Continuous Simplicial Neural Networks
Authors: Aref Einizade
Dorina Thanou
Fragkiskos D. Malliaros
Jhony H. Giraldo
Venue: NeurIPS 2025 poster ()
Keywords: Simplicial Neural Networks
Partial Differential Equations
Over-smoothing
Stability
Primary area: general_machine_learning

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.

Continuous Simplicial Neural Networks
Aref Einizade
Dorina Thanou
LTCI, Télécom Paris
EPFL, Lausanne, Switzerland dorina. thanou@epfl ch
Institut Polytechnique de Paris
Fragkiskos D. Malliaros CentraleSupélec, Inria
Jhony H. Giraldo LTCI, Télécom Paris Institut Polytechnique de Paris hoanrgggraaldooteeeccmmmpppep
Université Paris Saclay
fragki skos hwt.:a/aaaaareotttrrleeee.......
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) simplicial complexes offer 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 's stability under simplicial perturbations. Furthermore, we investigate the over- swoorssoothing phenomenon common issue geometric deep deemning-demonsstatiingg that COSIMO offers better control this effect than discrete SNNs. Our experiments on real-world datasets demonstrate that COSIMO achieves competitive performance compared state-of-the-art SNNs in complex and noisy environments. The implementation codes are available in http:://ghhhbbbooom////E///222Sf
Introduction
Graph representation learning provides a powerful framework for modeling structured data. In this context, graph neural networks (GNNs) have gained significant attention I extending neural network architectures to graph-structured data. By capturing complex relationships between nodes, GNNs have been successfully applied various domains, including computationalldiiiaall pathology 5). social network analysis [6], drug discovery [7]. materials modeling [8], and computer vision 9 However, traditional GNNs primarily focus on pairwise interactions between nodes, limiting their ability to model ordeer-order relationships in complex systems such as biological networks [11]. To overcome this limitation, researchers have explored more expressive mathematical structures, such as abstract simplicial complexes [12), generalizing graphs by incorporating multi-way connections and their Laplacians by introducing the Hodge decomposition theory [3]
An abstract simplicial complex is combinatorial structure composed of sets that are closed under subset operations. For instance, dinee-dimensional simplicial complex includes tetrahedrons (four- element sets), triangles (three-element sets), edges (two-clement sets), and vertices (one-clement sets), as illustrated Fig. [ Graphs correspond simplicial 1-complexes, containing only nodes and edges, while point clouds (sets of unconnected nodes) can be seen as simplicial O-complexes.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Building upon the mathematical foundation of abstract simpli- cial complexes, simplicial neural networks (SNNs) [4] have emerged as a powerful approach for learning on higher-order structures. Most existing SNNs rely on discrete simplicial fil-
Trinngl
Edge
Vertex
ters [4 15 and their variations [16 17 [8] to process data
Termhedron
defined on simplicial complexes. However, a fundamental ques- tion remains largely unexplored: how can we design continuous SNNs? Continuous models play a crucial role in capturing real- world dynamics evolving structured data [9]. Compared to their discrete counterparts, continuous convolutional models offer several advantages, including: i) better control of over-
Figure 1: Example of an abstract simplicial complex.
smoothing [9], preventing excessive feature homogenization across the structure, and ii) greater robustness to structural perturbations 20 21] Despite these benefits, the best of our knowledge, no continuous filtering method has been proposed for learning on simplicial complexes. To address this gap, we introduce the continuous simplicial neural network (COSIMO) model, the first method for modeling and learning over the continuous dynamics of simplicial complexes with higher-order connections. We analyze COSIMO both theoretically and empirically, demonstrating its effectiveness in learning from simplicial complex data. Our main contributions can be summarized as follows:
We introduce COSIMO, new approach that uses partial differential equations (PDEs) simplicial complexes to enable continuous information flow through these structures.
We establish theoretical stability guarantees for COSIMO, demonstrating its robustness to struc- tural perturbations,
We provide a detailed analysis of over-smoothing both discrete and continuous SNNs, showing that COSIMO achieves better control on the rate of convergence to the over-smoothing state.
We validate COSIMO through experiments on both synthetic and real-world datasets, showing its competitive performance against state-of-the-art (SOTA) methods.
Related Work
The introduction of topological signal processing over simplicial complexes [22 13 has significantly advanced lopological methods in machine learning. highlighting the benefits of these data structures [1], and contributing to the emerging field of topological deep learning [23 24] The evolution of
the SNN architectures has followed a trajectory similar to that of GNNs, with the following stages: i) the establishment of principles in topological signal processing over simplicial complexes (22), if)
the formulation of simplicial filters [12, and iii) the application these filters to create SNNs 14].
Research on learning methods for simplicial complexes has explored various approaches. Rodden- berry and Segarra (25) were among the first to develop neural networks that operate on the graph edges this, Ebli et al. [4 incorporated the Hodge Laplacian, simplices. Other studies [26] 17 refined this approach by separating the lower and upper Laplacians, two components of the Hodge Laplacian that capture connections between simplices of different dimensions. Keros er al. (27) further extended the framework in (26] detect topological holes, while Chen al. [28] integrated node and edge operations for link prediction. More recently, attention mechanisms have been incorporated into simplicial networks (29 30].
using simplicial representations. Building
a generalization of the graph Laplacian, to extend GNNs to higher-dimension
Most of these studies focus on learning within individual simplicial levels without explicitly modeling the incidence relations (interactions between simplices of different dimensions) inherent in simplicial
complexes [18]. The inclusion of these relations was explored in [18, 17 B1I. which proposed convolutional-lik. architectures that were later unified under the simplicial complex convolutional neural network (SCCNN) framework [1]]. Simultaneously, other studies extended message-passing 32 to simplicial complexes by leveraging both adjacency and incidence structures
from GNNs
16 33].
Unlike GNNs, the theoretical understanding of SNNs is still developing. For example, Roddenberry et al. (26) analyzed how permutation-symmerrt neural networks preserve equivariance under permutation and orientation changes- important property also supported by SCCNNs [18]. Another study examined message-passing in simplicial complexes through the lens of the Weisfeiler- Lehman test, applied to simplicial complexes derived from clique expansions of graphs [16]. A

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spectral formulation based on the simplicial complex Fourier transform was also proposed in [ Other critical yet underexplored theoretical aspects of SNNs, including their stability and over- smoothing behaviors, have been analyzed only in the context of SCCNNs [18]. For example, Yang et al. [88] analyzed the steady state solution of the diffusion PDE on simplicial complexes, showing that SCCNNs are more robust to over-smoothing compared to their non-Hodge-awand counterparts.
In contrast to previous methods, COSIMO leverages continuous dynamies in both the lower and upper Hodge Laplacians. Although the discrete SCCNN also decouples the Hodge Laplacians, it faces two challenges: i) the Hodge filters' order must be manually tuned, and ii) the model has limited control over over-smoot,ing, a common issue in deep GNNs and SNNs. COSIMO addresses these challenges by introducing lower and upper Hodge-aware PDEs on simplicial complexes. enabling differentiability of the convolutional operation w.K.f the simplicial receptive fields, providing greater flexibility and robustness in learning.
3 Preliminaries
3.1 Notation and Simplicial Complexes
Notation. Calligraphic letters like X' denote sets with X being their cardinality. Uppercase boldface letters such as B represent matrices, and lowercase boldface letters like x denote vectors. Similarly, tr(-) represents the trace of a matrix, || is the f--norm of vector, and (.) designates transposition.
Simplicial complex. A simplicial complex a set of finite subsets of another set V that is closed under restriction, i.e., V s E X, if s S then sk E X. Each element of X' is called a simplex. Particularly, if |s*] = k I, we call s a k-simplex. A face of s* is subset with cardinality k, while a coface of * is a k + 1-simplex that has s as a face [22]. We refer to the 0-simplices as nodes, the 1-simplices as edges, and the 2-simplices as triangles. For higher-order simplices, we use the term k-simplices. The notation N's represents the collection of k-simplices of X. If X. = 0 V > d, we say X is a simplicial complex of dimension d. For example, simple graph is a simplicial complex of dimension one and can be represented as G = (No, X1), i.e., the set of nodes and edges.
We use incidence matrices BN E 11 11- to describe the incidence relationships between k 1-simplices (faces) and k-simplices. For example, B₁ and B2 are node-to-edge and bllge-to-triangeee incidence matrices, respectively. Simplicial complexes are defined with some orientation, and therefore the value (i.i. is either -1 or if the simplee is incident to the k- k- 1-simplex j. depending on the orientation, and O otherwise. Please notice that Bo is not defined. We define the k-Hodge Laplacians as L; B, B. + B I where Lg,d B, B, is the lower Laplacian. Lk. = B B k++ is the upper Laplacian, Lo = B, is the graph Laplacian. Discrete SNNs [18 17 define their convolution operations matrix polynomials of the Hodge Laplacians over simplicial signals, i.e., signals defined over the simplicial complex.
Simplicial signal. We define a k-simplicial signal as function in Xx as 2h Xx + R. Therefore, we can define one-dimensional k-simplicial signal as X2 E IR Kel, We can calculate how X5. varies w.E.f the faces and cofaces of k-simplices by B and B 1 [18]. For example, in node signal X
Xo, B, Xo computes its gradient as the difference between adjacent nodes, and in an edge signal x1, B computes its divergence 18]
Dirichlet energy. The Dirichlet energy E(:) quantifies the smoothness of simplicial signal with respect to the Hodge Laplacian (18), where lower energy value indicates a smoother signal.
Definition 3.1 (from [8).. The Dirichlet energy of simplicial signal x₃ can be stated as:
E(x:) :- x L ||Baxx + |
This definition generalizes the Dirichlet energy from graphs to simplicial complexes, measuring how similar the values assigned to adjacent simplices are, with higher energy indicating larger variation.
Simplicial filters. For k-simplicial signal xx. simplicial filter is function f: RRl - RRAA as:
Ta Σ
To
f(x)) =
L
or
+
3
Lk
Xk:
1=0
3

