Title: Continuous Simplicial Neural Networks
Venue: NeurIPS 2025
Accepted: yes
Keywords: Simplicial Neural Networks

Partial Differential Equations

Over-smoothing

Stability
Abstract: Simplicial complexes provide a powerful framework for modeling higher-order interactions in structured data, making them particularly suitable for applications such as trajectory prediction and mesh processing. However, existing simplicial neural networks (SNNs), whether convolutional or attention-based, rely primarily on discrete filtering techniques, which can be restrictive. In contrast, partial differential equations (PDEs) on simplicial complexes offer a principled approach to capture continuous dynamics in such structures. In this work, we introduce continuous simplicial neural network (COSIMO), a novel SNN architecture derived from PDEs on simplicial complexes. We provide theoretical and experimental justifications of COSIMO's stability under simplicial perturbations. Furthermore, we investigate the over-smoothing phenomenon—a common issue in geometric deep learning—demonstrating that COSIMO offers better control over this effect than discrete SNNs. Our experiments on real-world datasets demonstrate that COSIMO achieves competitive performance compared to state-of-the-art SNNs in complex and noisy environments. The implementation codes are available in https://github.com/ArefEinizade2/COSIMO.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors develop a continuous-time formulation for training neural networks on simplical networks using discrete PDE theory. They show theoretically that the method prevents over-smoothing and is stable to perturbations to the simplex structure. The method is compared against a wide range of baselines and improves performance across multiple benchmarks.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- Making use of the simplical structure with learning based method is an important problem, and I am glad to see research in this direction.\n- The method outperforms previous state of the art SCCNN, while also reducing runtime and GPU memory\n\nWeaknesses:\n- The connection between theorems the theorems and practical implications are not always clearly explained (see questions)\n\nQUESTIONS:\n- Theorem 4.3: the proof does not seem to take into account that B_k is an integer matrix. This would mean ||E_k|| is always at least 1, which breaks the “eps_k is small” assumption in eq 19.\n- Theorem 4.3: it is not clear to me why being robust to perturbations of B_k is a useful property. Could you provide an example of a practical scenario where such perturbations might appear?\n- Section 6.3: The analysis of over-smoothing is limited to randomly initialized networks. It is therefore not clear how this phenomenon impacts the performance of trained networks. In principle, the network could learn to compensate for the initial over-smoothing during the training process. To complete the analysis, it would be interesting to repeat at least one of the experiments in section 6.1 for various settings ot t, to see if the “over-smoothed” variants actually have degraded performance.\n- what value of t was used for the experiments?\n- Is eq 8 is a definition or does it follow directly from eq 7. Please provide a proof/justification.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-27"}

comments: SUMMARY:
This paper presents a continuous simplicial neural network (SNN) architecture, COSIMO, derived from partial differential equations (PDEs) on simplicial complexes. Rather than relying on fixed, discrete polynomial filters of the Hodge Laplacians, COSIMO models feature propagation as coupled heat‐diffusion processes over the lower and upper Laplacians, yielding dynamic receptive fields that adapt continuously to the underlying topology. Each COSIMO layer applies exponential Hodge filters—computed via eigen‐decompositions—followed by learnable linear projections and nonlinearities. The authors prove that COSIMO is stable under simplicial perturbations and provides formal control over over-smoothing, and they demonstrate its effectiveness on trajectory prediction, partial mesh regression, and node/graph classification, where it outperforms state-of-the-art discrete SNNs and GNNs.

STRENGTHS AND WEAKNESSES:
### Strengths
- Cast simplical complexes into the PDEs framework and provide allows continuous information flow between structural data. 
- Bring theoretical analysis and proof to ensure the stability of the proposed work and explicit bound on over-smoothing rate.
- Different types of tasks demonstrates the effectiveness of this method
### Weakness
The eigenvalue decomposition may hinder the actual applicability of this method

QUESTIONS:
How do you choose the number of eigenpairs K for each Hodge Laplacian in practice? Have you evaluated accuracy versus cost trade‐offs as K varies?

