SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\delta E (\mathbf {q}, \mathbf {p}) = \left( \begin{array}{c} \mathbf {1} \otimes \nabla f _ {E} \left(s (\mathbf {q})\right) \\ d _ {0} d _ {0} ^ {\intercal} \mathbf {1} \otimes \nabla g _ {E} \left(s \left(d _ {0} d _ {0} ^ {\intercal} \mathbf {p}\right)\right) \end{array} \right), \quad \delta S (\mathbf {q}, \mathbf {p}) = \left( \begin{array}{c} \mathbf {0} \\ d _ {1} ^ {\intercal} d _ {1} \mathbf {1} \otimes \nabla g _ {S} \left(s \left(d _ {1} ^ {\intercal} d _ {1} \mathbf {p}\right)\right) \end{array} \right),
$$
from which the desired  $\mathbf{A}$ -gradients follow directly.
Finally, we can show that the degeneracy conditions for metriplectic structure are satisfied by the network in Section 4.
Theorem A.16. The degeneracy conditions  $\mathbf{L}\nabla S = \mathbf{M}\nabla E = \mathbf{0}$  are satisfied by the metriplectic bracket in Section A.6.
Proof. This is a direct calculation using Theorem 3.1 and Proposition A.6. In particular, it follows that
$$
\begin{array}{l} \mathbf {L} \nabla S = \left( \begin{array}{c c} \mathbf {0} & - d _ {0} ^ {*} \\ d _ {0} & \mathbf {0} \end{array} \right) \left( \begin{array}{c} \mathbf {0} \\ \mathbf {A} _ {1} ^ {- 1} d _ {1} ^ {\intercal} d _ {1} \mathbf {1} \otimes \nabla g _ {S} (s (d _ {1} ^ {\intercal} d _ {1} \mathbf {p})) \end{array} \right) \\ = \left( \begin{array}{c} - \mathbf {A} _ {0} ^ {- 1} \left(d _ {1} d _ {0}\right) ^ {\intercal} d _ {1} \mathbf {1} \otimes \nabla g _ {S} \left(s \left(d _ {1} ^ {\intercal} d _ {1} \mathbf {p}\right)\right) \\ \mathbf {0} \end{array} \right) = \left( \begin{array}{c} \mathbf {0} \\ \mathbf {0} \end{array} \right), \\ \mathbf {M} \nabla E = \left( \begin{array}{c c} \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathbf {A} _ {1} d _ {1} ^ {*} d _ {1} \mathbf {A} _ {1} \end{array} \right) \left( \begin{array}{c} \mathbf {A} _ {0} ^ {- 1} \mathbf {1} \otimes \nabla f _ {E} \left(s (\mathbf {q})\right) \\ \mathbf {A} _ {1} ^ {- 1} d _ {0} d _ {0} ^ {\intercal} \mathbf {1} \otimes \nabla g _ {E} \left(s \left(d _ {0} d _ {0} ^ {\intercal} \mathbf {p}\right)\right) \end{array} \right) \\ = \left( \begin{array}{c} \mathbf {0} \\ \mathbf {A} _ {1} d _ {1} ^ {*} (d _ {1} d _ {0}) d _ {0} ^ {\intercal} \mathbf {1} \otimes \nabla g _ {E} \left(s \left(d _ {0} d _ {0} ^ {\intercal} \mathbf {p}\right)\right) \end{array} \right) = \left( \begin{array}{c} \mathbf {0} \\ \mathbf {0} \end{array} \right), \\ \end{array}
$$
since  $d_{1}d_{0} = 0$  as a consequence of the graph calculus. These calculations establish the validity of the energy conservation and entropy generation properties seen previously in the manuscript body.
Remark A.17. Clearly, this is not the only possible metriplectic formulation for GNNs. On the other hand, this choice is in some sense maximally general with respect to the chosen operators  $\mathbf{L}$ ,  $\mathbf{G}$ , since only constants are in the kernel of  $d_0$  (hence there is no reason to include a nodal term in  $S$ ), and only elements in the image of  $d_1^\top$  (which do not exist in our setting) are guaranteed to be in the kernel of  $d_0^\top$  for any graph. Therefore,  $\mathbf{M}$  is chosen to be essentially  $\mathbf{G}$  without the  $\Delta_0$  term, whose kernel is graph-dependent and hence difficult to design.
