SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Table 1: Prediction error  $(\times 10^{-2})$  on N-body dataset. The header of each column "p, s, h" denotes the scenario with  $p$  isolated particles,  $s$  sticks and  $h$  hinges. Results averaged across 3 runs.
|Train| = 500	|Train| = 1500
1,2,0	2,0,1	3,2,1	0,10,0	5,3,3	1,2,0	2,0,1	3,2,1	0,10,0	5,3,3
Linear	8.23±0.00	7.55±0.00	9.76±0.00	11.36±0.00	11.62±0.00	8.22±0.00	7.55±0.00	9.76±0.00	11.36±0.00	11.62±0.00
GNN [9]	5.33±0.07	5.01±0.08	7.58±0.08	9.83±0.04	9.77±0.02	3.61±0.13	3.23±0.07	4.73±0.11	7.97±0.44	7.91±0.31
TFN [39]	11.54±0.38	9.87±0.27	11.66±0.08	13.43±0.31	12.23±0.12	5.86±0.35	4.97±0.23	8.51±0.14	11.21±0.21	10.75±0.08
SE(3)-Tr. [29]	5.54±0.06	5.14±0.03	8.95±0.04	11.42±0.01	11.59±0.01	5.02±0.03	4.68±0.05	8.39±0.02	10.82±0.03	10.85±0.02
RF [17]	3.50±0.17	3.07±0.24	5.25±0.44	7.59±0.25	7.73±0.39	2.97±0.15	2.19±0.11	3.80±0.25	5.71±0.31	5.66±0.27
NC (RF)	2.84±0.01	2.35±0.04	4.32±0.08	6.67±0.26	7.14±0.25	2.91±0.11	1.76±0.01	3.23±0.01	5.09±0.05	5.34±0.03
NC+(RF)	2.92±0.10	2.34±0.03	4.28±0.05	7.02±0.14	7.06±0.33	2.87±0.05	1.78±0.01	3.24±0.02	5.06±0.06	5.23±0.29
EGNN [18]	2.81±0.12	2.27±0.04	4.67±0.07	4.75±0.05	4.59±0.07	2.59±0.10	1.86±0.02	2.54±0.01	2.79±0.04	3.25±0.07
NC (EGNN)	2.41±0.03	2.18±0.02	3.53±0.01	4.26±0.03	4.13±0.03	2.23±0.13	1.91±0.03	2.14±0.03	2.36±0.05	2.86±0.03
NC+(EGNN)	2.25±0.01	2.07±0.06	2.01±0.02	3.54±0.06	3.96±0.04	2.32±0.05	1.86±0.13	2.39±0.01	2.35±0.07	2.99±0.02
GMN [19]	1.84±0.02	2.02±0.02	2.48±0.04	2.92±0.04	4.08±0.03	1.68±0.04	1.47±0.03	2.10±0.04	2.32±0.02	2.86±0.01
NC (GMN)	1.55±0.07	1.58±0.02	2.07±0.03	2.73±0.02	2.99±0.03	1.59±0.03	1.18±0.05	1.47±0.01	1.66±0.01	1.93±0.03
NC+(GMN)	1.57±0.02	1.43±0.02	2.03±0.04	2.57±0.04	2.72±0.03	1.49±0.02	1.17±0.04	1.44±0.01	1.70±0.06	1.91±0.02
Table 2: Prediction error  $\left( {\times {10}^{-2}}\right)$  on MD17 dataset. Results averaged across 3 runs.
