}_1 &= (\mu + A z)\, x_1 - \omega\, x_2 \\ \dot{x}_2 &= \omega\, x_1 + (\mu + A z)\, x_2 + u \\ \dot{z} &= -\lambda\bigl(z - x_1^2 - x_2^2\bigr) \end{aligned}$$

with parameters $\mu = 0.1$, $\omega = 1.0$, $A = -0.1$, $\lambda = 1.0$.

The action $u$ enters through the $\dot{x}_2$ equation. Integration uses RK45 with $\Delta t = 0.01\,\text{s}$.

Spaces and limits

Property	Value
State dimension	3
Action dimension	1
$x_1, x_2$ range	$[-1, 1]$
$z$ range	$[0, 1]$
Action range	$[-10, 10]$
$\Delta t$	0.01 s
Max episode steps	2000 (20 s)

Cost function

$$c(x, u) = x^\top Q x + u^\top R u, \qquad Q = I_3, \quad R = I_1$$

Reference point: $x^* = \mathbf{0}$.

LQR linearisation

The LQR baseline uses the Jacobian linearised at the origin $(x_1, x_2, z) = (0, 0, 0)$:

$$A_{\text{lin}} = \begin{pmatrix} \mu & -\omega & 0 \\ \omega & \mu & 0 \\ 0 & 0 & -\lambda \end{pmatrix}, \qquad B_{\text{lin}} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$$

Usage

uv run -m koopmanrl.soft_koopman_value_iteration --env_id FluidFlow-v0
uv run -m koopmanrl.soft_actor_koopman_critic \
    --config_file configurations/sakc_fluid_flow_hparams.json

Notes

• The third state $z = x_1^2 + x_2^2$ is the amplitude squared; the system has a natural limit cycle at radius $\sqrt{-\mu / A} = 1$.
• Source: koopmanrl/environments/fluid_flow.py