B Experimental details and more results
This Appendix provides details regarding the experiments in Section 5, as well as any additional information necessary for reproducing them. We implement the proposed algorithms with Python and PYTORCH [71] that supports CUDA. The experiments are conducted on systems that are equipped with NVIDIA RTX A100 and V100 GPUs. For NODEs capabilities, we use the TORCHDIFFREQ library [16].
B.1 Damped double pendulum
The governing equations for the damped double pendulum can be written in terms of four coupled first-order ODEs for the angles that the two pendula make with the vertical axis  $\theta_{1},\theta_{2}$  and their associated angular momenta  $\omega_{1},\omega_{2}$  (see [72]),
$$
\dot {\theta} _ {i} = \omega_ {i}, \quad 1 \leq i \leq 2, \tag {2}
$$
$$
\dot {\omega} _ {1} = \frac {m _ {2} l _ {1} \omega_ {1} ^ {2} \sin (2 \Delta \theta) + 2 m _ {2} l _ {2} \omega_ {2} ^ {2} \sin (\Delta \theta) + 2 g m _ {2} \cos \theta_ {2} \sin \Delta \theta + 2 g m _ {1} \sin \theta_ {1} + \gamma_ {1}}{- 2 l _ {1} (m _ {1} + m _ {2} \sin^ {2} \Delta \theta)}, \tag {3}
$$
$$
\dot {\omega} _ {2} = \frac {m _ {2} l _ {2} \omega_ {2} ^ {2} \sin (2 \Delta \theta) + 2 (m _ {1} + m _ {2}) l _ {1} \omega_ {1} ^ {2} \sin \Delta \theta + 2 g (m _ {1} + m _ {2}) \cos \theta_ {1} \sin \Delta \theta + \gamma_ {2}}{2 l _ {2} (m _ {1} + m _ {2} \sin^ {2} \Delta \theta)}, \tag {4}
$$
where  $m_{1}, m_{2}, l_{1}, l_{2}$  are the masses resp. lengths of the pendula,  $\Delta \theta = \theta_{1} - \theta_{2}$  is the (signed) difference in vertical angle,  $g$  is the acceleration due to gravity, and
$$
\gamma_ {1} = 2 k _ {1} \dot {\theta} _ {1} - 2 k _ {2} \dot {\theta} _ {2} \cos \Delta \theta .
$$
$$
\gamma_ {2} = 2 k _ {1} \dot {\theta} _ {1} \cos \Delta \theta - \frac {2 (m _ {1} + m _ {2})}{m _ {2}} k _ {2} \dot {\theta} _ {2},
$$
for damping constants  $k_{1}, k_{2}$ .
Dataset. A trajectory of the damped double pendulum by solving an initial value problem associated with the ODE 2. The initial condition used is  $(1.0, \pi/2, 0.0, 0.0)$ , and the parameters are  $m_1 = m_2 = 1$ ,  $g = 1$ ,  $l_1 = 1$ ,  $l_2 = 0.9$ ,  $k_1 = k_2 = 0.1$ . For time integrator, we use the TorchDiffreq library [16] with Dormand-Prince 5 (DOPRI5) as the numerical solver. The total simulation time is 50 (long enough for significant dissipation to occur), and solution snapshots are collected at 500 evenly-spaced temporal indices.
To simulate the practical case where only positional data for the system is available, the double pendulum solution is integrated to time  $T = 50$  (long enough for significant dissipation to occur) and post-processed once the angles and angular momenta are determined from the equations above, yielding the  $(x,y)$ -coordinates of each pendulum mass at intervals of 0.1s. This is accomplished using the relationships
$$
x _ {1} = l _ {1} \sin \theta_ {1}
$$
$$
y _ {1} = - l _ {1} \cos \theta_ {1}
$$
$$
x _ {2} = x _ {1} + l _ {2} \sin \theta_ {2} = l _ {1} \sin \theta_ {1} + l _ {2} \sin \theta_ {2}
$$
$$
y _ {2} = y _ {1} - l _ {2} \cos \theta_ {2} = - l _ {1} \cos \theta_ {1} - l _ {2} \cos \theta_ {2}.
$$
The double pendulum is then treated as a fully connected three-node graph with positional coordinates  $\mathbf{q}_i = (x_i, y_i)$  as nodal features, and relative velocities  $\mathbf{p}_{\alpha} = (d_0 \mathbf{q})_{\alpha}$  as edge features. Note that the positional coordinates  $(x_0, y_0) = (0, 0)$  of the anchor node are held constant during training. To allow for the necessary flexibility of coordinate changes, each architecture from Section 4 makes use of a message-passing feature encoder before time integration, acting on node features and edge features separately, with corresponding decoder returning the original features after time integration.
