SlowGuess/NeurIPS_2023_OCR huggingface dataset:
By plugging  $q = \alpha \otimes \nabla_{\mathcal{G}}u$  into Eq. 4, the energy transport process can be reformulated as:
$$
\frac {\partial u}{\partial t} = - \phi^ {- 1} (\alpha \otimes d i v (\nabla_ {\mathcal {G}} u)) = - \phi^ {- 1} (\alpha \otimes \Delta u), \tag {5}
$$
where  $\Delta = div(\nabla_{\mathcal{G}})$  is the graph Laplacian operator. Since the PDE in Eq. 5 is equivalent to the E-L equation of the quadratic functional  $\mathcal{J}(u) = \min_u\int_{\mathcal{G}}\alpha \otimes |\nabla_{\mathcal{G}}u|^2 du$  (after taking  $\phi$
<sup>3</sup>Since the optimization schema has been switched to the layer-by-layer manner, the initialization  $x_0$  becomes  $x^{(l-1)}$  from the previous layer.
$^4\lambda$  can be either pre-defined or learned from the data.
away), a major issue is the over-smoothness in  $u$  that might result in vanishing flows. In this context, we propose to replace the  $\ell_2$ -norm integral functional  $\mathcal{J}(u)$  with TV-based counterpart  $\mathcal{J}_{TV}(u) = \min_u\int_{\mathcal{G}}\alpha \otimes |\nabla_{\mathcal{G}}u|du$ .
Renovate new E-L equation - graph neural transport equations. Following the min-max optimization schema in Eq. 1-3, we introduce an auxiliary matrix  $f$  to lift the undifferentiable barrier. After that, the minimization of  $\mathcal{I}_{TV}(u)$  boils down into a dual min-max functional  $\mathcal{J}_{TV}(u,f) = \min_u\max_f\int_{\mathcal{G}}\alpha \otimes f(\nabla_{\mathcal{G}}u)du$ . Since  $u(t)$  is a time series, it is difficult to derive the deterministic solutions (as Eq. 2-3) by letting  $\frac{\partial}{\partial u}\mathcal{J}_{TV} = 0$  and  $\frac{\partial}{\partial f}\mathcal{J}_{TV} = 0$ . Instead, we use Gâteaux variations to optimize  $\mathcal{I}_{TV}(u,f)$  via the following two coupled time-dependent PDEs (please see Section S1.2, Eq. S14 to Eq. S19, in Supplementary for details):
$$
\left\{ \begin{array}{l} \max  _ {f} \frac {d f}{d t} = \alpha \otimes \nabla_ {\mathcal {G}} u \\ \min  _ {u} \frac {d u}{d t} = \alpha \otimes d i v (f) \end{array} \right. \tag {6}
$$
Remarks. The solution to Eq. 6 is known as continuous max-flow and constitutes a continuous version of a graph-cut [1]. Since  $\alpha$  is a latent variable and potential energy  $u$  is given, the maximization of  $f$  opts towards maximizing the spreading flow through the lens of  $\alpha$ . In this regard, the mechanistic role of auxiliary matrix  $f$  is essentially the latent (maximized) spreading flows that satisfy  $u(t + 1)_i = u(t)_i + \sum_{j=1}^{N} f_{ij}(t)$ . The potential energy  $\hat{u}$  can be solved via a wave equation  $(u_{tt} = div(f_t) = \alpha^2 \otimes \Delta u)$ , where the system dynamics is predominated by the adjusted Lagrangian mechanics  $\alpha^2 \otimes \Delta u$ . By determining  $\alpha$  at a granularity of graph links, we devise a novel GAN model to predict the spreading flows  $f$  which not only offers explainability underlying the min-max optimization mechanism in Eq. 6 but also sets the stage to understand system dynamics through machine learning.