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 4
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-01

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe work proposes a continuous simplicial neural networks (COSIMO). They use simplicial complexes to capture high-order geometrical relationships for a more sophisticated understanding of geometrical objects. In addition, they utilize the theory of differential equations to obtain the continuous formulation of the method to control the oversmoothing effect, which is a central problem in GNNs. They provided a mathematical evaluation of robustness against geometrical noise and oversmoothing. The experiments validate the mathematical theory and demonstrate that the proposed method has a competitive performance against SotA methods.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n* The method is clearly explained and easy to follow.\n* The authors provided the mathematical guarantees for stability and oversmoothing, which is essential to the community.\n\nWeaknesses\n* The strength to use higher-order relationships is not clear. In addition, the ideas for continuous evolution are not new. For instance, the community knows Neural ODE [Chen+ 2018] and GRAND [Chamberlain+ 2021]. Therefore, the authors should compare at least GRAND or its variants to show the effectiveness of higher-order relationship quantitatively.\n* Evaluation of the tradeoff between oversmoothing and accuracy is not evaluated. It turned out that reducing the t (time) parameter in COSIMO suppresses oversmoothing, but it limits the lengths of interaction because it corresponds to short-time diffusion. Therefore, the reviewer recommends adding an evaluation of that tradeoff. Otherwise, if we do not care about the accuracy, it would be sufficient to stack only one GNN layer to suppress oversmoothing.\n* Stability analysis (Section 6.3) needs comparison with other methods, because the readers cannot judge the robustness of the proposed method in the current form.\n\nQUESTIONS:\nWhat would be potential applications where higher-order relationships are essential?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces COSIMO, a novel simplicial neural network rooted in Topological Deep Learning (TDL), designed to process data on simplicial complexes—mathematical structures that extend beyond graphs to model complex, higher-order relationships like nodes, edges, triangles, and beyond. TDL, as highlighted in the paper, leverages these topological structures to capture multi-way interactions, making it ideal for applications such as trajectory prediction and mesh processing. COSIMO advances TDL by shifting from the discrete, layer-based processing of traditional simplicial neural networks to a continuous framework, using partial differential equations based on Hodge Laplacians to model signal evolution across simplicial dimensions over time. This continuous approach enables COSIMO to dynamically handle interactions between lower-dimensional (e.g., edges) and higher-dimensional (e.g., triangles) structures, a core strength of TDL’s topological perspective.\n\nCOSIMO’s TDL framework incorporates independent diffusion processes for lower and upper simplicial structures and a coupled process to integrate them, enhancing expressivity for complex relational data. The paper provides rigorous theoretical analysis, proving COSIMO’s stability under simplicial perturbations and its ability to control over-smoothing—a challenge in TDL where deep models lose discriminative power—through adjustable receptive fields. Experimentally, COSIMO is evaluated on TDL-relevant tasks like ocean drift prediction, partial shape regression on the Shrec-16 dataset, and node/graph classification on social networks, showing superior performance compared to existing TDL methods, with balanced runtime and memory usage. The authors also explore parameter impacts, such as the number of branches, and confirm permutation equivariance, ensuring alignment with TDL’s emphasis on structural symmetries.\n\nSTRENGTHS AND WEAKNESSES:\nQuality\nThis paper’s got some solid stuff going for it. The theoretical backbone is impressive—they’ve worked out proofs (like Proposition 4.1 and Theorem 5.3 in the appendices) that show COSIMO is stable under simplicial perturbations and can handle over-smoothing, leaning on math like Hodge Laplacians. It’s the kind of rigor that makes you feel they’ve thought this through. Experimentally, they test COSIMO on tasks like ocean drift prediction and shape regression on the Shrec-16 dataset, and Figure 5 shows it performs well against other TDL models while keeping runtime and memory in check. I like that they’ve shared their code and datasets, which is a big plus for reproducibility, especially since you’re into digging into results. Section 4.3’s breakdown of computational complexity, using eigenvalue decomposition for Hodge filters, shows they’re thinking about practical implementation, too.