To elicit a fair comparison, both the NODE and  $\mathrm{NODE + AE}$  architectures are chosen to contain comparable numbers of parameters to the bracket architectures ( $\sim 30\mathrm{k}$ ), and all networks are trained for 100,000 epochs. For each network, the configuration of weights producing the lowest overall error during training is used for prediction.
Hyperparameters. The networks are trained to reconstruct the node/edge features in mean absolute error (MAE) using the Adam optimizer [73]. The NODEs and metriplectic bracket use an initial learning rate of  $10^{-4}$ , while the other models use an initial learning rate of  $10^{-3}$ . The width of the hidden layers in the message passing encoder/decoder is 64, and the number of hidden features for nodes/edges is 32. The time integrator used is simple forward Euler.
Network architectures. The message passing encoders/decoders are 3-layer MLPs mapping, in the node case, nodal features and their graph derivatives, and in the edge case, edge features and their graph coderivatives, to a hidden representation. For the bracket architectures, the attention mechanism used in the learnable coderivatives is scaled dot product. The metriplectic network uses 2-layer MLPs  $f_{E}, g_{E}, g_{S}$  with scalar output and hidden width 64. For the basic NODE, node and edge features are concatenated, flattened, and passed through a 4-layer fully connected network of width 128 in each hidden layer, before being reshaped at the end. The NODE+AE architecture uses a 3-layer fully connected network which operates on the concatenated and flattened latent embedding of size  $32 * 6 = 192$ , with constant width throughout all layers.
B.2 Mujoco
We represent an object as a fully-connected graph, where a node corresponds to a body part of the object and, thus, the nodal feature corresponds to a position of a body part or joint. To learn the dynamics of an object, we again follow the encoder-decoder-type architecture considered in the
(a)  $\mathbf{q}$
(b)  $\mathbf{p}$
Figure 5: [Double pendulum] Trajectories of  $\mathbf{q}$  and  $\mathbf{p}$ : ground-truth (solid lines) and predictions of the metriplectic bracket model (dashed lines). The evolution of the energy  $E$  and the entropy  $S$  over the simulation time. Note that slight fluctuations appear in  $E$  due to the fact that forward Euler is not a symplectic integrator.
(c)  $E$  and  $S$
double-pendulum experiments. First we employ a node-wise linear layer to embed the nodal feature into node-wise hidden representations (i.e., the nodal feature  $\mathbf{q}_i$  corresponds to a position of a body part or an angle of a joint.). As an alternative encoding scheme for the nodal feature, in addition to the position or the angle, nodal velocities are considered as additional nodal features, i.e.,  $\mathbf{q}_i = (q_i, v_i)$ . The experimental results of the alternative scheme is represented in the following section B.2.2.
The proposed dynamics models also require edge features (e.g., edge velocity), which are not presented in the dataset. Thus, to extract a hidden representation for an edge, we employ a linear layer, which takes velocities of the source and destination nodes of the edge as an input and outputs edge-wise hidden representations, i.e., the edge feature corresponds to a pair of nodal velocities  $\mathbf{p}_{\alpha} = (v_{\mathrm{src}(\alpha)}, v_{\mathrm{dst}(\alpha)})$ , where  $v_{\mathrm{src}(\alpha)}$  and  $v_{\mathrm{dst}(\alpha)}$  denote velocities of the source and destination nodes connected to the edge.
The MuJoCo trajectories are generated in the presence of an actor applying controls. To handle the changes in dynamics due to the control input, we introduce an additive forcing term, parameterized by an MLP, to the dynamics models, which is a similar approach considered in dissipative SymODEN [32]. In dissipative SymODEN, the forcing term is designed to affect only the change of the generalized momenta (also known as the port-Hamiltonian dynamics [74]). As opposed to this approach, our proposed forcing term affects the evolution of both the generalized coordinates that are defined in the latent space. Once the latent states are computed at specified time indices, a node-wise linear decoder is applied to reconstruct the position of body parts of the object. Then the models are trained based on the data matching loss measured in mean-square errors between the reconstructed and the ground-truth positions.