Develop a GAN model of flow prediction with TV-based Lagrangian Mechanics. The overall network architecture is shown in Fig. 4 (b), which consists of a generator (red solid box) and a discriminator module (blue solid box). Specifically, the generator  $(G)$  consists of (1) a GCN component [14] to optimize  $\hat{u}$  through the wave equation  $u_{tt} = \alpha^2\otimes \Delta u$  and (2) a FC layer to characterize the non-linear mapping function  $\hat{x} (t + 1) = \phi^{-1}(\hat{u} (t))$ . In contrast, the discriminator  $(D)$  is designed to (1) synthesize  $\alpha$  and (2) construct the future  $\tilde{u}_{t + 1}$  based on the current  $u_{t}$  and current estimation of spreading flow  $f = \alpha \otimes \nabla_{\mathcal{G}}u$  (orange dash box). To make the network architecture consistent between generator and discriminator modules, we include another GCN to map the synthesized  $\tilde{u} (t + 1)$  to the external force  $\tilde{x} (t + 1)$ . Note, since the working mechanism of this adversarial model underlines the min-max optimization in the energy transport equation, the nature of predicted spreading flows is carved by the characteristics of max-flow.
The driving force of our network is to minimize (1) the MSE (mean square error) between the output of the generator  $\hat{x}_{t + 1}$  and the observed regional features, (2) the distance between the synthesized regional features  $\tilde{x}_{t + 1}$  (from the discriminator) and the output of generator  $\hat{x}_{t + 1}$  (from the generator). In the spirit of probabilistic GAN [43], we use one loss function  $\mathcal{L}_D$  to train the discriminator  $(D)$  and another one  $\mathcal{L}_G$  to train the generator  $(G)$ :
$$
\left\{ \begin{array}{c} \mathcal {L} _ {D} = D \left(x _ {t + 1}\right) + \left[ \xi - D \left(G \left(x _ {t}\right)\right) \right] ^ {+} \\ \mathcal {L} _ {G} = D \left(G \left(x _ {t}\right)\right) \end{array} \right. \tag {7}
$$
where  $\xi$  denotes the positive margin and the operator  $[\cdot ]^{+} = \max (0,\cdot)$ . Minimizing  $\mathcal{L}_G$  is similar to maximizing the second term of  $\mathcal{L}_D$  except for the non-zero gradient when  $D(G(x_{t}))\geq \xi$ .
In this section, we evaluate the performance of the proposed GNN-PDE-COV framework with comparison to six graph learning benchmark methods on a wide variety of open graph datasets [32], as well as a proof-of-concept application of uncovering the propagation pathway of pathological events in Alzheimer's disease (AD) from the longitudinal neuroimages.
3.1 Datasets and experimental setup
Dataset and benchmark methods. We evaluate the new GNN models derived from our proposed GNN framework in two different applications. First, we use three standard citation networks, namely
Cora, CiteSeer, and Pubmed [32] for node classification (the detailed data statistic is shown in Table S2 of Supplementary). We adopt the public fixed split [40] to separate these datasets into training, validation, and test sets. We follow the experimental setup of [9] for a fair comparison with six benchmark GNN models (vanilla GCN [23], GAT [34], GCNII [9], ResGCN [25], DenseGCN [25], GRAND [6]). Since our  $DG$ -layer can be seamlessly integrated into existing GNNs as a plug-in. The corresponding new GNN models (with  $DG$ -layer) are denoted  $\mathrm{GCN}+$ ,  $\mathrm{GAT}+$ ,  $\mathrm{GCNII}+$ ,  $\mathrm{ResGCN}+$ ,  $\mathrm{DenseGCN}+$ , and  $\mathrm{GRAND}+$ , respectively.
