Davide Murari
A PhD student in numerical analysis at the Norwegian University of Science and Technology.
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
Symplectic Time Dependent Network
By Davide Murari
Symplectic Time Dependent Network
- 118
