Title: Uncertainty Estimation on Graphs with Structure Informed Stochastic Partial Differential Equations
Venue: NeurIPS 2025
Accepted: yes
Keywords: Graph Neural Network

Uncertainty estimation

Stochastic Differential Equations
Abstract: Graph Neural Networks (GNNs) have achieved impressive results across diverse network modeling tasks, but accurately estimating uncertainty on graphs remains difficult—especially under  distributional shifts. Unlike traditional uncertainty estimation, graph-based uncertainty must account for randomness arising from both the graph’s structure and its label distribution, which adds complexity. In this paper, making an analogy between the evolution of a stochastic partial differential equation (SPDE) driven by Mat\'ern Gaussian Process and message passing using GNN layers, we present a principled way to design a novel message passing scheme that incorporates spatial-temporal noises motivated by the Gaussian Process approach to SPDE. Our method simultaneously captures uncertainty across space and time and allows explicit control over the covariance kernel’s smoothness, thereby enhancing uncertainty estimates on graphs with both low and high label informativeness. Our extensive experiments on Out-of-Distribution (OOD) detection on graph datasets with varying label informativeness demonstrate the soundness and superiority of our model to existing approaches.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper uses a Graph Stochastic Partial Differential Equation to model uncertainty for node classification. The model can represent uncertainty through the stochasticity of a Wiener Process component guided by a Matern Graph Gaussian Process. It consistently outperforms previous models both on homophilic and heterophilic graphs in terms of out-of-distribution detection.\n\nSTRENGTHS AND WEAKNESSES:\n## Strenghts\n1. The approach is clearly motivated as the GSDN [1] model it builds upon assumes homophily also in the uncertainty by using Laplacian-based propagation to guide the uncertainty. Since heterophilic graphs violate this assumption, a learnable diffusion process is needed.\n2. The empirical evaluation of the paper is thorough and methodically sound and shows a clear and consistent improvement over previous work.\n3. The paper provides an insightful intuition for the learned covariance that can be interpreted as similar to rewiring the graph to connect nodes with high mutual information in their label distribution.\n\n## Weaknesses\n### Major\n1. While well-motivated, the approach in this paper is very similar to GSDN. As far as I understand, it only extends on GSDN by introducing pointwise learnable transformations of the eigenvalues of the Graph Laplacian.\n2. The paper is sometimes a bit difficult to follow, especially if the reader is not super familiar with Partial Stochastic Differential Equations. My understanding of the paper's flow is: 1) It motivates a spatial kernel different from the Laplacian due to possible high frequencies in the epistemic uncertainty. 2) It introduces pointwise transformations to the graph spectrum and shows that the resulting process is a Wiener process. 2b) This also holds when applying a Matern kernel using the transformed Laplacian. 3) The proposed model is the solution to the resulting GSPDE, realized through forcing. \n- I found it difficult to follow the red thread that clearly motivates each of these components. For example, in Theorem 3, where exactly does the pointwise transformation of $\\Delta$'s spectrum manifest? What is different from a version where no Matern Kernel (with transformed spectrum?) is used?\n- Later on (Eq. 11-13), it seems to me that there is no pointwise transformation of the graph's spectrum anymore and only the Matern kernel (i.e. where is $\\Phi$?). Can the authors clarify that?\n- Figure 1 shows the Matern kernel (which is used in this work but not a novel contribution) on a synthetic heterophilic graph. What is the takeaway from this figure in terms of the proposed model? Is this to motivate a different diffusion kernel than just the standard Laplacian? If so, how does this motivate the learnable transformation of the graph's spectrum?\n\n### Minor\n1. There are some grammatical errors and inconsistencies in the paper (e.g. whether references to equations, figures, etc. are capitalized). I recommend using a grammar-checking tool to fix those.\n2. The most recent baseline for UQ in node classification, GEBM [2], is cited but not compared against. Can the authors extend their evaluation to this baseline?\n3. Notation like the gradient $\\nabla$ and its adjoint $div$ could be briefly introduced for the graph domain.\n\n## References:\n\n[1]: Lin, Xixun, et al. \"Graph neural stochastic diffusion for estimating uncertainty in node classification.\" Forty-first International Conference on Machine Learning. 2024.\n[2]: Fuchsgruber, Dominik, Tom Wollschläger, and Stephan Günnemann. \"Energy-based Epistemic Uncertainty for Graph Neural Networks.\" The Thirty-eighth Annual Conference on Neural Information Processing Systems.\n\nQUESTIONS:\n1. Is $\\nu$ a learnable parameter? If not, how is it tuned?\n2. Why is a $\\nu$ needed to capture \"long-range\" interactions if the pointwise transformation of the graph's spectrum is already enabled through $\\Psi$?\n3. Is the term \"long-range\" interaction justified? From my understanding, what happens is that the kernel promotes high frequency signals (in terms of the graph spectrum), which need not be long-range per se. Would \"high-frequency\" patterns be a more suitable term?\n4. For the leave-out-class setting, the authors also consider some datasets with only two classes. Did the authors train the model only in one class in this setting? I find it very surprising that the model learns anything sensible in this case, as just predicting this one class for any input should minimize the training loss.\n5. How is the LI computed? Eq. 1 suggests that one needs access to $p(y_i | y_j)$, but this distribution is unknown as the data-generating process is unknown. Is this distribution estimated as MLE from the entire graph?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-02"}

comments: SUMMARY:
The authors propose to model message passing in a graph as a stochastic partial differential equation (SPDE) with the flexible covariance structure of the Brownian noise (here realized by a Matérn kernel). This allows for better control over defining the spatial correlation pattern that suits the graph structure, improving the uncertainty estimation and OOD detection performance.