B.2.1 Experiment details
We largely follow the experimental settings considered in [23].
Dataset. As elaborated in [23], the standard Open AI Gym [75] environments preprocess observations in ad-hoc ways, e.g., Hopper clips the velocity observations to  $[-10, 10]^d$ . Thus, the authors in [23] modified the environments to simply return the position and the velocity  $(q, v)$  as the observation and we use the same dataset, which is made publicly available by the authors. The dataset consists of training and test data, which are constructed by randomly splitting the episodes in the replay buffer into training and test data. Training and test data consist of  $\sim 40\mathrm{K}$  and  $\sim 300$  or  $\sim 85$  trajectories, respectively. For both training and test data, we include the first 20 measurements (i.e., 19 transitions) in each trajectory.
Hyperparameters. For training, we use the Adam optimizer [73] with the initial learning rate 5e-3 and weight decay 1e-4. With the batch size of 200 trajectories, we train the models for 256 epochs. We also employ a cosine annealing learning rate scheduler with the minimum learning rate 1e-6. For time integrator, we use the Torchdiffreq library with the Euler method.
Network architectures. The encoder and decoder networks are parameterized as a linear layer and the dimension of the hidden representations is set to 80. For attention, the scaled dot-product attention is used with 8 heads and the embedding dimension is set to 16. The MLP for handling the
forcing term consists of three fully-connected layers (i.e., input, output layers and one hidden layer with 128 neurons). The MLP used for parameterizing the "black-box" NODEs also consists of three fully-connected layers with 128 neurons in each layer.
B.2.2 Additional results
Figure 6 reports the loss trajectories for all considered dynamics models. For the given number of maximum epochs (i.e., 256), the Hamiltonian and double bracket models tend to reach much lower training losses (an order of magnitude smaller) errors than the NODE and Gradient models do. The metriplectic model produces smaller training losses compared to the NODE and gradient models after a certain number of epochs (e.g., 100 epochs for Hopper).
Figure 6: [Mujoco] Train MSE over epoch for all considered dynamics models. For the nodal feature, only the position or the angle of the body part/joint is considered.
In the next set of experiments, we provide not only positions/angles of body parts as nodal features, but also velocities of the body parts as nodal features (i.e.,  $\mathbf{q}_i = (q_i,v_i)$ ). Table 6 reports the relative errors measured in L2-norm; again, the Hamiltonian, double bracket, and metriplectic outperform other dynamics models. In particular, the metriplectic bracket produces the most accurate predictions in the Hopper and Swimmer environments. Figure 7 reports the loss trajectories for all considered models. Similar to the previous experiments with the position as the only nodal feature, the Hamiltonian and Double bracket produces the lower training losses than the NODE and Gradient models do. For the Hopper and Swimmer environments, however, among all considered models, the metriplectic model produces the lowest training MSEs after 256 training epochs.
Dataset	HalfCheetah	Hopper	Swimmer
NODE+AE	0.0848 ± 0.0011	0.0421 ± 0.0041	0.0135 ± 0.00082
Hamiltonian	0.0403 ± 0.0052	0.0294 ± 0.0028	0.0120 ± 0.00022
Gradient	0.0846 ± 0.00358	0.0490 ± 0.0013	0.0158 ± 0.00030
Double Bracket	0.0653 ± 0.010	0.0274 ± 0.00090	0.0120 ± 0.00060
Metriplectic	0.0757 ± 0.0021	0.0269 ± 0.00035	0.0114 ± 0.00067
Table 6: Relative errors of the network predictions of the MuJoCo environments on the test set, reported as avg±stdev over 4 runs.
B.3 Node classification
To facilitate comparison with previous work, we follow the experimental methodology of [61].
(a) HalfCheetah
(b) Hopper
Figure 7: [Mujoco] Train MSE over epoch for all considered dynamics models. For the nodal feature, along with the position or the angle of the body part/joint, the node velocities are also considered.
