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@@ -159,7 +159,7 @@ $$
 118 Symplectic methods: The flow map of a Hamiltonian system is symplectic, meaning that its Jacobian   
 119 $\begin{array} { r } { \dot { \Upsilon _ { \varphi } } : = \frac { \partial } { \partial y } \varphi _ { h , f } ( y ) } \end{array}$ satisfies $\Upsilon _ { \varphi } ^ { T } J \Upsilon _ { \varphi } = J$ , where $J$ is the same matrix as in $\textcircled { 2 }$ . As explained in $\mathbb { B } ,$ Ch.   
 120 VI.2], this is equivalent to the preservation of a projected area in the phase space of $[ q , p ] ^ { T }$ . Similarly,   
-121 a numerical integrator is symplectic if its Jacobian ⌥ := @@yn $\begin{array} { r } { \Upsilon _ { \Phi } : = \frac { \partial } { \partial y _ { n } } \Phi _ { h , f } ( y _ { n } ) } \end{array}$ satisfies $\Upsilon _ { \Phi } ^ { T } J \Upsilon _ { \Phi } = J$ . It is   
+121 a numerical integrator is symplectic if its Jacobian ⌥ := @@yn $\begin{array} { r } { \Upsilon _ { \Phi } : = \frac { \partial } { \partial y _ { n } } \Phi _ { h , f } ( y _ { n } ) } \end{array}$ satisfies $\Upsilon _ { \Phi } ^ { T } J \Upsilon _ { \Phi } = J$ . It is   
 122 possible to prove $\mathbb { B } ,$ Ch. VI.4] that a Runge–Kutta method is symplectic if and only if the coeffients   
 123 satisfy
 
@@ -420,9 +420,9 @@ Figure 4: Average of $\overline { { \rho } }$ over 10 trajectories. Shaded area
 Methods and test problems: We train HNNs using different integrators and methods in the inverse problem $\textcircled{6}$ . We use MIRK4 together with the MII method and compare to the implicit midpoint method, RK4 and MIRK4 applied as one-step methods, as well as ISO followed by Störmer–Verlet and RK4 integrated over multiple time-steps. The latter strategy, illustrated in Figure $\bigtriangledown ,$ was suggested in [10], where Störmer–Verlet is used. Separable networks $H _ { \theta } ( q , p ) = H _ { 1 , \theta } ( q ) + H _ { 2 , \theta } ( p )$ are trained on data from the Fermi–Pasta–Ulam–Tsingou (FPUT) problem and the Hénon–Heiles system. For the double pendulum, which is non-separable, a fully connected Flow roll-out H´enon-Hnetwork is used for all methods except Störmer– Flow roll-out H´enon-Heiles h = 0.1, FVerlet, which requires separability in order to be explicit. The Hamiltonians are described in Appendix 0.2 0.0A and all systems have solutions $y ( t ) \not \in \mathbb { R } ^ { 4 }$ .
 
 0.2 0.0 0.0After using the specified integrators in training, a   
-294 0.0  0.2proximated solutions are computed for each learned   
-295 vector field $f _ { \theta }$ 0.2 0.4using the Scikit-learn implementation   
-296 0.2   0.6of DOP853 [35], which is also used to generate   
+294 0.0  0.2proximated solutions are computed for each learned   
+295 vector field $f _ { \theta }$ 0.2 0.4using the Scikit-learn implementation   
+296 0.2   0.6of DOP853 [35], which is also used to generate   
 297 0.0 2.5 5.0 7.5 training data. The error is averaged over $M = { \mathfrak { M } } =$   
 298 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17points and we find what we call the flow error by