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where {av..... or, J and {Bu..... ................................ , } are the parameters of the polynomials, and Ta, Tu are the order of the polynomials [3]. Please notice that the well-known graph filter [35] is specific case of (2). In this case, the graph signal is given by Xo 12/a and since Bo is not defined, we have f(xo) E B,L, i.e, the classical graph filter with the graph Laplacian as the shift operator.
Pr.
3.2 Discrete Simplicial Neural Network
Analogous to a convolutional neural network, an SNN can be defined for simplicial complexes using simplicial filters [17]. However, notice that relying only on filters like in (2) ignores the connections among the adjacent simplices modeled by L,- Since different simplicial signals influence each other via the simplicial complex localities, previous works have defined simplicial filter banks [34.. The following discrete Hodge-aware filtering model adapted from SCCNNs (18).
One-dimensional case. Let the lower and upper projections of a simplicial signal x3 at layer / be xk,d B x E | and xkgu B2+1XE+1 E Rl respectively We define a simplicial
layer (with parameters 0 and ub [8] function g Re x R Rill IR
RE R given as:
x H + H + H H
where Hidd = Eimo of Kad Hin = Eino K and H E.= V/ -
En ahai +
Multi-dimensional case. Let (x R,d ( be the dimeensnani simplicial signal and its
lower and upper projections at layer 1. Let O k,d,s O V k.d. and V u. a be learnable linear projections in IR Fi- corresponding to the a and á parameters for the unidimensional case in Using 2 and 3
we can define an SNN layer for the multidimensional case follows [8]:
X - ETE To L. X Oha
Ele X W.ids
ET=
L kuu.
The discrete SNN in is analogous to the GNN case, where discrete powers of the Hodge Laplacians capture multi-hop diffusions in the simplicial signal and its lower and upper projections.
All the proofs of theorems, propositions, and lemmas of the paper are provided in Appendices
4 Continuous Simplicial Neural Network
Discrete SNNs provide flexibility in filtering simplicial signals through lower and upper projections. However, their information propagation remains fixed for each polynomial order, limiting adaptability. In this section, we introduce COSIMO, which enables a dynamic receptive field in each convolutional operation. We begin by formulating the PDEs that govern physics-ifformed dynamics over simplicial complexes. Next, we define the fundamental operations of COSIMO as the solutions to these PDEs. Finally, we provide a rigorous stability analysis of COSIMO, showing its robustness to topological perturbations in simplicial complexes.
4.1 PDEs in Simplicial Complexes
Our set of PDEs is inspired by heat diffusion on simplicial complexes, providing a natural extension of discrete SNNs. Intuitively, performing heat diffusion over the decoupled Hodge Laplacians enables information propagation at different rates within the continuous domain of the simplicial complex. This parallels the case in graphs 19). where continuous GNN formulations have been shown to generalize certain discrete GNNs 36). opening new possibilities for architectural design.
In our framework, considering both joint diffusion on *² and independent diffusion on k and A+1 allows for greater flexibility in modeling complex relationships. By enabling these dynamics to evolve at different rates, we can better adapt to the underlying topology. Motivated by these considerations, we model simplicial heat diffusion using system of PDEs on the Hodge Laplacians. Let tu and t,
In this work, the superseript refers to the layer index and should not be confused with exponentiation.

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Simplicial Complex
X2 R
X1,
X2
Desulto) -
Upper
Lower
-L. (ita)
Joint Dynamics
de 102
Rafe)
X1
1 (1a 00)
(00,
B,
x1(l:t)) = x1 (ts) + ((.) (t.) (1.)
Integrated Dynamics
La tu +
The PDE-based signal evolution on a simplicial complex, governed by independent diffusion processes the lower and upper Hodge Laplacians and coupled process integrating both spaces. The colors in the simplicial complexes represent the values the underlying simplicial signals.
Figure 2:
be the time variables governing the dynamics in the lower and upper Laplacians, respectively, We define these dynamics through the following system PDEs:
Independent lower dynamics: The signal evolution in the lower space follows a heat diffusion process:
ōxa,a(ta) atta
(5)
Ota
Independent upper dynamics: Similarly, the signal in the upper space evolves according to:
@xk,u(lu) L
(6)
Gt
Joint dynamics: The interaction between the lower and upper spaces is captured by:
TxL (ta: tw)
(x,
LK. (td, oo) - LL, tw),
Ita
Ol.,
where x (ta. o0) lim,, moo Xx (ta, tn) and x (oo, t.) lim Xx. (tas tw) state marginal stable solutions in upper and lower subspaces.
Integrated dynamics: The final solution by integration of the independent and joint dynamics is as:
(ta, tw) (ta)(ta) + (ta) + x (f.) + x (t..)
(8)
These dynamics describe the information flow across different simplicial levels, ensuring a principled integration of independent and coupled dynamics. Refer to Fig. for visual representation on s?.
Remark 4.1. With la l,, the joint dynamic PDE in 7 turns into (xx(t) L (1)
Lx, (t), and its steady state solution. L = O, lies in the kernel space of L [8.. justifying the need for independent lower and upper Hodge-aware PDEs.
4.2 COSIMO as a Solution the Simplicial PDEs
We propose COSIMO as solution to the descriptive sets of PDEs introduced in Section 4.1

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Proposition 4.2, The solution to the descriptive sets of PDEs in Section 4.1 is given by:
(ka)
(1.)
X4 (ta.fu)
(9)
XR,u (ha, lu) fabe, xx a (0)) + tubk Xx, (0)
tabk x (0, 0) e tub. (0,0,));
where (0). X (0), and X (0. 0) are the initial conditions for the PDEs.
Using ( and extending the solution to the multidimensional case, we propose the I-th layer of COSIMO as (considering (-) nonlinearity and (X) X K. X x, as the initial conditions):
te O t tu x O u + te taben x Whid etube, X = (e-ba k.d
kuu
(10)
where {O k.d: ku V k.d V ( are learnable linear projections in 11-1
Remark 4.3. One possible approach to make our model more expressive is the aggregation of M
learnable branches. Let p(m)) (X;4, θ V be the m-th branch of layer as the right-hand side
of the COSIMO model in (10). where X = [X X kdd X
{(m) = {ta (m))
ta ((m)) ). (m)
((m) (mm) @ }. and y (m) T k,d
T (.((m)
}. Considering AGG(.) as well-defined aggregation
k.d
function, e.g., multilayer perceptron, the output for the 1-th layer can be stated as:
X = A
(X:t, θ ))
(11)
4.3 Computational Complexity of COSIMO
For the efficient implementation of exponential Hodge filters, we benefit from the eigenvalue de- composition (EVD) of the Hodge Laplacians (37 21 Precisely, for a Laplacian L E RNXN with
the eigenvalues Å S A S ... S AN-1, after performing EVD, L VAV the exponential Laplacian filtering on input X E R.F and learnable weight W E FxF can be implemented as:
F times
XW V(K) (K)
(K) X))W,
(12)
where Ā e= (e- AN and V(K) N is built by choosing the K most dominant wwwpnaall---igenveeee pairs of L, and @ states the element-wiss multiplication. As K gets closer to N, the estimation is accurate. Using the implementation in (12. the computational complexity of the EVD decreases from ( to O(KN²). Thus, the computational complexity of calculating the Hodge-aware exponential Hodge filters in (9) can be reduced from O( 15) to O I[ (d) + k(") + K2)), where K("). k("), and Kg are the most dominant k
egeevvalue-eigeveeetee of Lk,d, Lg,u. and LR, respectively. The empirical analysis regarding the trade-off on runtime, computational complexity, and performance is presented in the Appendix]]
4.4 Stability Analysis
Here, we study the robustness of our model against simplicial perturbations. We model these perturbations structural inaccuracies in the incidence matrices, given by the additive error models
B, B& + Eg and B B&+1 + E where || S cg and |E+tall S C&+1 represent the perturbation errors. Building upon these additive models on the continuous simplicial filtering operations in (9) the following theorem bounds the COSIMO's stability:
Theorem 4.4. Given the additive simplicial perturbation models B B Ex and B B 1 + Es+1. where ||E S E and EE++ S ex+1, the error between true and perturbed targets in (.. iee, 8x. == [ (.)) (t E. All, is bounded as:
8x, S LadkRaetaou.. (aoeeae(((xxx))) ([)x + (0, 0)ll) + L e buben (I)x + O) (13)
where 8x.d 2 Ammac (LR,))ex + 2 8k.a 2 Amaze (L
2 2
Theorem 4.4 shows how the robustness of the model influenced by the maximum eigenvalues of Lg and Lk..: as well as by the Hodge receptive fields l, and l,, Furthermore, the error bounds of e and 66+11 play a critical role in determining 6R,d and 5R,m, which ultimately control the stability. The following corollary simplifies Theorem 4.4 under the assumption of sufficiently small error bounds.

