Symplectic Time Dependent Network

Coordinates for the phase space \((q_1,p_1)\in\mathbb{R}^{2n}\)

We want to solve the initial value problem

\begin{cases} \dot{q}_1(t) = \partial_p H(q_1(t),p_1(t))\\ \dot{p}_1(t) =- \partial_q H(q_1(t),p_1(t)) + F(q_1(t),p_1(t))\\ q_1(0)=q_0,\,\,p_1(0)=p_0 \end{cases}

And we rely on the correspondent conservative formulation in the variables \((q,p):=(q_1,q_2,p_1,p_2)\in\mathbb{R}^{2n}\)

\begin{cases} \begin{bmatrix} \dot{q}(t) \\ \dot{p}(t) \end{bmatrix} = \mathbb{J}\nabla \tilde{H}(q(t),p(t))\in\mathbb{R}^{4n},\\ q(0) = (q_0,q_0),\,\,p(0)=(p_0,p_0) \end{cases}

We approximate \(t\mapsto (q_1(t),q_2(t),p_1(t),p_2(t))\) by composing maps of the following form

\begin{bmatrix} q^j \\ p^j \end{bmatrix} \mapsto (q^{j+1},p^{j+1}):=\Phi^t_j(q^j,p^j):=\begin{bmatrix} p^j+\eta_j(t)-\eta_j(0) \\ -q^j+\nabla V_j(t,p^j) - \nabla V_j(0,p^j)\end{bmatrix}\\ j=1,...,L.

\nabla V_j(t,p) = M_j^T\Sigma(M_jp + L_jt + b_j)\in\mathbb{R}^{2n},\\ V_j(t,p)=1^T\Gamma(M_jp + L_jt + b_j)

\eta_j(t) = A_jt + a_j\in\mathbb{R}^{2n}

Preserving the physical limit means having \(q_1(t)=q_2(t)\) and \(p_1(t)=p_2(t)\)

This can be obtained by constraining suitably the weights.

\begin{bmatrix} q \\ p \end{bmatrix} \mapsto \Phi^t_j(q,p):=\begin{bmatrix} p+\eta_j(t)-\eta_j(0) \\ -q+\nabla V_j(t,p) - \nabla V_j(0,p)\end{bmatrix}\\ =:\begin{bmatrix} p+\bar{\eta}_j(t) \\ -q+\nabla \bar{V}_j(t,p)\end{bmatrix}

Additionally, \(\Phi_j^t\) is symplectic by construction for every \(t\):

\Phi^t_j(q,p)=\begin{bmatrix} p+\bar{\eta}_j(t) \\ -q+\nabla \bar{V}_j(t,p)\end{bmatrix}

D\Phi_j^t(q,p) = \begin{bmatrix} 0_n & I_n \\ -I_n & \nabla^2\bar{V}_j(t,p)\end{bmatrix}

\begin{bmatrix} 0_n & I_n \\ -I_n & \nabla^2\bar{V}_j(t,p)\end{bmatrix}^T\mathbb{J} \begin{bmatrix} 0_n & I_n \\ -I_n & \nabla^2\bar{V}_j(t,p)\end{bmatrix}\\ =\begin{bmatrix}I_n & 0_n \\ -\nabla^2 \bar{V}_j(t,p) & I_n\end{bmatrix} \begin{bmatrix} 0_n & I_n \\ -I_n & \nabla^2\bar{V}_j(t,p)\end{bmatrix}\\ =\begin{bmatrix} 0_n & I_n \\ -I_n & 0_n \end{bmatrix}=\mathbb{J}

\mathcal{N}(t,q_0,q_0,p_0,p_0) := \Phi_L^t \circ ... \circ \Phi_1^t(q_0,q_0,p_0,p_0)\\ =:(q_1(\theta,t),q_2(\theta,t),p_1(\theta,t),p_2(\theta,t))\\ =(q(\theta,t),p(\theta,t))

We then find \(\theta\) by minimising

Where we collect in \(\theta\) all the free parameters in the model.

\mathcal{L}(\theta):=\frac{1}{N}\sum_{i=1}^N \left\|\frac{d}{dt}q_1(\theta,t_i)-\partial_{p_1}\tilde{H}(q(\theta,t_i),p(\theta,t_i))\right\|^2+\\ \frac{1}{N}\sum_{i=1}^N \left\|\frac{d}{dt}p_1(\theta,t_i)+\partial_{q_1}\tilde{H}(q(\theta,t_i),p(\theta,t_i))\right\|^2

0 < t_0 < t_1 < ... < t_N < T