Aspirin	Benzene	Ethanol	Malonaldehyde
RF	EGNN	GMN	RF	EGNN	GMN	RF	EGNN	GMN	RF	EGNN	GMN
Raw	10.94±0.01	14.41±0.15	10.14±0.03	103.72±1.29	62.40±0.53	48.12±0.40	4.64±0.01	4.64±0.01	4.83±0.01	13.93±0.03	13.64±0.01	13.11±0.03
NC	10.87±0.01	9.63±0.01	9.50±0.06	63.22±0.01	55.05±1.60	41.62±1.16	4.62±0.01	4.63±0.01	4.83±0.01	12.93±0.03	12.82±0.01	12.97±0.01
NC+	10.87±0.01	9.60±0.02	9.40±0.09	63.04±0.05	54.76±0.74	41.26±0.15	4.62±0.01	4.62±0.01	4.83±0.01	12.93±0.03	12.81±0.02	12.92±0.01
Naphthalene	Salicylic	Toluene	Uracil
RF	EGNN	GMN	RF	EGNN	GMN	RF	EGNN	GMN	RF	EGNN	GMN
Raw	0.50±0.01	0.47±0.02	0.40±0.01	1.23±0.01	1.02±0.02	0.91±0.01	10.93±0.04	11.78±0.07	10.22±0.08	0.64±0.01	0.64±0.01	0.59±0.01
NC	0.39±0.01	0.37±0.01	0.38±0.01	1.09±0.01	0.84±0.01	0.86±0.01	10.85±0.01	10.64±0.11	10.17±0.04	0.60±0.01	0.57±0.01	0.56±0.01
NC+	0.39±0.01	0.37±0.01	0.37±0.02	1.09±0.01	0.84±0.01	0.86±0.01	10.85±0.01	10.52±0.01	10.14±0.01	0.60±0.01	0.57±0.01	0.56±0.02
were identical. The results reported in the main experiments were based on a model of NC (2) for a balance of training cost and performance.
We reported the performance of NC w/ additional points (denoted by  $\mathbf{NC}^{+}$ ) and w/o additional points (the relaxed version, denoted by NC).
We leveraged three datasets towards different aspects of reasoning dynamics to exhaust the performance of NC. We used the N-body simulation benchmark proposed in [19]. Compared with the previous ones, this benchmark has more different settings, e.g., number of particles and training data. Furthermore, it also considers some physical constraints (e.g., hinges and sticks) widely existing in the real world. For molecular dynamics, We used MD17 [28] that consists of the molecular dynamics simulations of eight different molecules as a scenario. We followed [19] to construct the training/testing sets. We also considered reasoning human motion. Follow [19], the dataset was constructed based on 23 trials in CMU Motion Capture Database [27].
As  $\mathrm{NC}^+$  requires additional supervised data, we accordingly extracted more intermediate data points between the input and target from each dataset. It is worth noting that this operation was tractable as the sampling frequency of the datasets was much higher than our requirement.
We reported the main experimental results w.r.t. three datasets in Tables 1, 2, and 3, respectively. Overall,  $\mathrm{NC}^+$  significantly improved the performance of all baselines on almost all datasets and across all settings, which empirically demonstrates the generality as well as the effectiveness of the proposed method. Remarkably, NC (without extra data) achieved slightly low results but still had great advantages over performance in comparison with three baselines.
Specifically, The average improvement on N-body datasets was most significant, where we observe a near  $20\%$  decrease in prediction error for all three baseline methods. For example, NC (RF) and  $\mathrm{NC^{+}}$
Table 3: Prediction error  $(\times 10^{-2})$  on motion capture. Results averaged across 3 runs.
GNN	TFN	SE(3)-Tr.	RF	EGNN	GMN
Raw	67.3±1.1	66.9±2.7	60.9±0.9	197.0±1.0	59.1±2.1	43.9±1.1
NC	N/A	N/A	N/A	164.8±1.7	55.8±6.1	30.0±1.4
NC+	N/A	N/A	N/A	162.6±0.8	51.3±4.0	29.6±0.8
(a) N-body (1, 2, 0)
(b) N-body (5, 3, 3)
Figure 2: A detailed comparison of NCGNN without extra data on 5 datasets, w.r.t.  $k$ , average of 3 runs, conducted on a V100. The lines with dot, circle, and triangle denote the results of GMN, EGNN, and RF, respectively. The first, second, and third rows are the results of prediction error, training time of one epoch, and total training time, respectively.
(c) MD17 (Aspirin)
(d) MD17 (Uracil)
(e) Motion Capture
(RF) greatly outperformed the original RF and even had better results than the original EGNN under some settings (e.g., 3,2,1). This phenomenon also happened on the molecular dynamics datasets, where we find that  $\mathrm{NC^{+}}$  (EGNN) not only defeated its original version, but also the original GMN and even  $\mathrm{NC^{+}}$  (GMN) on many settings.