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Corollary 4.5. When the error bounds and ck+1 are sufficiently small, the ervor between the true and perturbed targets is given by ox, O(es) + O((x+1)
Corollary 4.5
demonstrates stability of the proposed network against small simplicial perturbations, generalizing the stability results on the continuous GNNs 20
Understanding Over-smoothing in SNNs
We first comprchensively analyze the over-smoothing problem in discrete SNNs.. Next, study over-smoothing in COSIMO highlighting the key differences with discrete SNNs. In both cases, we focus the Dirichlet energy convergence zero, making the simplicial signals non-discriminative. In this section, we set Fi F for all I.
5.1 Over-smoothing in Discrete SNNs
Based on the discrete SNN in 4 with T2 = T. = (and zeroing weights for i 0) and Definition 3.1 the following theorem characterizes the over-smoothing properties of the discrete SNN.
Theorem 5.1. In the discrete SNN in 4 and nonlinearity functions ReLU(-) or LeakyReLUC) the Dirichlet energy of the simplicial signals at the + 1-th layer is bounded by the Dirichlet energy of the 1-th layer and some structural and architectural characteristics as:
E 1 112 max E(X%)) + s X (Ex (Ex-I(X)-1) + Ex+1 (X++(X111)) 2F &maa X (IX + k+1
X
(14)
where
Amar
max; (Lx,d,, Amar (LR,))
and
8
V /maxe,, (11 / 11 aalll 1I).
The upper bound in (14 is composed of three terms. Unlike GNN counterparts that depend solely on E(X)) [38. this bound includes two additional terms, the second and third, which help prevent the upper bound from vanishing exponentially. This robustness has been previously analyzed in the context of SCCNNs [8] (more details in Appendix M.2]. However, under some practically justified conditions, the upper bound in (14) can converge zero, i.e., the over-smoothing phenomenon, as described in the next corollary:
Corollary 5.2. In if 7mas < min fainfsshh (2FFs(2Fs) sir. s - }..
then lim E(X)) + 0.
With the assumption in Corollary 5.2 we should modify Ámxx to control the upper bound in 14] This involves modifying the structural properties of the simplicial complex. Therefore, preventing discrete SNNs from converging to the over-smoothing state is not straightforward.
5.2 Over-smoothing in COSIMO
The next theorem characterizes the counterpart of Theorem 5.1
the continuous settings.
Theorem 5.3. Considering the continuous Hodge-aware framework in I0 and nonlinearity funccionnitfunncions functions
ReLU(()) or LeakyReLU(() and defining ip = ming {tadmin (L a), Eyg Amin uk, it holds:
E(X!1)) 1+1 S s. (
s.c 24p 7maa (E(X) E(X(+1))
(15)
(e..s. +e 22). N
X (IXX) + || + 2F...e -P, - IX
where s maxi, { kall 0 k. Il.
[T] k.d
| k.ll)
We observe that the first two terms in 15) tend to converge to zero as in (4) when stacking multiple layers. However, the third term might have different behavior described in the following corollary.
Corollary 5.4. The upper bound in (1)) exponentially converges zero by stacking layers if
In (8) < min { In (1 + e20), 24 In (Amar)s 4 - In (2F(1 ())))))) In (2F )mma))))
7

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Table I: Accuracies in trajectory prediction on synthetic and ocean-drifte datasets.
Table 2: MSE in regression on partial deformable shapes on the Shrec- 16 dataset.
Method
synthetic t
ocean-drifts
Method
small +
full
SNN
[4
0,655 = 0.02
0.525 + 0.06
HSN 4S
0.138 = 0.001
0.133 +: 0.001
SCoNe 22
0.631 + 0.03
0.490 + 0.08
SCACMPS
0.137 - 0.011
0.432 0.001
SCNN 17
0.677 + 0.02
0,530 + 0.08
SAN II
0.052 = 0.011
0.075
Bunch 31
0.623 + 0.04
0.460 0.06
SCCNN I8
0.020 = 0.003
0.063 + 0.003
SCCNN
18
0.652 + 0.04
0.545 0.08
COSIMO
0.010 - 0.004
0.027 1 0.007
COSIMO
0.659 0.04
0,550 0.06
Assuming tu - tu t in 9 (then yp mmmin (L&)) and considering hyperparameter, one heuristic to prevent over-smoothing in COSIMO is stated in the next proposition.
Proposition 5.5. If In (8) > 2Gp In (Amar) (violating one of the conditions in Corollary 5.4. then
In (s) + (Lx)),), where ky (Ix) is the finite condition number 39 of the k-simplex.
<<
Theorem 5.3 aligns with the main takeaways in the GNN literature
40 where increasing the
graph receptive field leads to an increase in the mixing rate of the node features, leading to a faster convergence to the swotthng state. We observe from Proposition [5.5] that decreasing the simplicial receptive field can alleviate over-smoothing, which is a key difference from the discrete case discussed in Section 5.1 We experimentally validate this claim in Section Besides stability and over-smoothing, we show the permutation equivariance property of COSIMO in Appendix ! Remark 5.6. Due to the differentiability of our model in (TO), the simplicial receptive fields { & can be treated as learnable parameters [37,, which is the case with all of our experiments except Section 6.2 This provides a significant advantage over the discrete SNN formulation in (3), where the graph filter orders (Ta, T.} must be manually tuned, leading additional computational cost.
6 Experiments and Results
We evaluate COSIMO against SOTA methods in applications of trajectory prediction, mesh regression,
node and graph classification. We compare COSIMO with a wide range of graph and simplicial
models, including GCN [ GraphSAGE 4T, GIN
32 [17], Bunch 31. HSN 43 SCACMPS 24,. SAN
GAT 42), SNN 14), SCoNe 26). SCNN
SaNN 44 GSAN 45) and SCCNN (18).
Then, we experimentally validate the theoretical claims in this paper. More details on the datasets and implementation are outlined in the Appendix K and L. respectively.
6.1 Real-world Applications
Trajectory prediction. Trajectory prediction involves forecasting paths within simplicial complexes. To evaluate the effectiveness of COSIMO, we assess its performance on two datasets: a synthetic simplicial complex and the ocean-drifts dataset from 26 46). As shown in Table T COSIMO, SCCNN, and Bunch, which incorporate inter- inter-simplicial couplings, do not outperform SCNN on the synthetic dataset. This likely because the imput data assigned nodes and triangles is zero, as noted in 266, making inter-simplicial couplings ineffective. However, in the ocean-drif drifts dataset, where higher-order information plays more significant role, incorporating higher-order convolutions-ar in COSIMO and SCCNN-improved the average accuracy.
Regression on partial deformable shapes. The Shrec-16 benchmark [47] extends prior mesh classification datasets to meshes with missing parts, where each class has a full template in a neutral pose for evaluation. To increase complexity, all shapes were sampled to 10K vertices before introducing missing parts in two regular and irregular ways. This results in a dataset of 599 shapes across eight classes (humans and animals). The dataset is divided into a training set (199 shapes) and a test set (400 shapes). The main task here is to regress the correct mesh class under missing parts. We compare COSIMO against SOTA methods on two small and full versions of the dataset. As shown in T COSIMO achieves the lowest mean square error (MSE) in mesh regression on both dataset versions, outperforming all baselines. Notably, its superior performance on both small and full versions highlights its adaptability to different amounts of data, e.g., even limited data.
8