Second, we apply the GAN model in Section 2.2.2 to predict the concentration level of AD-related pathological burdens and their spreading pathways from longitudinal neuroimages. Currently, there is no in-vivo imaging technique that can directly measure the flow of information across brain regions. Here, our computational approach holds great clinical value in understanding the pathophysiological mechanism involved in disease progression [22]. Specifically, we parcellate each brain into 148 cortical surface regions and 12 sub-cortical regions using Destrieux atlas [12]. The wiring topology of these 160 brain regions is measured by diffusion-weighted imaging [3] and tractography techniques [17]. The regional concentration levels AD pathology including amyloid, tau, and fluorodeoxyglucose (FDG) and cortical thickness (CoTh) are measured from PET (positron emission tomography) and MRI (magnetic resonance imaging) scans [20]. We use a total of  $M = 1$ , 291 subjects from ADNI [28], each having longitudinal imaging data (2-5 time points). The details of image statistics and pre-processing are shown in Sec. S2.1.2. Since we apply the flow prediction model to each modality separately, we differentiate them with  $X$ -FlowNet ( $X$  stands for amyloid, tau, FGD, and CoTh).
Experimental setup. In the node classification task, we verify the effectiveness and generality of  $DG$ -layer in various number of layers ( $L = 2, 4, 8, 16, 32, 64, 128$ ). All baselines use their default parameter settings. Evaluation metrics include accuracy, precision and F1-score. To validate the performance of  $X-FlowNet$ , we examine (1) prediction accuracy (MAE) of follow-up concentration level, (2) prediction of the risk of developing AD using the baseline scan, and (3) understand the propagation mechanism in AD by revealing the node-to-node spreading flows of neuropathologies.
The main results of graph node classification and flow prediction are demonstrated in Section 3.2 and 3.3, respectively. Other supporting results such as ablation study and hyper-parameter setting are shown in Section S2 of the Supplementary document.
3.2 Experimental results on graph node classification
We postulate that by mitigating the over-smoothing issue, we can leverage the depth of GNN models to effectively capture complex feature representations in graph data. As shown in Table 1, we investigate the graph node classification accuracy as we increase the number of GNN layers by six benchmark GNN models and their corresponding plug-in models (indicated by  $+^{\prime}$  at the end of each GNN model name) with the  $DG$ -layer. The results demonstrate that: (1) the new GNN models generated from the GNN-PDE-COV framework have achieved SOTA in Cora  $(86.30\%)$  by GCNII $+$ , Citeseer  $(75.65\%)$  by GRAND $+$ , and Pubmed  $(80.10\%)$  by GCNII $+$ ; (2) all of new GNN models outperforms their original counterparts with significant improvement in accuracy; (3) the new GNN models exhibit less sensitivity to the increase of model depth compared to current GNN models; (4) the new GNN models are also effective in resolving the gradient explosion problem [26] (e.g., the gradient explosion occurs when training GAT on all involved datasets with deeper than 16 layers, while our  $\mathrm{GAT + }$  can maintain reasonable learning performance even reaching 128 layers.)
It is important to note that due to the nature of the graph diffusion process, graph embeddings from all GNN models (including ours) will eventually become identical after a sufficiently large number of layers [11]. However, the selective diffusion mechanism (i.e., penalizing excessive diffusion across communities) provided by our GNN-PDE-COV framework allows us to control the diffusion patterns and optimize them for specific graph learning applications.
3.3 Application for uncovering the propagation mechanism of pathological events in AD
First, we evaluate the prediction accuracy between the ground truth and the estimated concentration values by our  $X$ -FlowNet and six benchmark GNN methods. The statistics of MAE (mean absolute error) by  $X$ -FlowNet, GCN, GAT, GRAND, ResGCN, DenseGCN and GCNII, at different noise levels on the observed concentration levels, are shown in Fig. 5 (a). It is clear that our  $X$ -FlowNet consistently outperforms the other GCN-based models in all imaging modalities.
Table 1: Test accuracies \((\%)\) on citation networks. We show the mean value, the quota of increase \((\uparrow)/decrease(\downarrow)\) after adding \(DG\) layer. Statistical significance is determined from 50 resampling tests. \(*\) means statistically significance with \(p \leq 0.05\), \(\*\* denotes \(p \leq 0.01\). The missing results are due to the huge consumption of GPU memory for large graphs (DenseGCN) or gradient explosions (GAT) or non-convergence (GRAND). The best performance of baselines is denoted in blue, while the best performance after adding the \(DG\) layer is denoted in red.