4.4 Impact of estimation step  $k$
We conducted experiments to explore how the performance changes with respect to the estimation step  $k$ . The experimental results are shown in Figure 2.
For all three baselines, the performance improved rapidly as the growth of step  $k$  from 0 to 2, but this advantage gradually vanished as we set larger  $k$ . Interestingly, the curve of training time showed an inverse trend, where we find that the total training time of NC (5) with GMN was almost two times slower than the NC (1) version. Therefore, we set  $k = 2$  in previous experiments to get the best balance between performance and training cost.
By comparing the prediction errors of different baselines, we find that GMN that considers the physical constraints (e.g., sticks and hinges) usually had better performance than the other two methods, but the gap significantly narrowed with the increase of step  $k$ . For example, on MD17 and motion datasets, EGNN obtained similar prediction errors for  $k \geq 2$  while demanding less training time. This phenomenon demonstrates that predicting the integration with multiple estimations lowers the requirement for model capacity.
Overall, learning with NC significantly reduced the prediction errors of different models. The additional training time was also acceptable in most cases.
4.5 A Comparison between NC and  $\mathbf{NC}^{+}$
The only difference between NC and  $\mathrm{NC^{+}}$  was the use of the regularization loss in  $\mathrm{NC^{+}}$  to learn the intermediate velocities. Therefore, we designed experiments to compare the intermediate results and the final predictions of NC and  $\mathrm{NC^{+}}$  with GMN as the backbone model.
(a) NC(2)
(b) NC(3)
(c) NC(5)
Figure 3: The prediction errors of intermediate velocities on valid set, w.r.t. training epoch. The blue and green lines denote the prediction errors of NC and  $\mathrm{NC}^+$ , respectively.
Figure 4: Visualization of the intermediate velocities w.r.t.  $k$ . The red, blue, and green lines denote the target, prediction of NC, and prediction of  $\mathrm{NC}^{+}$ , respectively.
Figure 5: Consecutive predictions on the Motion dataset. The red, blue, and green lines denote the target, the prediction of NC (0), and the prediction of  $\mathrm{NC^{+}}$  (2), respectively. The interval between two figures is identical to the setting in the main experiment.
We first tracked the average intermediate velocity prediction errors on valid datasets in terms of training epochs. The results are shown in Figure 3, from which we observe that the prediction errors were not huge for both two methods, especially  $\mathrm{NC}^{+}$  obtained highly accurate prediction for the intermediate velocities with different  $k$ . For NC, we actually find that it implicitly learned to model the intermediate velocities more or less, even though its prediction errors gradually improved or dramatically fluctuated during training. This observation empirically demonstrates the effectiveness of the intermediate velocities in predicting the final state of the system. We also reported the results on MD17 and N-body datasets in Appendix C.
We then visualized the intermediate velocities and the final predictions in Figure 4. The intermediate velocities were visualized by adding the velocity with the corresponding coordinate at the intermediate time point. From Figure 4, we can find that  $\mathrm{NC}^{+}$  (green ones) learned very accurate predictions with only a few non-overlapped nodes to the target nodes (red ones). For the loose version NC, it learned good predictions for the backbone part, but failed on the limbs (especially the legs). The motion of the main backbone was usually stable, while that of the limbs involved swing and twisting. We also conducted experiments on the other two types of datasets, please see Figure 8 in Appendix C.
4.6 On the Time Evolution of Dynamic Systems
In previous experiments, we show that  $\mathrm{NC}^+$  was capable of learning highly accurate predictions of intermediate velocities. This inspired us to realize the full potential of  $\mathrm{NC}^+$  by producing a series of predictions given only the initial state as input. For each time point (including the intermediate time points), we used the most recent  $k + 1$  points of averaged data as input to predict the next velocity and
Table 4: Multi-step prediction results  $(\times 10^{-2})$  on the N-body and MD17 datasets. ADE and FDE denote average displacement error and final displacement error, respectively. The results of four baseline methods are from [49].