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Table 3: Accuracy results on node classification (NC) and graph classification (GC) tasks.
Method
high-school f (NC)
senate-bills (NC)
proteins 1 (GC)
GCN E
0.40 + 0.04
0.67 0.06
0.58 + 0.05
GraphSAGF.
0.27 + 0.05
0.54 + 0.03
0.61 + 0.03
GIN 32
0.18 1 0.04
0.53 1 0.04
0.61 i 0.03
GAT 42
0.34 1 0.05
0.50 1 0.04
0.57 1 0.06
SCNN 17
0.81 + 0.01
0.62 + 0.05
0.61 = 0.03
SCCNN 18
0.88 + 0.04
0.64 0.09
0.69 = 0.06
SAN 29
0.86 + 0.04 0.04
0.53 0.09
0.64 + 0.05
SaNN 44
0.83 + 0.03
0.61 0.08
0.77 + 0.02
GSAN 45
0.88 1 0.05
00M
0.77 1 0.04
COSIMO
0.90 - 0.05
0.69 E 0.08
0.79 + 0.01
0.01
0.1
=
0.5
Diriebes Energy
10
10
10-51
100
100
50
100
Number Layers
Number of Layers
Number of Layers
Number Layers
Actual COSIMO
Theorem COSIMO
Theorem discrete SNN
Figure 3: Over-smoothing results of discrete SNNs and COSIMO across different layer depths.
Node classification (NC). In this task, the objective is to infer categorical labels associated with O-dimensional simplices (nodes) embedded a simplicial complex on two benchmark datasets:
high-school [48 49]. and benate-bills [48 50 51. We partition each dataset chronologically.
reserving the initial 80% for training the encoder and the final 20% for evaluation, as in the literature
Table shows the results for node classification. We observe that COSIMO outperforms previous simplicial and graph-based SOTA methods.
44
52
Graph classification (GC). This task entails assigning discrete labels to entire graphs that result from a clique-lifting process applied to simplicial complexes conducted on the standard proteins dataset [53] Model performance is assessed by computing the average classification accuracy over ten in Table 3] Similar to the NC task, COSIMO obtains superior results by leveraging the higher-order information. In fact, the Hadge-aware SNNs mostly effective in cases of existing higher-order information (e.g., on the edges, triangles, etc.). But, the poor performance of GNNs is probably due relying only the data O-order simplices, i.e., the nodes, and neglecting the other simplices like edge flows or triangular dynamics.
stratified folds 44
52 45
6.2 Over-smoothing Analysis
The goals of this section are twofold: (i) validate Theorems 5.1 and 5.3 and (ii) to study the behavior of discrete SNNs in 4 and COSIMO in (10) when facing over-smoothing. For the discrete SNN, consider Ta Tu =1 (i 1) in 4 For COSIMO in (TP, we explore different scenarios by setting the receptive fields Ls = L, where 1. E (10-2 10-1 1021, 0.5). In both cases, the are generated from normal distributions. Figure 3 shows the left-hand side (LHS) and right-hand side (RHS) of Theorems 5. and [53 averaged over 50 random realizations with number layers varying from 100. These results validate Theorems 5.1 and 5..3. confirming that the LHSs are upper bounded by the RHSs. We also observe that adjusting t in COSIMO provides control over the over-smoothing rate, i.e how quickly the output of the SNN converges to zero Dirichlet energy. Specifically, setting t = 10- results in slower over-smoothing rate in COSIMO compared to the discrete SNN. contrast, increasing leads to faster over-smoothing rate in COSIMO than in the discrete SNN. This shows that variations in the continuous receptive fields in (10p directly influence the rate of convergence to the over-smoothing state. Additional analysis has been provided in the Appendix M
linear projections with hidden units Fi =

---

6.3 Stability Analysis
We generate 2-order simplicial complexes, following the approach in [26]: i) we uniformly sample N = 30 random points from the unit square and construct the Delaunay triangulation, ii)
Synthetic Data
Prediction Errr
0.4
remove triangles contained within predefined disk regions, and iti) we use the generative model in (9) with k = 1, la 1, and tu 2, generating (X% E [ 1 from normal probability distribu- tions. After extracting the incidence matrices B, and Bz, we include noise by varying their respective signal- to-noise ratios (SNRs) in {-5,0, 0, 10, 20} dB. For each setting, train COSIMO and evaluate its prediction per- formance, averaged over 30 random realizations. Figure Presents the results, including standard deviations. The results confirm that the model's 's overall stability arises from the stability on upper and lower subspaces, validating theoretical findings of Theorem
0.3
0.2
SNR:==3
SNR;=0
0.1
SNR:-10 10
SNR:-20
-
20
SNR,
Figure 4: Stability analysis under vary- ing SNRs.
f
7 Conclusion and Limitations
In this paper, we have introduced COSIMO, a novel Hodge-aware model for filtering simplicial signals that addresses the limitations of discrete SNNs by incorporating dynamic receptive fields. We provided rigorous thcoretical analyses of the stability and over-smoothing behavior of our model, offering our theoretical findings, demonstrating that COSIMO is not only stable but also allows for effective control over the over-smoothing rate through its continuous receptive fields. Our experimental results highlight the superiority of COSIMO over existing SOTA SNNs. particularly in challenging trajectory prediction, regression of partial shapes, node and graph classification tasks.
insights into its performance. Through extensive experiments, validated
Regarding future work, although we discussed in detail how to reduce the computational complexity of EVD operations, we will seek to alleviate the need for performing EVDs and potential alternatives to ease this requirement, like non-negative matrix factorization and Cholesky decomposition. Precisely, we will explore the direct implicit Fuler method (54) for matrix exponential approximations to reduce the EVD computational complexity at the expense of higher runtime.
Acknowledgments
This research was supported by DATAIA Convergence Institute as part of the <Programme d' d'Investissemen d' Avenirs, (ANR-17-COL 17-CCNV-000) operated by the center Hi! PARIS. This work was also partially supported by the GuroTTech Universities Alliance, and the ANR French National Research Agency under the JCJC projects DeSNAP (ANR-24-CE23-189500 and GraphIA (ANR- 20-CE23-0000-01).
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13

---

A Proof of Proposition 4.2
Proof. By considering the proposed solution in 9 as follows:
(ta)
Xk. (tn)
Xx (x(ta)
(tdt, tu) ttabkdd x
tabedd x (0. 0) + e -t b.bai xx (0.0).
(16)
the simplified generative PDE of this formulation can be expressed as:
@xk,alta) LiNde - tabe, (0) = -L,,ax,,(la),
dld (tt, Lkkue -t ba (0) LL,uxhu(tu)
Ttu
xb(ldslhi) e -labke xx (0,0) -tab,," xx (0,0))
(17)
- lim xx<lad, = e bu.b,, x t00 Oxn (la, bu) Lgg xx (0,0). @xa(labbu) - LI Cu (0,
(0,0), 0), lim xx (lds bw) = -tabe, xx xx (0.0),
400
ota
Ota
Therefore.
Xe(lailu)
lime X (ta.ti
(ta,tattul tu)
Oxn (tanttatt tu)
+
Lond
xxlta; 00)
x00 lu)
(18)
Ohn
which concludes the proof.
L
B Proof of Theorem 4.4
Proof. Using the additive perturbation models and the triangular inequality principle, one can write:
[La_d Lu,all s 2 | E
Ex
S ZE
/ Amax (L c
(19)
[[EE LR S 2|| B || | || + || Ex+ || S 22 (IR,)) + Ex++1
Now, for a Laplacian perturbation model on L, one can write 155 20
- c 1 S < Lll |le LL [e
(20)
for a positive constant p. Then, by the definitions òx.d = 2 / Amax (Lx,a) + et and ō,.u = 2cx+1 / Amax (L..)) + e the bound on the exponential Hodge filters can be obtained as:
62.d
(2(0)
||e- Iteeee - - lebbed S to Lk.d Lual
le teLand (Ila ta tabeed
(21)
(u(0)=
|le te-bb - tubb, bu Le.u Lu
tubu elltaalu, -L.,)l S eetdeen
Next, by considering the solution in (9. the perturbation bound can be expressed as:
(Ia, (u) - Xx ku)
lee tatued -tatheedd (0)ll |le
-le-tuuun -u tubnn (0)]
(22)
(lle-talee -tatiia -tabedd + lle tu Lu E -tute. II) || (0, O)
< tndx.detannn (1xx,)))) + | 0)1) + tude eluun ([]x,, (0) + (0, ()l)).).
14

---

C Proof of Corollary 4.5
Proof. Based on the proof stated in Appendix we can bound | Lxll as:
L La = [RRd Lx + Lk,a - Lu.
- 2 BBll ER + Eg EEll + 2| Bx+ill Ex+1 + Ex+ |
(23)
$ 2cg V Amax (Ln,a) + ox 2cx+ V Anaa (IRgc) + cK+
en. c O(ex) + O(ex+1)-
Using this direction, can be easily seen that:
[Lad
Lx s 2 | + | E& S 200 200/ Amax (Lx,d) + 6x
ef small ((Ex)
LL,, - Lu,ull 5 2 |E+IIl [E Ex+Ill
(24)
ex if is small
2 Amax
(
Now, for a sample Laplacian with |[ - L|| = O(),, one can write:
||e | - 1| S
Lll ||e |e Olete = O(c),
(25)
for a positive constant
[55 20
Therefore, by adapting 25 on can write:
|le tabud taterdd
s Lall &aalTRRd Li,d ll |le futun Welle Jelleu(Enu Luaill
OEettee ) O(Ex),
(26)
|le le-uutu E (ubb,
< t L.ll - LR.ull tabi [leelleeete Ielltutter buw!l
O(c+1itue-ntt O(c+21).
By considering 22) and 26 the proof is completed.
D Proof of Theorem 5.1
Proof. First, by stating the Dirichlet with E(.), we need the following lemmas from 38]
Lemma D.1. (Lemma 3.2 in 38
F XW) < || (X).
Lemma D.2. (Lemmna 3.3 in 38 ). For ReLU and Leaky-Reil ReLU nonlinearities E(o(X)) S E(X).
Lemma D.3. (Von Neumann's 's trace inequality (566) For two square matrices A and B of size and singular values 0i (A) and (B)), respectively, tr (r( < Em (A)o, & m |||||||
Lemma D.4. Since BgBa+1 O, can obtain L, and Lg thu Lk Similar deductions can be obtained for Lk.a-
Lemma D.5. Since Xx,d B, Xx and Ltnd B, Ba and LE- B one can obtain Ea (Xx.d) (r(X]_, [X[_BB.(B]] B. B Xx-1) Amar (I - E, (Xa-1), Similar deductions can be obtained for Lg.3-
15