\n\nOn the other side, the experimental details could use more meat. They talk about setups in Section 6 and appendices, but stuff like exact hyperparameters or how they chose the optimizer is a bit fuzzy, which might make it tricky to replicate. The datasets are decent for TDL, but it’s mostly Shrec-16 and ocean drift—adding something like a biological network could show COSIMO’s got broader chops. Also, the eigenvalue decomposition they rely on can get heavy for big simplicial complexes. They mention this, but I wish they’d tossed in a few more ideas for making it less resource-hungry.\n\nClarity\nThe paper’s laid out nicely, with a clear path from the intro to preliminaries (Section 3), model details, and experiments. For someone familiar with TDL or graph neural networks, it’s pretty approachable—they explain simplicial complexes and Hodge Laplacians well enough, and Figure 1’s visual of a simplicial complex helps a lot. The theoretical bits, like how they tackle over-smoothing (Figures 3 and 6), are straightforward, with equations like the PDE-based signal evolution in Figure 2 giving a good sense of the architecture, which I know you care about. Figures like Figure 7, showing how tweaking the branch parameter affects results, make the practical side easy to grasp.\n\nThat said, the math-heavy parts can feel a bit thick if you’re not already familiar with TDL or algebraic topology. Terms like “Hodge Laplacians” or “permutation equivariance” get some context, but a simple example or two could make them less daunting for a wider audience, maybe not in the main text. Also, a lot of the good content, like proofs and dataset details, is stashed in the appendices, which is standard but leaves the main text a tad thin on quick takeaways. I had to switch back and forth more than I’d prefer. And while they show how parameters like the receptive field or branch count impact performance, they don’t give much guidance on picking those for new TDL problems, which could leave practitioners scratching their heads.\n\nSignificance\nCOSIMO feels like a meaningful step for TDL. Its continuous, PDE-based approach shakes up the usual discrete SNNs, making it easier to model dynamic interactions in simplicial complexes, which is super relevant as TDL gains traction. The applications they mention in Appendix O—like ocean navigation, social network analysis, and protein classification—hint at how TDL can tackle real-world challenges, which is exciting. Their focus on over-smoothing (Section 5) remains a standing problem in TDL and graph models, and they back it up with theory and results. \n\nBut I wonder if some might see it as a solid increment. It builds on existing ideas like graph neural diffusion and simplicial filters, so it’s not reinventing the wheel. The limitations section, covering things like computational costs or over-squashing, is there but feels a bit tacked on in the appendices. A deeper look at when COSIMO might not work well, like on sparse complexes, would make its impact clearer. Also, its focus on simplicial complexes might make it a niche player for the broader machine learning crowd, which could limit its splash outside TDL circles.\n\n Originality\nThe continuous framework is where COSIMO stands out. Using PDEs to model signal evolution on simplicial complexes is a fresh angle for TDL, moving past the fixed filters of traditional SNNs. Their use of Hodge Laplacians in a neural network setup is pretty creative, pulling from topological signal processing [19] but giving it a deep learning twist. The permutation equivariance proof in Appendix J is a nice touch, too, ensuring the model plays nicely with the symmetries of simplicial complexes, which isn’t always covered in TDL papers.\n\nThat said, it’s not entirely out of left field. COSIMO leans on concepts from TDL and graph neural networks, like diffusion models  and over-smoothing work , so it’s more of a smart remix than a totally new idea. The tasks they test—trajectory prediction, shape regression—are standard TDL fare, so there’s no big surprise in the applications. A novel TDL use case could’ve pushed the originality further. Also, the overlap with graph neural diffusion might make some people wonder how much is truly new versus adapting those ideas to simplicial complexes.