N-body (Springs)	MD17 (Aspirin)	MD17 (Benzene)	MD17 (Ethanol)	MD17 (Malonaldehyde)
ADE	Accuracy	ADE	FDE	ADE	FDE	ADE	FDE	ADE	FDE
LSTM [45]	-	53.5	17.59	24.79	6.06	9.46	7.73	9.88	15.14	22.90
NRI [50]	-	93.0	12.60	18.50	1.89	2.58	6.69	8.78	12.79	19.86
GroupNet [51]	-	-	10.62	14.00	2.02	2.95	6.00	7.88	7.99	12.49
EqMotion [49]	16.8	97.6	5.95	8.38	1.18	1.73	5.05	7.02	5.85	9.02
NC (EqMotion)	16.2	98.9	4.74	6.76	1.15	1.63	5.02	6.87	5.19	8.07
coordinate. This strategy produced much more stable predictions than the baseline NC (0) – directly using the last output as input to predict the next state.
The results are shown in Figure 5, from which we can find that both the baseline (blue lines) and  $\mathrm{NC^{+}}$  (green lines) had good predictions at the initial steps. However, due to the increasing acculturated errors, the baseline soon collapsed and only the main backbone was roughly fit to the target. By contrast,  $\mathrm{NC^{+}}$  was still able to produce promising results. The deviations mainly concentrated on the limbs. Therefore, we argue that  $\mathrm{NC^{+}}$  can produce not only the highly accurate one-step prediction, but also the relatively reliable consecutive predictions. We also uploaded the .gif files of examples on all three datasets in Supplementary Materials, which demonstrated the consistent conclusion.
We also compared our method with the EGNN methods for multi-step prediction [49-52]. Specifically, multi-step prediction takes a sequence of states with fixed interval as input and predicts a sequence of future states with same length and interval. As recent multi-step prediction methods are still based on sequence or encoder-decoder architectures, it would be interesting to examine the applicability of NC to them. To this end, we adapted NC ( $k = 2$ ) to the source code of the best-performing method EqMotion [49] and used identical settings for the hyper-parameters.
The experimental results are shown in Table 4, where we can observe that NC (EqMotion) outperformed the base model EqMotion on all datasets and across all metrics. It is worth noting that the architecture and parameter-setting of the EGNN module were identical to EqMotion and NC (EqMotion), which clearly demonstrates the effectiveness of the proposed method. We have also released the source code of NC (EqMotion) in our GitHub repository.
We conducted ablation studies to investigate the effectiveness of the proposed Newton-Cotes integration. We implemented two alternative methods: NC (coeffs  $= 1$ ), where we set all the coefficients for summing the predicted velocities as 1; and depth-varying ODE, where we employed common neural ODE solvers from [40, 41] to
Table 5: Ablation study results  $\left( {\times {10}^{-2}}\right)$  .
N-body (1,2,0)	MD17 (Aspirin)	Motion
GMN	1.84	10.14	43.9
NC (coeffs= 1)	1.78	9.64	48.4
depth-varying ODE	1.82	10.08	42.2
NC	1.55	9.50	30.0
address our problem. We used an identical GMN as the backend EGNN model and conducted a simple hyper-parameter search for the learning rate and ODE solver.
The results are presented in Table 5. The two alternative methods methods generally performed better than the baseline GMN. The performance of depth-varying GDE was comparable to that of NC (coeffs = 1), but both were lower than that of the original NC. This observation further supports the effectiveness of our method.
In this paper, we propose NC to tackle the problem of reasoning system dynamics. We show that the existing state-of-the-art methods can be derived to the basic form of NC, and theoretically/empirically prove the effectiveness of high order NC. Furthermore, with a few additional data provided,  $\mathrm{NC}^{+}$  is capable of producing promising consecutive predictions. In future, we plan to study how to improve the ability of  $\mathrm{NC}^{+}$  on the consecutive prediction task.
We would like to thank all anonymous reviewers for their insightful and invaluable comments. This work is funded by NSFCU19B2027/NSFC91846204/NSFC62302433, National Key R&D Program of China (Funding No.SQ2018YFC000004), Zhejiang Provincial Natural Science Foundation of China (No.LGG22F030011) and sponsored by CCF-Tencent Open Fund (CCF-Tencent RAGR20230122).