---

Then.
F(x = Ir (x(+1 L x(+1) + tr(x
L xxxx +
$ a s (L ( (.a) + 2F max (L
x. 2- x 22 + X (L ) Fidl (xx)
$1 ( E(X k,10) 2F max (k,.a) [ 22
X
S LL Er (x((x
1 s Mmx (Liked) Mmx L2 1 Bu [ +
2F
3.5
(L
X
- l1e + s (L a) Ed (x(+)
min
+ A (Lkk ) (
2F
3.5
x ll + 8 8 (I En (X²s
b+1
max
I ST max Eu (X 1) 2F 2
X
2|
s
E
[
Xmx max
Fel x+11) + 2F max
X
X
2+1
X2
max
F. (X
3 s max
(E 10
2F's
3.5
X
|| ll2 + || (12) + E E(X²)
(27)
E Proof of Corollary 5.2
Proof. Using the results of Theorem 5.1 if the constraints of s\ < 1, 2Fs
< 1, and
sX < are simultancously satisfied, by stacking layers, their multiplications converge making the RHS Theorem 5.1 converge zero as well. Holding the mentioned conditions together completes the proof.
F Proof of Theorem
Proof. First, consider that E(x) can be stated by X, i.e., the Graph Fourier Transform (GFT) (where {} eigenvalues of the Laplacian L), follows (8]:
35 of
F(x) Lx Σ A.
(28)
Next, taking À as the smallest nonzero eigenvalue of the Laplacian L, the following lemma describes the behavior heat kernel in the most basic scenario of over-smoothing.
Lemma F.I. We have:
E(e Lx) <e- E(x))
(29)
Proof. By showing the EVD of VAVT and e Ve-AVi
we have:
Ele Ele-Lxx) xx)
Ve AV-
VAV
VTT
(30)
N
N
=x
L
X
Σ
Aai
2h.
<e-22
-2
Σ
A
= 22
-xxx).
1=1
Note that we excluded the zero eigenvalues because they do not engage in the calculation of the Dirichlet energy.
D
16

---

Leveraging from Lemmas D.I D.5 and F.I and also the triangle principle in inequalities, one can simply derive the following usetul inequalities:
Balentabu. X k Oka
k,a) I Eg(e tab X
6 k
2 Ea(c Eale-taba X v k.d) I Ea(e laband X k,lll
3 Calletaben X
4 k,a) S Ea W kull2
tr(Xked T e etabb Lo tab x 0
T
FFIlX T Lu tabn X || 1 wk.d 1. lla
(31)
5
ur(Xked kid e -tabe Lube X
k.d
FF.l c e-labe Ln tabe X ll2 I O' llz
6 tr(x): T -labe. LK,dE La X v O T )
F e l. t.b Xtlla | k, 0 k.d lle
Then, building upon 31 and for obtaining upper bound 11x tr(x)+ bxxxx!11)
(x(+11 1+1 X(11)
first elaborate on the first term:
"xx L
=w x a ta be x ui xx tabe
+ +a(k 'x) tulkk to be w. 2ur "xh Te La Calik v(.a)
+ 2210 xxha -aaaaai tu x 20 "xl" -aahaa
B x
2ea Xha mlll + 20 8a0x) Robdi
+ F2. e a Il willll p.ll wa
(d)
2. ( |
2e B. ex a. Ex0x)) +. 2.P... -. aol XX
+ F.F.s.. will |X IX
|1 2.
s.e Amm x sife 1). Fia.te - - all 2. F.a. "nn [Ix; 11
(32)
and similarly for the second term
tr(x/-1 xxxxx S. -2 Aam Ea aee ().. 2.Raa(e 4 3........111. 111 2.R. F P.m...11 [xx]1 12
(33)
Therefore, the final upper bound on E(x(+1) can be expressed in combined form as
E( s.e" Amm + (E(X)_-) E(x!+1))
(34)
+ *e 2F 1)E({)) + 2F 20 IXIXX il (x) Ama X2,11
where
== min (taaAmia (LR,d): to Amin (I
5 Amax max (Amax (L Amax (Lb,
(35)
{ (1|Or.all 1|0 w.ll vi "alll Ill p "..ll
max kit
Therefore, the proof is completed.
17

---

Shrec 16 (Full)
Shree 16 (Small)
SCACMPS
HSN
SCACHPS
2000
Runt me (G0 (eecanda)
SAT
HEW
300
Error
SAT
SCCNN
SCONN
200
COS
100
324
GPU memory
(GB)
GPU
(GB)
Figure 5: Comparison of the performance error and GPU memory usage (in GB) across runtime (in seconds) (in both color and circle size) on both the Small and Full versions of the Shrec- 16 dataset. The proposed COSIMO method has a good trade-off between runtime and memory usage while performing considerably better.
G Proof of Corollary 5.4
It follows similar to the justifications mentioned in Section E for the second and third terms of the results of Theorem
H Proof of Proposition 5.5
Proof. We start from In (s) > 24 In (Amaa). By considering the fact that Amia (Lx.d) < min(>mia( SaminLLg,. (Lk,)), Amin M[in(L())), [I, assuming to tu replacing &p t-min (Ig),, and the definition
ky (Ix) = Ana (mb (with Aminn (Lx) * 0) 391, one can write:
(Tx)
In (s) 22 In (Imax) + +t In (sxx 2A (Is)
In (SAmax) 2dmux (L&) In (sxmux) + ke (I+)). (Im)
2 (Ls)
(36)
I Runtime and Memory Usage Comparison in the Performance Trade-off
We provide additional comparison results of runtime (in seconds) and memory usage (in GB) both Small and Full versions of the Shrec-16 dataset compared to the SOTA in Fig. S In general and as observed from these results, the proposed COSIMO method enjoys a trade-off between runtime and memory usage while performing considerably better. More precisely, the proposed COSIMO method ranked in the middle of the SOTA runtime and memory usage while providing significantly better accuracy performance. Note that, COSIMO these metrics are heavily relying on the number of selected eigenvalue-eigenvedd pairs in simplicial subspaces and therefore it can be optimized in data-driven manner.
Permutation Equivariance Property of COSIMO
Property (Permutation equivariance 16)). Consider simplicial complex X' characterized by boundary operators = =(Bx) [B&]-=] Let
{P represent a sequence of permutation matrices, where each Pk is size [&g x [& and corresponds to the chain complex dimensions {Cx K ensuring P E [| We define the permuted boundary operator as
[PB] P B.P.T.
simplicial convolutional network (SCN) with the learnable weight matrix W said be permuta- tion equivariant if, for any such transformation P, the following holds:
SCNw.B(c,) Pt SCNw.PB(P,co)
(37)
18

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Based on the above-mentioned properties, we show that COSIMO governs them in the following proposition.
Proposition J.1. The COSIMO model stated in exhibits the property of Permutation Equivariance.
Proof. First, by considering PLg.d = (Px PP = La and similarly PL = P Lg a Pk. and also PX = PP..BBPPPPP (P--Xx--) P and similarly PX K, the permuted exponential expansion be written follows:
P x = X.
PA NOSTMOW,rB((P -II-- PECKE Pk+ICK+1))
P P x O + +Ppe tube X k +P taaaaa -
+PI Tek. X(- (
= o -o X O' + t La x x
kiu ku + + Vel + e uuuben x 1 kk,)
-OOSMOO.n((CC--. ca. cx+1)),
(38)
which completes the proof.
K
More Details on the Datasets
Trajectory prediction. is important note that trajectory prediction in this context involves identifying candidate node within the neighborhood of the target node, process influenced by node degree. Given that the average node degree is 5.24 in the synthetic dataset and 4.81 in the ocean-drifte dataset, random guess would achieve approximately 20% accuracy. The high standard deviations observed, particularly in the icean-drife dataset, may be attributed to its limited size. For more information, please refer to (26).
Regression on partial deformable shapes. The sampling the shapes were in regular cuts, where template shapes were sliced at six orientations, producing 320 partial shapes, and irregular holes, where surface erosion was applied based on area budgets (40%. 70%, and 90%), yielding 279 shapes. This creates to a dataset of 599 shapes across eight classes (humans and animals), with varying missing areas (10%-60%).
Node classification. To provide more details about the NC datasets, we have outlined their statistics Table
L Implementation Details
to implement previous SOTA methods. For accessing and processing real-world datasets, we employ Torch TopoNetX [58]. For the experiments on trajectory prediction, we use the aggregation of M branches discussed in Remark [433] We use
In certain cases, we mostly use TopoModelX 57 58
cross-validane cross-validation for tuning the possible hyperparameters with the selected values provided in Table 5 Detailed hyperparameter configurations for both synthetic and real-world datasets are provided For experimental results synthetic
in the code in and ocean-drs crean-drifts. we followed the experimental settings from reference papers 266 18]. The experiments were conducted
A100 NVIDIA GPU with 40 GB of memory.
Number of selected eigenvalue-eigenveeoor K. In our framework, the selection of an appropriate K can be approached in two ways: (1) supervised and (2) unsupervised. In the supervised setting which serves as the primary approach in this work - K is determined through cross vals-vvaliotionn over reasonable range of values to identify the configuration yielding the best performance. For the unsupervised case, alternative strategies can be employed. For example, we explore the use of spectral entropy (59 as proof of concept to assess its potential effectiveness, which is defined as follows:
n
H =
pi log where Pi
(39)
Σ, X,
19