STRENGTHS AND WEAKNESSES:
**Strength:**
- The idea is well-motivated and theoretically sound.
- Proposed method outperforms state-of-the-art uncertainty estimation baselines in graphs.

**Weakness:**
- No significant weakness
- To better gain insights in the results presented in Figure 2(b), it would be useful to see the some kind of LI distribution of nodes in the graphs having high LI (e.g Citeseer), medium LI (Pubmed) and low LI(Questions).

QUESTIONS:
- High correlation between nodes (low $\nu$) comes with high node variance. In such case, what is the strategy to balance those? And how careful should we be to choose the OOD uncertainty threshold in cases with low $\nu$ ?

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-07

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a novel Structure Informed Stochastic Partial Differential Equation (SISPDE) framework for uncertainty estimation on graphs. The key innovation lies in modeling spatially correlated noise via a Matérn-Gaussian-Process-driven SPDE, which generalizes prior Q-Wiener processes by explicitly controlling covariance smoothness through parameter ν. This enables superior uncertainty quantification under both high and low label informativeness (LI) scenarios. Theoretical contributions include proofs for the Φ-Wiener process properties (Theorems 1–2), existence/uniqueness of SPDE solutions (Theorem 3), and Chebyshev approximation bounds (Theorem 4). Experiments on 8 graph datasets demonstrate state-of-the-art OOD detection, especially on low-LI graphs where traditional GNNs struggle.\n\nSTRENGTHS AND WEAKNESSES:\nQuality \n\n​​Strength:​​ Rigorous theoretical grounding with 4 theorems and appendices. Extensive experiments (8 datasets, 3 OOD types, 5 seeds) validate claims.\n\n​​Strength:​​ Matérn kernel’s ν adapts to LI – low ν captures long-range dependencies in heterophilic graphs (Figure 2b), addressing a critical gap.\n​​\n\nWeakness:​​ Computational cost of Cholesky decomposition for Matérn kernel not deeply analyzed (runtime in Fig 4 but scalability limits unclear).\n\n​​Clarity \n\n\n​​Strength:​​ Well-structured with clear analogies between SPDEs and GNN message passing (Section 2.3–2.4).\n​\n\n​Weakness:​​ Appendix-heavy proofs may disrupt flow; some notation (e.g., Φ-Wiener process in Def 2) requires multiple reads.\n\n\nSignificance \n\n\n​​Strength:​​ Unifies stochastic processes (GNSD), graph kernels (Matérn GP), and PDE-based GNNs (GRAND) into a flexible framework.\n​​\n\nStrength:​​ Achieves SOTA on low-LI graphs (Table 1), where uncertainty estimation is most challenging (Section 4.2).\n\n\n\nOriginality \n\n\n​​Strength:​​ First to integrate SPDE-driven spatially correlated noise with tunable smoothness for graph uncertainty. The physics-inspired approach (Figure 1b) is novel.\n\nQUESTIONS:\n1)​​Kernel Generalization:​​ Why not explore non-Matérn kernels (e.g., RBF)? If ν is pivotal for LI adaptation (Fig 2b), does this limit applicability to non-Matérn covariances?\n\n​​2)Smoothness vs. LI:​​ Can you establish a quantitative relationship between optimal ν and dataset LI (e.g., via regression)? Fig 2b implies heuristic tuning.\n\n​​3)Scalability:​​ For large graphs (e.g., >10^6 nodes), does Chebyshev approximation (Theorem 4) suffice? Have you tested scalability vs. graph size?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a new method of uncertainty estimation on graphs by making an analogy between the evolution of SPDE and message passing. Through introducing the noise process, the proposed method enables GNNs to quantify uncertainty and to achieve detection tasks on graphs. The core innovation is proposing a new noise form based on graph Gaussian Process to learn the spatial-temporal covariance of noise. Experimental results demonstrate the effectiveness of the proposed methods.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n1. Exploring the integration of SPDEs with GNNs is a highly interesting and promising direction. The authors propose a novel form of SPDE and provide a theoretical analysis of it.\n2. The authors provide a detailed descriptions of graph SPDE and related works, which clarifies the logical flow of the paper and enhances its readability.\n\nWeaknesses\n1. The novelty of the proposed model appears to be limited. The combination of SPDEs and GNNs has already been explored in prior works [1,2], and this paper mainly makes adjustments to the noise term. Moreover, the dependence of the noise can also be learned through the stochastic forcing term, rather than simply adding independent noise. Therefore, the authors need to place greater emphasis on this aspect.\n2. The computation of graph Gaussian processes is typically very complex. Therefore, the authors should clarify the computational complexity of the proposed model and compare its efficiency with that of other methods.\n3. The connection between label informativeness and the proposed method in this paper is not clearly emphasized. It is necessary to further highlight the relationship between label informativeness and the proposed approach.\n\n[1] Graph Neural Stochastic Diffusion for Estimating Uncertainty in Node Classification. Xixun Lin et al. ICML 2024.\n\n[2] Uncertainty Modeling in Graph Neural Networks via Stochastic Differential Equations. Richard Bergna et al. ICLR 2025.\n\nQUESTIONS:\n1. The paper mentions uncertainty estimation but does not explain how the two types of uncertainty are accurately learned after formulating the modeling equation. Therefore, how are epistemic uncertainty and aleatoric uncertainty actually learned?\n\n2. Besides OOD detection, does uncertainty estimation serve any other purposes?\n\nLIMITATIONS:\nNA", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}