Dataset	Model	L=2	L=4	L=8	L=16	L=32	L=64	L=128
Cora	GCN	81.30	79.90	75.70	25.20	20.00	31.80	20.80
GCN+	82.70**	82.70**	82.30**	70.60**	67.80**	66.00**	59.90**
GAT	82.40	80.30	57.90	31.90	-	-	-
GAT+	82.60	80.50	69.70**	66.00**	63.60**	54.60**	45.70**
GRAND	80.00	82.64	82.74	83.45	81.83	80.81	79.19
GRAND+	81.93**	83.45**	82.95	84.27**	83.15*	81.52**	80.10**
ResGCN	76.30	77.30	76.20	77.60	73.30	31.90	31.90
ResGCN+	77.80**	78.70**	78.80**	78.60**	76.90**	34.90	33.60**
DenseGCN	76.60	78.50	76.00	-	-	-	-
DenseGCN+	78.00**	78.70	76.90**	-	-	-	-
GCNII	76.40	81.90	81.50	84.80	84.60	85.50	85.30
GCNII+	84.70**	84.80**	84.70**	85.20**	85.40**	86.30**	85.60**
Citeseer	GCN	70.20	62.50	62.90	21.00	17.90	22.90	19.80
GCN+	72.90**	67.30**	72.00**	54.70**	50.30**	48.40**	46.60**
GAT	71.70	58.60	26.60	18.10	-	-	29.30**
GAT+	73.00**	69.50**	47.60**	31.80**	31.30**	30.60**	29.30**
GRAND	71.94	72.58	73.87	75.00	75.16	72.90	69.52
GRAND+	72.26*	73.55**	75.16**	75.65*	75.52*	74.52*	72.26**
ResGCN	67.10	66.00	63.60	65.50	62.3	18.80	18.10
ResGCN+	68.00**	67.60**	66.00**	66.00**	65.80**	24.00**	24.30**
DenseGCN	67.40	64.00	62.20	-	-	-	-
DenseGCN+	67.80*	66.60**	64.70**	-	-	-	-
GCNII	66.50	67.80	69.30	71.60	73.10	71.40	70.20
GCNII+	72.40**	73.30**	73.80**	73.40**	73.80**	74.60**	73.90**
Pubmed	GCN	78.50	76.50	77.30	40.90	38.20	38.10	38.70
GCN+	79.80**	79.10**	78.20**	77.40**	76.20**	75.10**	73.00**
GAT	77.40	72.20	77.80	40.70	-	-	-
GAT+	77.90*	77.30**	78.50*	73.50**	68.20**	66.80**	63.50**
GRAND	77.07	77.94	78.29	79.93	79.12	-	-
GRAND+	78.03**	78.34*	80.21**	80.08**	79.32	-	-
ResGCN	76.30	77.30	76.20	77.60	73.30	31.90	31.90
ResGCN+	77.80**	78.70**	78.80*	78.60**	76.90**	76.80**	32.00
DenseGCN	75.80	76.10	75.80	-	-	-	-
DenseGCN+	76.10	76.70*	77.50**	-	-	-	-
GCNII	77.30	78.80	79.50	79.70	79.90	0.7980	79.70
GCNII+	78.40**	80.10**	80.00*	80.10*	80.00*	80.00*	80.10*
Second, we have evaluated the potential of disease risk prediction, which can be regarded as a graph classification problem. We assume that we have baseline amyloid, tau, FDG, and CoTh scans, and evaluate the prediction accuracy, precision and F1-score of various models in forecasting the risk of developing AD. We consider two dichotomous cases: one included only AD vs. CN groups and the other involved AD/LMCI vs. CN/EMCI. The results of the mean of 5-fold cross-validation are shown in Table 2. Our GNN-PDE-COV model not only achieved the highest diagnostic accuracy but also demonstrated a significant improvement (paired  $t$ -test  $p < 0.001$ ) in disease risk prediction compared to other methods. These results suggest that our approach holds great clinical value for disease early diagnosis.
Third, we examine the spreading flows of tau aggregates in CN (cognitively normal) and AD groups. As the inward and outward flows shown in Fig. 5(b), it is evident that there are significantly larger amount of tau spreading between sub-cortical regions and entorhinal cortex in CN (early sign of AD onset) while the volume of subcortical-entorhinal tau spreading is greatly reduced in the late stage of AD. This is consistent with current clinical findings that tau pathology starts from sub-cortical regions and then switches to cortical-cortical propagation as disease progresses [24]. However, our Tau-FlowNet offers a fine-granularity brain mapping of region-to-region spreading flows over time, which provides a new window to understand the tau propagation mechanism in AD etiology [13].
Figure 5: (a) Prediction accuracy by  $X$ -FlowNet and six benchmark GNN models w.r.t. various noise levels. (b) The subcortical→cortical tau flows are profound in CN. But in AD, there is a diminished extent of such flows.
Table 2: The performance of disease risk prediction. Note: RGCN denotes ResGCN, DGCN denotes DenseGCN. \* denotes the significant improvement (paired  $t$  -test:  $p <   0.001$  ). Blue: Tau, red: amyloid, orange: FDG, green: CoTh.
Tau	Unit (%)	GCN	GCN+	GAT	GAT+	GCNII	GCNII+	RGCN	RGCN+	DGCN	DGCN+	GRAND	GRAND+
AD/LMCI vs. CN/EMCI	Precision	80.15	90.03(*)	69.91	86.18(*)	83.93	90.03(*)	84.64	89.46(*)	84.03	91.58(*)	87.95	88.22(*)
Accuracy	82.30	88.74(*)	81.05	87.50(*)	83.79	88.75(*)	86.03	90.00(*)	85.54	91.25(*)	88.75	90.12(*)
F1-score	75.55	84.49(*)	72.87	84.72(*)	78.82	84.45(*)	83.15	88.54(*)	82.45	91.39(*)	88.14	89.44(*)
AD vs. CN	Precision	89.29	91.92(*)	87.26	90.13(*)	83.65	88.52(*)	92.61	95.72(*)	92.61	95.91(*)	91.77	95.76(*)
Accuracy	86.64	90.91(*)	84.86	88.41(*)	76.84	86.36(*)	91.07	95.45(*)	91.07	95.65(*)	90.91	95.45(*)
F1-score	85.64	90.26(*)	83.99	87.16(*)	71.51	84.68(*)	90.45	95.32(*)	90.45	95.55(*)	88.86	95.38(*)
Amyloid	Unit (%)	GCN	GCN+	GAT	GAT+	GCNII	GCNII+	RGCN	RGCN+	DGCN	DGCN+	GRAND	GRAND+
AD/LMCI vs. CN/EMCI	Precision	76.36	83.78(*)	67.73	71.79(*)	60.87	60.01( )	72.53	83.21(*)	74.92	60.17(*)	79.00	79.93(*)
Accuracy	76.40	79.44(*)	75.43	77.57(*)	74.31	76.64(*)	75.99	78.50(*)	76.92	77.57(*)	80.37	81.31(*)
F1-score	70.33	72.58(*)	67.66	69.39(*)	63.57	67.31(*)	70.66	70.67(*)	72.68	67.77(*)	79.25	79.63(*)
AD vs. CN	Precision	81.58	88.37(*)	81.54	87.98(*)	70.59	79.98(*)	85.75	93.09(*)	83.87	90.56(*)	65.53	89.62(*)
Accuracy	80.77	88.10(*)	80.78	88.10(*)	75.02	81.24(*)	85.56	92.86(*)	85.29	90.48(*)	80.95	87.80(*)
F1-score	78.14	87.68(*)	78.07	87.98(*)	65.87	77.42(*)	85.34	92.92(*)	82.30	90.27(*)	72.43	88.22(*)
FDG	Unit (%)	GCN	GCN+	GAT	GAT+	GCNII	GCNII+	RGCN	RGCN+	DGCN	DGCN+	GRAND	GRAND+
AD/LMCI vs. CN/EMCI	Precision	68.43	69.29(*)	55.86	59.29(*)	60.08	70.94(*)	50.45	55.14(*)	50.45	55.14(*)	51.38	56.25(*)
Accuracy	73.17	76.00(*)	72.17	77.00(*)	71.78	74.54(*)	70.98	74.26(*)	70.98	74.26(*)	71.10	75.00(*)
F1-score	63.94	68.15(*)	62.15	66.99(*)	61.02	69.07(*)	58.96	63.29(*)	58.96	63.29(*)	59.42	64.29(*)
AD vs. CN	Precision	81.11	87.25(*)	61.90	62.33(*)	74.31	81.06(*)	59.77	80.57(*)	66.84	81.77(*)	70.91	72.24(*)
Accuracy	82.17	84.62(*)	72.82	78.95(*)	79.55	82.05(*)	73.35	79.58(*)	73.87	80.11(*)	84.21	86.32(*)
F1-score	79.40	82.04(*)	64.23	69.66(*)	73.88	80.99(*)	62.77	75.98(*)	63.92	76.58(*)	76.99	78.06(*)
CoTh	Unit (%)	GCN	GCN+	GAT	GAT+	GCNII	GCNII+	RGCN	RGCN+	DGCN	DGCN+	GRAND	GRAND+
AD/LMCI vs. CN/EMCI	Precision	74.85	76.23(*)	62.63	67.15(*)	62.63	74.71(*)	62.63	68.77(*)	62.63	64.59(*)	63.81	68.77(*)
Accuracy	80.68	82.32(*)	79.10	79.34(*)	79.10	80.37(*)	79.10	82.93(*)	79.10	80.37(*)	79.88	82.93(*)
F1-score	73.55	75.93(*)	69.89	70.44(*)	69.89	72.72(*)	69.89	75.19(*)	69.89	71.62(*)	70.94	75.19(*)
AD vs. CN	Precision	83.45	85.77(*)	71.24	72.80(*)	76.14	79.84(*)	65.04	80.62(*)	65.04	81.52(*)	71.24	74.37(*)
Accuracy	84.79	87.16(*)	81.50	85.32(*)	81.50	83.49(*)	80.59	83.52(*)	80.59	82.42(*)	84.40	86.24(*)
F1-score	82.02	83.69(*)	75.06	78.56(*)	74.17	81.07(*)	71.95	78.48(*)	71.95	76.13(*)	77.27	79.87(*)
In this work, we present the GNN-PDE-COV framework to re-think and re-design GNN models with great mathematical insight. On top of this, we devise the selective inductive bias to address the over-smoothing problem in GNN and develop new GNN model to predict the pathology flows in-vivo via longitudinal neuroimages. Future work may involve exploring innovative graph regularization techniques and conducting further validation on a broader range of graph-based learning tasks.
This work was supported by the National Institutes of Health AG070701, AG073927, AG068399, and Foundation of Hope. Data collection and sharing for this project was funded by the Alzheimer's Disease Neuroimaging Initiative (ADNI) (National Institutes of Health Grant U01 AG024904) and DOD ADNI (Department of Defense award number W81XWH-12-2-0012).