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A The Derivations of NC (0) for Other EGNN Models
In this section, we first illustrate how the other two baseline models (i.e. RF [17] and GMN [19]) can be derived to NC (0), and then discuss multi-layer EGNN.
A.1 RF
The overall equivariant convolution process of RF is similar to that of EGNN. The main difference is that RF does not use the node feature. Instead, it leverages the L2 norm of the velocity as an additional feature to update the predicted velocity:
$$
\hat {\mathbf {v}} _ {i} ^ {0} = \operatorname {N o r m} \left(\mathbf {v} _ {i} ^ {0}\right) \left(\phi_ {v} \left(\mathbf {h} _ {i}\right) \mathbf {v} _ {i} ^ {0} + \frac {1}{N - 1} \sum_ {j \neq i} \left(\mathbf {x} _ {i} ^ {0} - \mathbf {x} _ {j} ^ {0}\right) \mathbf {m} _ {i j} ^ {0}\right), \tag {19}
$$
$$
\hat {\mathbf {x}} _ {i} ^ {T} = \mathbf {x} _ {i} ^ {0} + \hat {\mathbf {v}} _ {i} ^ {0} T, \tag {20}
$$
where  $\mathrm{Norm}(\mathbf{v}_i^0)$  denotes the L2 norm of the input velocity  $\mathbf{v}_i^0$ . Therefore, RF can be also viewed as a NC (0) model.
A.2 GMN
The main difference between GMN and EGNN is that GMN detects the sub-structures in the system and process the particles in each special sub-structure independently. Specifically, GMN re-formulates the original EGNN (i.e., Equations (2-4)) as follows:
$$
\mathbf {m} _ {i j} ^ {0} = \phi_ {e} \left(\mathbf {h} _ {i}, \mathbf {h} _ {j}, \left\| \mathbf {x} _ {i} ^ {0} - \mathbf {x} _ {j} ^ {0} \right\| ^ {2}, e _ {i j}\right), \tag {21}
$$
$$
\hat {\mathbf {v}} _ {k} ^ {0} = \phi_ {v} \left(\sum_ {i \in \mathcal {S} _ {k}} \mathbf {h} _ {i}\right) \mathbf {v} _ {k} ^ {0} + \sum_ {i \in \mathcal {S} _ {k}} \phi_ {k} \left(\mathbf {x} _ {i} ^ {0}, \mathbf {m} _ {i j} ^ {0}\right), \tag {22}
$$
$$
\hat {\mathbf {v}} _ {i} ^ {0} = \operatorname {F K} \left(\hat {\mathbf {v}} _ {k} ^ {0}\right), \tag {23}
$$
$$
\hat {\mathbf {x}} _ {i} ^ {T} = \mathbf {x} _ {i} ^ {0} + \hat {\mathbf {v}} _ {i} ^ {0} T, \tag {24}
$$
where  $\hat{\mathbf{v}}_k^0$  is the predicted velocity of the sub-structure. GMN then uses it to calculate the velocity of each particle by a function FK which can be either learnable or based on the angles and relative positions in the sub-structure [19].
A.3 Multi-layer EGNN
In current EGNN methods, the input coordinate  $\mathbf{x}_i^0$  and velocity  $\mathbf{v}_i^0$  are regarded as constant features and used in different layers. For example, in the multi-layer version EGNN,  $\mathbf{m}_{ij}^0$  will be formulated as follows:
$$
\mathbf {m} _ {i j} ^ {0, l} = \phi_ {e} \left(\mathbf {h} _ {i} ^ {l - 1}, \mathbf {h} _ {j} ^ {l - 1}, \left\| \mathbf {x} _ {i} ^ {0} - \mathbf {x} _ {j} ^ {0} \right\| ^ {2}, e _ {i j}\right), \tag {25}
$$
where  $\mathbf{m}_{ij}^{0,l}$  is  $\mathbf{m}_{ij}^{0}$  in layer  $l$  and  $\mathbf{h}_i^{l-1}$  denotes the hidden feature in layer  $l-1$ . Evidently, only the hidden features are updated within different layers. The overall formulation of multi-layer layer is akin to the single-layer EGNN.
B Proofs of Things
Proof. The objective of the existing methods for a single system can be defined as: