Korteweg-De Vries

Here we apply the PK-NN to control the forced Korteweg-De Vries (KdV) equation

\[ \frac{\partial \eta(t, x)}{\partial t} + \eta(t, x) \frac{\partial \eta(t, x)}{\partial x} + \frac{\partial^3 \eta(t, x)}{\partial x^3} = w(t, x), \]

where \(\eta(t, x)\) is the state, \(w(t,x) = \sum\limits_{i = 1}^3v_i(x) \sin(\pi u_i(t))\) is the forcing term. Control parameters at \(t_n\) are \(\mathbf{u}_n = (u_{1,n}, u_{2,n}, u_{3,n})^T \in [-1, 1]^3\). The functions \(v_i(x) = e^{-25(x - c_i)^2}\) are fixed spatial profiles with \(c_1 = - \frac{\pi}{2}, c_2 = 0, c_3 = \frac{\pi}{2}\). We consider periodic boundary conditions on the spatial variable \(x \in [-\pi, \pi]\), and we discretize with a spatial mesh of \(128\) points.

We consider a tracking problem involving one of the following two observables: the mass \(\int_X \eta(t, x)\mathrm{d} x\) and the momentum \(\int_X \eta^2(t,x)\mathrm{d} x\). Given a reference trajectory \(\{r_n\}\), the tracking problem refers to a Bolza problem with \(\Phi \equiv 0\) and \(L_n(m, \mathbf{u}) = |m - r_n|^2\).

Solving via PK-NN¶

Training data are generated from \(1000\) trajectories of length \(200\) samples. The initial conditions are a convex combination of three fixed spatial profiles and written as

\[ \eta(0, x) = b_1 e^{-(x - \frac{\pi}{2})^2} + b_2(- \sin(\frac{x}{2})^2) + b_3 e^{-(x + \frac{\pi}{2})^2}, \]

with \(b_i > 0\) and \(\sum\limits_{i = 1}^3 b_i = 1\), \(b_i\)'s are randomly sampled in \((0,1)\) with uniform distribution. The training controls \(u_i(t)\) are uniformly randomly generated in \([-1, 1]\).

The dictionary designed for the two tracking problems is of the form

\[ \Psi(\eta) = (1, \int_X \eta(t, x)\mathrm{d} x, \int_X \eta^2(t, x)\mathrm{d} x, \mathrm{NN}(\eta))^T \]

with \(3\) trainable elements.