[1] Appleton, B., Talbot, H.: Globally minimal surfaces by continuous maximal flows. IEEE Transactions on Pattern Analysis and Machine intelligence 28(1), 106-118 (2005)
[2] Arnol'd, V.I.: Mathematical methods of classical mechanics, vol. 60. Springer Science & Business Media (2013)
[3] Bammer, R.: Basic principles of diffusion-weighted imaging. European Journal of Radiology 45(3), 169-184 (2003)
[4] Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM Journal on Imaging Sciences 2(1), 183-202 (2009)
[5] Bronstein, M.: Beyond message passing: a physics-inspired paradigm for graph neural networks. The Gradient (2022)
[6] Chamberlain, B., Rowbottom, J., Gorinova, M.I., Bronstein, M., Webb, S., Rossi, E.: Grand: Graph neural diffusion. In: International Conference on Machine Learning. pp. 1407-1418. PMLR (2021)
[7] Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. Journal of Mathematical Imaging and Vision 40(1), 120-145 (2011)
[8] Chan, T., Esedoglu, S., Park, F., Yip, A.: Total variation image restoration: Overview and recent developments. Handbook of Mathematical Models in Computer Vision pp. 17-31 (2006)
[9] Chen, M., Wei, Z., Huang, Z., Ding, B., Li, Y.: Simple and deep graph convolutional networks. In: International Conference on Machine Learning. pp. 1725-1735. PMLR (2020)
[10] Chen, R.T., Rubanova, Y., Bettencourt, J., Duvenaud, D.K.: Neural ordinary differential equations. Advances in Neural Information Processing Systems 31 (2018)
[11] Coifman, R.R., Lafon, S.: Diffusion maps. Applied and Computational Harmonic Analysis 21(1), 5-30 (2006)
[12] Destrieux, C., Fischl, B., Dale, A., Halgren, E.: Automatic parcellation of human cortical gyri and sulci using standard anatomical nomenclature. Neuroimage 53(1), 1-15 (2010)
[13] Dujardin, S., Fernandes, A., Bannon, R., Commins, C., De Los Santos, M., Kamath, T.V., Hayashi, M., Hyman, B.T.: Tau propagation is dependent on the genetic background of mouse strains. Brain Communications 4(2), fcac048 (2022)
[14] Eliasof, M., Haber, E., Treister, E.: Pde-gcn: novel architectures for graph neural networks motivated by partial differential equations. Advances in Neural Information Processing Systems 34, 3836-3849 (2021)
[15] Figueiredo, M.A., Bioucas-Dias, J.M., Nowak, R.D.: Majorization-minimization algorithms for wavelet-based image restoration. IEEE Transactions on Image processing 16(12), 2980-2991 (2007)
[16] Gao, H., Ji, S.: Graph u-nets. In: International Conference on Machine Learning. pp. 2083-2092. PMLR (2019)
[17] Glasser, M.F., Rilling, J.K.: Dti tractography of the human brain's language pathways. Cerebral Cortex 18(11), 2471-2482 (2008)
[18] Guo, Z., Zhang, Y., Teng, Z., Lu, W.: Densely connected graph convolutional networks for graph-to-sequence learning. Transactions of the Association for Computational Linguistics 7, 297-312 (2019)
[19] Hamilton, W., Ying, Z., Leskovec, J.: Inductive representation learning on large graphs. Advances in Neural Information Processing Systems 30 (2017)
[38] Xhonneux, L.P., Qu, M., Tang, J.: Continuous graph neural networks. In: International Conference on Machine Learning. pp. 10432-10441. PMLR (2020)
[39] Xu, K., Li, C., Tian, Y., Sonobe, T., Kawarabayashi, K.i., Jegelka, S.: Representation learning on graphs with jumping knowledge networks. In: International Conference on Machine Learning. pp. 5453-5462. PMLR (2018)
[40] Yang, Z., Cohen, W., Salakhudinov, R.: Revisiting semi-supervised learning with graph embeddings. In: International Conference on Machine Learning. pp. 40-48. PMLR (2016)
[41] Zhang, M., Chen, Y.: Link prediction based on graph neural networks. Advances in Neural Information Processing Systems 31 (2018)
[42] Zhang, Y., Gao, S., Pei, J., Huang, H.: Improving social network embedding via new second-order continuous graph neural networks. In: Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining. pp. 2515-2523 (2022)
[43] Zhao, J., Mathieu, M., LeCun, Y.: Energy-based generative adversarial network. International Conference on Learning Representations (2017)
[44] Zhu, M., Wright, S.J., Chan, T.F., et al.: Duality-based algorithms for total-variation-regularized image restoration. Computational Optimization and Applications 47(3), 377 (2010)