(c) Swimmer
B.3.1 Experiment details
Datasets. We consider the three well-known citation networks, Cora [58], Citeseer [59], and Pubmed [60]; the proposed models are tested on the datasets with the original fixed Planetoid training/test splits, as well as random train/test splits. In addition, we also consider the coauthor graph, CoauthorCS [61] and the Amazon co-purchasing graphs, Computer and Photo [62]. Table 7 provides some basic statistics about each dataset.
Dataset	Cora	Citeseer	PubMed	CoauthorCS	Computer	Photo
Classes	7	6	3	15	10	8
Features	1433	3703	500	6805	767	745
Nodes	2485	2120	19717	18333	13381	7487
Edges	5069	3679	44324	81894	245778	119043
Table 7: Dataset statistics.
Hyperparameters The bracket architectures employed for this task are identical to those in Section 4 except for that the right-hand side of the Hamiltonian, gradient, and double bracket networks is scaled by a learnable parameter Sigmoid  $(\alpha) > 0$ , and the matrix  $\mathbf{A}_2 = \mathbf{I}$  is used as the inner product on 3-cliques. It is easy to verify that this does not affect the structural properties or conservation character of the networks. Nodal features  $\mathbf{q}_i$  are specified by the datasets, and edge features  $\mathbf{p}_{\alpha} = (d_0\mathbf{q})_{\alpha}$  are taken as the combinatorial gradient of the nodal features. In order to determine good hyperparameter configurations for each bracket, a Bayesian search is conducted using Weights and Biases [76] for each bracket and each dataset using a random 80/10/10 train-valid/test split with random seed 123. The number of runs per bracket was 500 for CORA, CiteSeer, and PubMed, and 250 for CoauthorCS, Computer, and Photo. The hyperparameter configurations leading to the best validation accuracy are used when carrying out the experiments in Table 4 and Table 5.
Specifically, the hyperparameters that are optimized are as follows: initial learning rate (from 0.0005 to 0.05), number of training epochs (from 25 to 150), method of integration (rk4 or dopri5), integration time (from 1 to 5), latent dimension (from 10 to 150 in increments of 10), pre-attention mechanism  $\tilde{a}$  (see below), positive function  $f$  (either exp or Squareplus), number of pre-attention heads (from 1 to 15, c.f. Remark A.14), attention embedding dimension (from  $1\times$  to  $15\times$  the number of heads), weight decay rate (from 0 to 0.05), dropout/input dropout rates (from 0 to 0.8), and the MLP activation function for the metriplectic bracket (either relu, tanh, or squareplus). The pre-attention is chosen
from one of four choices, defined as follows:
$$
\tilde {a} \left(\mathbf {q} _ {i}, \mathbf {q} _ {j}\right) = \frac {\left(\mathbf {W} _ {K} \mathbf {q} _ {i}\right) ^ {\intercal} \mathbf {W} _ {Q} \mathbf {q} _ {j}}{d}
$$
scaled dot product,
$$
\tilde {a} \left(\mathbf {q} _ {i}, \mathbf {q} _ {j}\right) = \frac {\left(\mathbf {W} _ {K} \mathbf {q} _ {i}\right) ^ {\intercal} \mathbf {W} _ {Q} \mathbf {q} _ {j}}{\left| \mathbf {W} _ {K} \mathbf {q} _ {i} \right| \left| \mathbf {W} _ {Q} \mathbf {q} _ {j} \right|}
$$
cosine similarity,
$$
\tilde {a} \left(\mathbf {q} _ {i}, \mathbf {q} _ {j}\right) = \frac {\left(\mathbf {W} _ {K} \mathbf {q} _ {i} - \overline {{\mathbf {W} _ {K} \mathbf {q} _ {i}}}\right) ^ {\intercal} \left(\mathbf {W} _ {Q} \mathbf {q} _ {j} - \overline {{\mathbf {W} _ {Q} \mathbf {q} _ {j}}}\right)}{\left| \mathbf {W} _ {K} \mathbf {q} _ {i} - \overline {{\mathbf {W} _ {K} \mathbf {q} _ {i}}} \right| \left| \mathbf {W} _ {Q} \mathbf {q} _ {j} - \overline {{\mathbf {W} _ {Q} \mathbf {q} _ {j}}} \right|},
$$
$$
\tilde {a} \left(\mathbf {q} _ {i}, \mathbf {q} _ {j}\right) = \left(\sigma_ {u} \sigma_ {x}\right) ^ {2} \exp \left(- \frac {\left| \mathbf {W} _ {K} \mathbf {u} _ {i} - \mathbf {W} _ {Q} \mathbf {u} _ {j} \right| ^ {2}}{2 \ell_ {u} ^ {2}}\right) \exp \left(- \frac {\left| \mathbf {W} _ {K} \mathbf {x} _ {i} - \mathbf {W} _ {Q} \mathbf {x} _ {j} \right| ^ {2}}{2 \ell_ {x} ^ {2}}\right),