---

Table 4: Node classification dataset statistics.
Dataset
Simplex
# O-simplicies
# I-simplicies
#2-simplicies
Order
high-school
Group of people
327
5818
2370
senate-bills
Co-sponsors
294
6974
3013
Table 5: Hyperparameter details for each dataset; Ir and Mepoohh are the learning rate and number of epochs, respectively.
Hyperparam
syathetic ocean-drifta Shrec-small Shrec-full high-school senate-bills proteins
lr
5 10
x 10-0-
10-22
10-2
10-3
10-2
10-3
Optimizrr
ADAM
ADAM
ADAM
ADAM
ADAM
ADAM
ADAM
Batch size
100
100
256
512
256
256
256
Depacts
1000
1000
100
100
700
100
30
M
Table 6: Node classification accuracy across selected optimal value of eigenvalue-rigenvecooo (K) pairs by unsupervised (Spectral entropy) and supervised (cross-validation) methods on the high-school dataset.
Spectral entropy
Cross validation
K
Acc (%)
92.0
91.9
10.0
0
AUie Sheone
16"
10
Dinchee Enengy
15-1
thermem
10
10
10-4
20
10-17
Ze: 12
16:15
10-1)
20
20
Londen
Figure wro-sing results COSIMO across different layer depths. Each subplot corresponds to different parameter t in COSIMO, showing the evolution of the Dirichlet energy as a function of the number of layers.
One can choose K where the entropy contribution of the next eigenvalues becomes negligible.
We experimentally apply supervised and unsupervised methods for the selection of K on the hi high-school node classification dataset in Table 6 Note that the cross-validation method se- lects ~ 3% of eigenvalues for Lg, which is fairly consistent with the spectral entropy methodology (~ 4.5%).
M
A Deeper Look at Over-Smoothing
M.1
The effect of varying the Hodge receptive field
To study the effect of varying the Hodge receptive field t on the effective number of layers, we first generate 100 random realizations of simplices with some random (fille)) holes in them using the approach introduced in (26). Then, by fixing the number of hidden features to 4, we varied the number of layers and monitored the actual and theoretical bounds in Theorem 5.3 averaged over the random realizations. The results are shown in Fig. 6 We considered threshold of 10-10 as over-smoothing occurrence threshold. As observed, increasing reduces the effective number of layers from approximately 40 to 15. Besides, the difference between the actual and theoretical bounds gradually decreases and approaches zero, demonstrating the descriptive results of Theorem
5.3
20

---

55.0
34.3
Accurccy (%)
53.5
53.0
52.0
51.5
Number Branches
Figure 7: Sensitivity analysis on the number branches (M) of COSIMO.
Note that decreasing t too much negatively impact the topology of the simplicial complex, potentially leading to issues like over-squashing of information [60], degrading performance. A thorough analysis of this variation is beyond the scope of the paper and is explored in future work.
M.2
Connection with related work [18
we consider simplified filtering framework as x (I La
Lx)x W + + W X W
L ()),, we can elaborate on the first in (). for E(X tr(x(t1 L + tr(x
term
tr(X(+1 LR = tr([ T x (I Lx Lu x W')) + ttr(W's Xh. L wh) + 2tr((W' x (I LR.a)l LR. ak W
(40)
< Ss.2 [I - Ea + F.s F.s.Amar (I
+.FF.s.Amer (I (e..))-maz (Lv.d) [x.lll
Similarly for the second term, have:
tr Ix(x!+1 1+1 (
S S. (I L En (X²) + F.s. u.), ||
(41)
+ F.s.Aa (I (. 1
| ||X
So, in combined manner, we can write:
1(x(11)
2.\2 man. -I- (() + F.s. A.s.Amar (Lg, [ + F.s.Amaw (IR,))
(42)
+ 2.F. s.mmaa (I L (Lx).11 || + (x).
If we consider the simplified framework with F' x wo La )xf + wixk,d w2Xka which was proposed in [1], the bound in (43) takes the form of:
E(x+1)
< 2.12 maze (I-La) Lx)E(x)) + s.Aaaz (Ln 1|x s.Aman (Lm,aLLR, (LR,,))|[x((.ll
(43)
+ 2.8.Amaa L Amma (Lx))||xx|II + [|x
which partially coincides with the smor-sngg bound obtained [18, with an additional fourth term missing in the previous work (18).
N Sensitivity on the Number of Branches (M) of COSIMO
F the accuracy results on the Ocean trajectory prediction task different values of M. It is observed that increasing M up to 3 enhances the expressivity of COSIMO, stated in Remark However, continuing increase beyond results performance degradation, likely due to overfitting.
21

---

Table 7: Experimental validation of the stability Theorem 4.4
the practical gap between left and right
sides in 13) in Theorem 4.4
SNR;
- SNR,
0
SNR;
10
SNR1 20
SNR, --5
12.63
9.55
8.74
8.62
2
5.72
2.71
1.88
1.79
SNR2 -10
4.11
1.13
0.31
0.18
SNR2 20
3.86
0.91
0.11
0.01
Experimental Validation of the Stability Theorem 4.4
Following the SNR setting of Figure Table Thoos
shows the difference between the right and left bounds Theorem 4.4 (13). We observe that, as the SNR, and SNR₂ increase, the gap tends get tighter.
Broader Impact
In the paper, have shown the applicability of the proposed approach, i.e., COSIMO, wide range of real-world societal applications, including ocean drift prediction, regression of the uncompleted mesh shapes, social data analysis, and protein structure classification. This can illustrate the societal impact of the COSIMO for improvement in different humanistic aspects like sea navigation, computer vision, social network development, and medical informatics.
22