\n\nQUESTIONS:\n1. Provide Detailed Experimental Specifications\nQuestion/Suggestion: Could the authors provide a comprehensive description of the experimental setup, including specific hyperparameter values (e.g., learning rate, optimizer type, batch size), data splits, and training protocols, either in the main text or a dedicated appendix section? The current details in Section 6 and appendices (e.g., Appendix K) are incomplete, lacking specifics on how hyperparameters were chosen or tuned, which hinders reproducibility. I think since this one of the first papers : TDL+PDE, this will be very important for reproducibility. \n\nGuidance for Action: Include a table or subsection listing all hyperparameters for each dataset (e.g., Shrec-16, ocean drift), specifying values for learning rate, optimizer (e.g., Adam, SGD), batch size, number of epochs, and any regularization techniques. Clarify the data split ratios (e.g., train/validation/test) and describe the hyperparameter tuning process (e.g., grid search, random search). If space is limited, a supplemental table or a pointer to the open-access code with these details would suffice.\n\n\n2. Clarify Distinction from Graph Neural Diffusion Models\nQuestion/Suggestion: The paper draws parallels with graph neural diffusion models [16, 37], but the unique contributions of COSIMO’s continuous PDE-based framework on simplicial complexes are not fully delineated. Could the authors explicitly clarify how COSIMO differs from or improves upon graph neural diffusion approaches, particularly in terms of architecture and theoretical advantages for TDL?\n\nGuidance for Action: Add a subsection in Section 4 or the introduction comparing COSIMO’s PDE-based architecture (e.g., use of Hodge Laplacians for lower, upper, and coupled dynamics) to graph neural diffusion models (e.g., GRAND [37]). Highlight specific architectural differences (e.g., handling higher-order simplices vs. graph nodes/edges) and theoretical benefits (e.g., stability or expressivity for TDL tasks). A table contrasting key features (e.g., input structure, dynamics, computational complexity) or a paragraph discussing how COSIMO extends diffusion to simplicial complexes would be effective.\n\n\n 3. Expand Discussion of Limitations and Failure Cases\nQuestion/Suggestion: The discussion of limitations (e.g., computational complexity, over-squashing in Appendix O) is brief and mostly relegated to appendices. Could the authors expand this in the main text, detailing specific scenarios where COSIMO might underperform (e.g., sparse or noisy simplicial complexes) and proposing potential mitigations?\n\nGuidance for Action: In the conclusion or a dedicated subsection, discuss at least two specific failure cases (e.g., performance on sparse simplicial complexes with few higher-order simplices, or sensitivity to noisy input signals). For each, explain why COSIMO might struggle (e.g., due to reliance on dense Hodge Laplacian computations) and suggest mitigations (e.g., sparse matrix techniques, noise-robust regularization). Reference empirical or theoretical evidence, such as sensitivity analyses (e.g., Figure 7) or related TDL work [54], to ground the discussion. Aim for 1–2 paragraphs in the main text, with additional details in appendices if needed.\n\n\n 4. Provide Practical Guidance for Parameter Selection\nQuestion/Suggestion: The paper shows how parameters like the receptive field (tau) and branch count (M) impact performance (e.g., Figure 7), but offers little guidance on selecting these for new TDL tasks. Could the authors provide practical recommendations for choosing these parameters, supported by empirical or theoretical insights?\n\nGuidance for Action: Add a paragraph or subsection in Section 5 or the experiments (Section 6) outlining a strategy for selecting tau and M. For example, suggest starting values based on dataset characteristics (e.g., simplicial complex size, density of higher-order simplices) and adjusting based on validation performance. Reference Figure 7’s results (e.g., M=3 as a sweet spot for ocean drift) and theoretical insights (e.g., tau’s role in controlling over-smoothing from Theorem 5.3). If possible, provide a heuristic or algorithm (e.g., cross-validation over a range of tau values) and test it on one dataset to demonstrate efficacy.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-05"}