$$
Network architectures. The architectures used for this experiment follow that of GRAND [49], consisting of the learnable affine encoder/decoder networks  $\phi, \psi$  and learnable bracket-based dynamics in the latent space. However, recall that the bracket-based dynamics require edge features, which are manufactured as  $\mathbf{p}_{\alpha} = (d_0 \mathbf{q})_{\alpha}$ . In summary, the inference procedure is as follows:
$$
\mathbf {q} (0) = \phi (\mathbf {q})
$$
(nodal feature encoding),
$$
\mathbf {p} (0) = d _ {0} \mathbf {q} (0)
$$
(edge feature manufacturing),
$$
\left(\mathbf {q} (T), \mathbf {p} (T)\right) = \left(\mathbf {q} (0), \mathbf {p} (0)\right) + \int_ {0} ^ {T} \left(\dot {\mathbf {q}}, \dot {\mathbf {p}}\right) d t, \quad \text {(l a t e n t d y n a m i c s)}
$$
$$
\tilde {\mathbf {q}} = \psi (\mathbf {q} (T)),
$$
(nodal feature decoding)
$$
\mathbf {y} = c (\tilde {\mathbf {q}}).
$$
(class prediction)
Training is accomplished using the standard cross entropy
$$
H \left(\mathbf {t}, \mathbf {y}\right) = \sum_ {i = 1} ^ {| \mathcal {V} |} \mathbf {t} _ {i} ^ {\mathsf {T}} \log \mathbf {y} _ {i},
$$
where  $\mathbf{t}_i$  is the one-hot truth vector corresponding to the  $i^{\mathrm{th}}$  node. In the case of the metriplectic network, the networks  $f_{E},g_{E},g_{S}$  are 2-layer MLPs with hidden dimension equal to the latent feature dimension and output dimension 1.
B.3.2 Additional depth study
Here we report the results of a depth study on Cora with the Planetoid train/val/test split. Table 9 shows the train/test accuracy of the different bracket architectures under two different increased depth conditions, labeled Task 1 and Task 2, respectively. Task 1 refers to fixing the integration step-size at  $\Delta t = 1$  and integrating to a variable final time  $T$ , while Task 2 instead fixes the final time  $T$  to the value identified by the hyperparameter search (see Table 8) and instead varies the step-size  $\Delta t$ . Notice that both tasks involve repeatedly composing the trained network and hence simulate increasing depth, so that any negative effects of network construction such as oversmoothing, oversquashing, or vanishing/exploding gradients should appear in both cases. For more information, Table 10 provides a runtime comparison corresponding to the depth studies in Table 9.
Observe that every bracket-based architecture exhibits very stable performance in Task 2, where the final time is held fixed while the depth is increased. This suggests that our proposed networks are dynamically stable and effectively mitigate negative effects like oversmoothing which are brought by repeated composition and often seen in more standard GNNs. Interestingly, despite success on Task 2, only the gradient and metriplectic architectures perfectly maintain or improve their performance during the more adversarial Task 1 where the final time is increased with a fixed step-size. This suggests that, without strong diffusion, the advection experienced during conservative dynamics has the potential to radically change label classification over time, as information is moved through the feature domain in a loss-less fashion.
Remark B.1. It is interesting that the architecture most known for oversmoothing (i.e., gradient) exhibits the most improved classification performance with increasing depth on Task 1. This is perhaps due to the fact that the gradient system decouples over nodes and edges, while the others
do not, meaning that the gradient network does not have the added challenge of learning a useful association between the manufactured edge feature information and the nodal labels. It remains to be seen if purely node-based bracket dynamics exhibit the same characteristics as the node-edge formulations presented here.
CORA networks	Trainable Parameters	Integration Time
Hamiltonian	60723	1.49625
Gradient	160772	14.82404
Double Bracket	30718	5.36151
Metriplectic	104088	7.53107