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Reviews:
[{"review_id": "gk8Olo0p36", "reviewer": "Reviewer_hz14", "summary": "The 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.", "questions": "1. 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.", "limitations": "Yes", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Quality\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.", "was_revised": true, "final_justification": "All my concern have been thoroughly addressed by the authors. I increased my score accordingly as they did additional experiments and justified a few more claims theoretically.", "rebuttal": "We appreciate the reviewer for providing constructive comments on our paper. We'll include the feedback in the camera-ready paper. In the following, we have addressed the reviewer's concerns:\n\n**Q1**: Firstly, we clarify that we used cross-validation for tuning the hyperparameter. For experimental results on $\\texttt{synthetic}$ and $\\texttt{ocean-drifts}$, we followed the experimental settings from reference papers [1,2]. In addition to adding a comprehensive table of details like the following table for the camera-ready, we will publicly release the codes (which have already been uploaded alongside our submission) and refer to them to make these settings as clear as possible.\n\n|$\\textbf{Hyperparam}$|$\\texttt{synthetic}$|$\\texttt{ocean-drifts}$|$\\texttt{Shrec-small}$|$\\texttt{Shrec-full}$|$\\texttt{high-school}$|$\\texttt{senate-bills}$|$\\texttt{proteins}$|\n|---|---|---|---|---|---|---|---|\n|$lr$|$5\\times10^{-3}$|$5\\times10^{-2}$|$10^{-2}$|$10^{-2}$|$10^{-3}$|$10^{-2}$|$10^{-3}$|\n|Optimizer|ADAM|ADAM|ADAM|ADAM|ADAM|ADAM|ADAM|\n|Batch size|100|100|256|512|256|256|256|\n|$n_\\text{epochs}$|1000|1000|100|100|700|100|30|\n|$M$|3|3|1|1|1|1|1|\n\n**Q2:** In order to make the connection between the proposed COSIMO and reference Graph-PDE counterparts [3,4,10], we first highlight the generalization aspect of simplicial complexes over graphs. For instance, let's consider a simple case of a simplicial complex containing nodes, edges, and triangles. In this case, by relying on graph structure, we neglect edge and triangle dynamics. Therefore, we only deal with $\\mathbf{L}_0$ (in the nodes space with $k=0$), and equations (6) and (7) in our framework are canceled. On the other hand, since $k=0$, only the lower node space is present and equation (5) in our framework turns to $\\frac{\\partial \\mathbf{x}(t)}{\\partial t}=-\\mathbf{L}_0\\mathbf{x}(t)$. Then, by assuming the usage of the normalized version of this Laplacian, *i.e.*, $\\hat{\\mathbf{L}}_0=\\mathbf{I}-\\mathbf{A}$ with $\\mathbf{A}$ being the normalized adjacency matrix, this equation turns to $\\frac{\\partial \\mathbf{x}(t)}{\\partial t}=(\\mathbf{A}-\\mathbf{I})\\mathbf{x}(t)$, which is equivalent to eq. (2) in GRAND [4] and eq. (7) in CGNN [3] frameworks in the isotropic linear cases. Therefore, COSIMO generalizes the CGNN and GRAND models by considering edge and triangular dynamics. Apart from these theoretical/architectural differences, we have also experimentally compared COSIMO against GRAND [4] and Graph-coupled oscillator networks (GraphCON) [10] in the following table, showcasing the superior performance by relying on higher-order dynamics. \n\n|$\\textbf{Method}$|$\\texttt{high-school}$|$\\texttt{senate-bills}$|\n|---|---|---|\n|GCN|0.40$\\pm$0.04|$\\underline{0.67\\pm0.06}$|0.58$\\pm$0.05|\n|GAT|0.34$\\pm$0.05|0.50$\\pm$0.04|0.57$\\pm$0.06|\n|GraphSage|0.27$\\pm$0.05|0.54$\\pm$0.03|0.57$\\pm$0.06|\n|GIN|0.18$\\pm$0.04|0.53$\\pm$0.04|0.57$\\pm$0.06|\n|GRAND|0.16$\\pm$0.07|0.52$\\pm$0.10|\n|GraphCON|0.39$\\pm$0.08|0.58$\\pm$0.04|\n|SCCNN|$0.81\\pm0.01$|0.62$\\pm$0.05|0.61$\\pm$0.03|\n|SAN|$\\underline{0.86\\pm0.04}$|0.53$\\pm$0.09|0.64$\\pm$0.05|\n|COSIMO|$\\textbf{0.90$\\pm$0.05}$|$\\textbf{0.69$\\pm$0.08}$|$\\textbf{0.79$\\pm$0.01}$|\n\n**Q3:** Apart from the outlined possible limitations in the conclusion section, we also point out the case of *i)* eventually facing over-smoothing in Figure 3, and *ii)* an extremely noisy regime (for example, $SNR_1=-5$ and $SNR_2=-5$ in Figure 4). To mitigate these limitations in these cases, it has been shown that equipping the underlying set of PDEs with **source terms** [5] can effectively improve the stability and robustness to over-smoothing. Therefore, the corresponding equations of our framework, *i.e.*, independent and joint dynamics, turn to:\n\n$\\frac{\\partial \\mathbf{x}\\_{k,d}(t\\_d)}{\\partial t\\_d} = -\\mathbf{L}\\_{k,d}\\mathbf{x}\\_{k,d}(t\\_d)\\rightarrow\\frac{\\partial \\mathbf{x}\\_{k,d}(t\\_d)}{\\partial t\\_d} = -\\mathbf{L}\\_{k,d}\\mathbf{x}\\_{k,d}(t\\_d)+\\mathbf{E}\\_d,$\n\n$\\frac{\\partial \\mathbf{x}\\_{k,u}(t\\_u)}{\\partial t\\_u} = -\\mathbf{L}\\_{k,u}\\mathbf{x}\\_{k,u}(t\\_u)\\rightarrow\\frac{\\partial \\mathbf{x}\\_{k,u}(t\\_u)}{\\partial t\\_u} = -\\mathbf{L}\\_{k,u}\\mathbf{x}\\_{k,u}(t\\_u)+\\mathbf{E}\\_u,$\n\n$ \\frac{\\partial \\mathbf{x}\\_{k}(t\\_d, t\\_u)}{\\partial t\\_d} + \\frac{\\partial \\mathbf{x}\\_{k}(t\\_d, t\\_u)}{\\partial t\\_u} = -\\mathbf{L}\\_{k,d}\\mathbf{x}\\_{k}(t\\_d, \\infty) - \\mathbf{L}\\_{k,u}\\mathbf{x}\\_{k}(\\infty, t\\_u)\\rightarrow\\frac{\\partial \\mathbf{x}\\_{k}(t\\_d, t\\_u)}{\\partial t\\_d} + \\frac{\\partial \\mathbf{x}\\_{k}(t\\_d, t\\_u)}{\\partial t\\_u} = -\\mathbf{L}\\_{k,d}\\mathbf{x}\\_{k}(t\\_d, \\infty) - \\mathbf{L}\\_{k,u}\\mathbf{x}\\_{k}(\\infty, t\\_u)+\\mathbf{E}\\_{d,u},$\n\nwhere $\\mathbf{E}\\_d$, $\\mathbf{E}\\_u$, and $\\mathbf{E}\\_{d,u}$ are source terms. Although this modification might bring the above-mentioned advantages, the solutions pose more computational burden, as detailed in different frameworks in [5].\n\n**Q4:** As stated in Remark 5.6, and due to the differentiability of our framework *w.r.t.* the simplicial receptive fields, in all the experimental results (except the over-smoothing analysis in Section 6.2), the values of $t$ are learned during the training process, *i.e.*, $t$ are parameters, not hyperparameters. We considered $t$ as hyperparameters only in the over-smoothing analysis in Section 6.2 to illustrate the effect of different values for $t$ on the over-smoothing phenomenon. In general, we use cross-validation over a range of possible values to tune the hyperparameters of COSIMO. Regarding the hyperparameter $M$, we also exploited a multi-branch architecture in trajectory prediction tasks in which we have already provided its sensitivity and accuracy trade-off in the Appendix, *i.e.*, Figure 7, by using a cross-validation scheme on the corresponding range for $M$. Furthermore, one can select a proper $K$ using supervised (cross-validation) or unsupervised methodologies. We used the cross-validation approach in the paper. Regarding the unsupervised methodologies, one can exploit the following alternative strategies:\n\n1.  **Eigengap heuristic** [6]: The number $K$ is often found using the eigengap heuristic:\n\n$K = \\text{argmax}\\_{i}{(\\lambda\\_{i+1}-\\lambda\\_i)}$.\n\nA large gap indicates a natural cutoff point.\n\n2. **Energy-based Criterion** [7]: The Laplacian EVD can be viewed like PCA, such that one can choose enough eigenvectors to explain a certain percentage of *spectral energy*. Therefore, the total spectral energy is defined as:\n\n$E\\_{total}:=\\sum\\_{i=1}^{n}{\\lambda\\_i}$.\n\nOne can thus choose the smallest $K$ such that $\\frac{\\sum_{i=1}^{K}{\\lambda_i}}{E_{total}}\\ge\\eta$, where $\\eta\\in[0,1]$ is a threshold like 90\\%.\n\n3. **Spectral Gap Ratio or Knee Detection** [8]: Instead of just picking the largest gap, use a normalized eigengap ratio:\n\n$\\text{gap ratio}(K):=\\frac{\\lambda_{K+1}-\\lambda_{K}}{\\lambda_{K}}$.\n\nThis method accounts for the scaling of eigenvalues and can find relative jumps.\n\n4. **Spectral Entropy-Information-Theoretic Criteria** [9]: The spectral entropy can be defined as:\n\n$H:=-\\sum\\_{i=1}^{n}{p\\_i\\log{p\\_i}}; \\text{ where } p\\_i=\\frac{\\lambda\\_i}{\\sum\\_{j}{\\lambda\\_j}}$.\n\nTherefore, one can choose $K$ where the entropy contribution of the next eigenvalues becomes negligible.\n\nApart from introducing and listing the relevant methods, we have also experimentally applied these strategies on a subset of the $\\texttt{high-school}$ node classification dataset in the following table, accompanied by the accuracy performance. Note the supervised methodology picked $\\sim5.1\\\\%$ of eigenvalues for $\\textbf{L}_1$, which is consistent with the unsupervised methods ($\\sim5.5\\\\%$).\n\n**Table: Selected optimal values of eigenvalue-eigenvector $(K)$ pairs from $\\sim$ 5800 ones for $\\mathbf{L}_1$ by different relevant methods.**\n| |Eigengap | Energy-based | Knee detection | Spectral entropy|Cross-Validation|\n|---|---|---|---|---|---|\n|K|322 | 312 | 289 |312|300|\n|Acc|91.7 | 92.0 | 91.6 |92.0|91.9|\n\nIn conclusion and by observing results on various datasets, choosing 5-10$\\\\%$ of eigenvalue-eigenvector pairs and number of branches ($M$) from 1-3 seems to provide a fair compromise of performance on downstream tasks.\n\n---\nReferences:\n\n[1] M. Roddenberry, et al, “Principled simplicial neural networks for trajectory prediction,” ICML 2021\n\n[2] M. Yang, et al, “Convolutional learning on simplicial complexes,” arXiv 2025\n\n[3] L.-P. Xhonneux, et al, “Continuous graph neural networks,” ICML 2020\n\n[4] B. Chamberlain, et al, “Grand: Graph neural diffusion,” ICML 2021\n\n[5] A. Han, et al, “From continuous dynamics to graph neural networks: Neural diffusion and beyond,” TMLR 2024\n\n[6] J. Shi, et al, “Normalized cuts and image segmentation,” IEEE TPAMI, 2000\n\n[7] M. Belkin, et al, “Laplacian eigenmaps for dimensionality reduction and data representation,” Neural computation, 2003\n\n[8] I. M. Johnstone, “On the distribution of the largest eigenvalue in principal components analysis,” The Annals of statistics, 2001\n\n[9] M. De Domenico, et al, “Spectral entropies as information-theoretic tools for complex network comparison,” Physical Review X, 2016\n\n[10] T. K. Rusch, et al, “Graph-coupled oscillator networks,” ICML 2022"}, {"review_id": "16ziTizh33", "reviewer": "Reviewer_yoZd", "summary": "The 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.", "questions": "What would be potential applications where higher-order relationships are essential?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "The authors addressed all the concerns raised by the reviewer. Therefore, I would recommend acceptance.", "rebuttal": "We appreciate the reviewer for providing constructive comments on our paper. We'll include the feedback in the camera-ready paper. In the following, we have addressed the reviewer's concerns:\n\n**W1:** To complement the experimental analysis, we compare COSIMO against GRAND and Graph-coupled oscillator networks (GraphCON) [1] in the following table. We observe the superiority of COSIMO over GNNs, which only rely on the node space. This is a general observation in the comparison of SNNs vs. GNNs when higher-order information is available.\n\n|$\\textbf{Method}$|$\\texttt{high-school}$|$\\texttt{senate-bills}$|\n|---|---|---|\n|GCN|0.40$\\pm$0.04|$\\underline{0.67\\pm0.06}$|0.58$\\pm$0.05|\n|GAT|0.34$\\pm$0.05|0.50$\\pm$0.04|0.57$\\pm$0.06|\n|GraphSage|0.27$\\pm$0.05|0.54$\\pm$0.03|0.57$\\pm$0.06|\n|GIN|0.18$\\pm$0.04|0.53$\\pm$0.04|0.57$\\pm$0.06|\n|GRAND|0.16$\\pm$0.07|0.52$\\pm$0.10|\n|GraphCON|0.39$\\pm$0.08|0.58$\\pm$0.04|\n|SCCNN|$0.81\\pm0.01$|0.62$\\pm$0.05|0.61$\\pm$0.03|\n|SAN|$\\underline{0.86\\pm0.04}$|0.53$\\pm$0.09|0.64$\\pm$0.05|\n|COSIMO|$\\textbf{0.90$\\pm$0.05}$|$\\textbf{0.69$\\pm$0.08}$|$\\textbf{0.79$\\pm$0.01}$|\n\n**W2:** We provide comprehensive results on the $\\texttt{high-school}$ node classification dataset averaged over three random seeds in the following table regarding the tradeoff between over-smoothing and accuracy. We observe that the proposed COSIMO can maintain the performance even with an increasing number of layers up to 16 in terms of accuracy and Dirichlet energy. We also observe that COSIMO is more robust against over-smoothing than SCCNN, where extremely smoothness happens after $n_l=4$ and leads to severely reduced performance. The key observation and trend here are also fairly consistent with the results in Figure 3.\n\n**Acc:**\n|**Method** | $n_l$=1 | $n_l$=2 | $n_l$=4 | $n_l$=8 | $n_l$=16 | $n_l$=32|\n|---|---|---|---|---|---|---|\n|SCCNN | 57 | 81 | 43 | 16 | 12 | 10 |\n|COSIMO | 79.33 | 86.33 | 91.33 | 91.33 | 88.00 | 53.67|\n\n**Dirichlet Energy:**\n|**Method** | $n_l$=1 | $n_l$=2 | $n_l$=4 | $n_l$=8 | $n_l$=16 | $n_l$=32|\n|---|---|---|---|---|---|---|\n|SCCNN | 0.88| 0.80| 0.75| 0.53 | 0.42 | 0.07|\n|COSIMO | 0.94 | 0.92 | 0.92 | 0.92 | 0.88 | 0.19|\n\n**W3:** We have repeated the experiment in Section 6.3, including comparisons with the baseline methods SCCNN and SAN, and provided the results in the following table in terms of prediction error. We observe that the baselines are vulnerable in moderate to high amounts of noise (-5 or 0 db SNRs). In comparison, COSIMO has superior and more robust performance compared to the baselines even in the low-noisy regime (SNRs of 10 or 20 db).\n\n**SNR$_2$=-5:**\n|**Method** | SNR$_1$=-5 | SNR$_1$=0 | SNR$_1$=10 | SNR$_1$=20|\n|---|---|---|---|---|\n|SCCNN | 625.15 | 612.814 | 639.24 | 482.98|\n|SAN | 78281.68 | 32423.05 | 8984.80 | 4246.06|\n|COSIMO | 1.30 | 1.29 | 1.10 | 1.14|\n\n**SNR$_2$=0:**\n|**Method**| SNR$_1$=-5 | SNR$_1$=0 | SNR$_1$=10 | SNR$_1$=20|\n|---|---|---|---|---|\n|SCCNN | 191.45 | 57.84 | 46.44 | 49.54|\n|SAN | 526.45 | 1.65 | 1.00 | 1.09|\n|COSIMO | 1.34 | 1.16 | 1.03 | 0.92|\n\n**SNR$_2$=10:**\n|**Method** | SNR$_1$=-5 | SNR$_1$=0 | SNR$_1$=10 | SNR$_1$=20|\n|---|---|---|---|---|\n|SCCNN | 243.41 | 28.06 | 8.07 | 10.16|\n|SAN | 128.13 | 0.81 | 0.92 | 0.84|\n|COSIMO | 1.14 | 1.13 | 0.76 | 0.69|\n\n**Q1:** Based on the provided results regarding over-smoothing, stability, and performance, we argue that relying on simplicial complexes is meaningful when higher-order information is available. This is because SNNs, like COSIMO, can benefit from the higher-order dynamics. One practical example is the $\\texttt{high-school}$ dataset, where friendship groups of multiple people (maximum cliques) are modeled as simplices. The description of the dataset has been explicitly mentioned on the official website as follows:\n\n\"*This is a temporal higher-order network dataset, which here means a sequence of timestamped simplices where each simplex is a set of nodes. The dataset is constructed from interactions recorded by wearable sensors by people at a high school. The sensors record interactions at a resolution of 20 seconds (recording all interactions from the previous 20 seconds). Nodes are the people, and simplices are maximal cliques of interacting individuals from an interval.*\"\n\nIt is clear from our experimental results that relying on the typical graph structure (only node space) cannot leverage the information dynamics lying on the simplices in this dataset, so the performance of classical GNNs is not competitive against SNNs.\n\nTo explore more diverse real-world applications from computer graphics to drug discovery and biotechnology, we kindly refer the reviewer to this recent ICML position paper [2]. \n\n---\nReferences:\n\n[1] T. K. Rusch, et al, “Graph-coupled oscillator networks,” ICML 2022\n\n[2] Papamarkou, et al. \"Position: Topological deep learning is the new frontier for relational learning\". ICML 2024."}, {"review_id": "h6ASa0Gckk", "reviewer": "Reviewer_tWza", "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.", "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", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths\n- Cast simplical complexes into the PDEs framework and provide allows continuous information flow between structural data. \n- Bring theoretical analysis and proof to ensure the stability of the proposed work and explicit bound on over-smoothing rate.\n- Different types of tasks demonstrates the effectiveness of this method\n### Weakness\nThe eigenvalue decomposition may hinder the actual applicability of this method", "was_revised": true, "final_justification": "Since my only concern that applying eigen-decomposition may consume the majority part of the computational efficiency is addressed, I will raise the score. Meanwhile, I am not an expert in this field, my confidence will remain low.", "rebuttal": "We appreciate the reviewer for providing constructive comments on our paper. We'll include the feedback in the camera-ready paper. In the following, we have addressed the reviewer's concerns:\n\n**W1 and Q1:** First, we emphasize that performing EVD on the Hodge Laplacians is done once and as a preprocessing step, so they can be precomputed. Additionally, as will be presented in the following, one can rely on a small subset of eigenvalue-eigenvector pairs, and then, the computational complexity of performing EVD is reduced from $\\mathcal{O}(N^3)$ to $\\mathcal{O}(N^2K)$. In our framework, one can select a proper $K$ using supervised (cross-validation) or unsupervised methodologies. We used the cross-validation approach in the paper. Regarding the unsupervised methodologies, one can exploit the following alternative strategies:\n\n1.  **Eigengap heuristic** [1]: The number $K$ is often found using the eigengap heuristic:\n\n$K = \\text{argmax}\\_{i}{(\\lambda\\_{i+1}-\\lambda\\_i)}$.\n\nA large gap indicates a natural cutoff point.\n\n2. **Energy-based Criterion** [2]: The Laplacian EVD can be viewed like PCA, such that one can choose enough eigenvectors to explain a certain percentage of *spectral energy*. Therefore, the total spectral energy is defined as:\n\n$E\\_{total}:=\\sum\\_{i=1}^{n}{\\lambda\\_i}$.\n\nOne can thus choose the smallest $K$ such that $\\frac{\\sum_{i=1}^{K}{\\lambda_i}}{E_{total}}\\ge\\eta$, where $\\eta\\in[0,1]$ is a threshold like 90\\%.\n\n3. **Spectral Gap Ratio or Knee Detection** [3]: Instead of just picking the largest gap, use a normalized eigengap ratio:\n\n$\\text{gap ratio}(K):=\\frac{\\lambda_{K+1}-\\lambda_{K}}{\\lambda_{K}}$.\n\nThis method accounts for the scaling of eigenvalues and can find relative jumps.\n\n4. **Spectral Entropy-Information-Theoretic Criteria** [4]: The spectral entropy can be defined as:\n\n$H:=-\\sum\\_{i=1}^{n}{p\\_i\\log{p\\_i}}; \\text{ where } p\\_i=\\frac{\\lambda\\_i}{\\sum\\_{j}{\\lambda\\_j}}$.\n\nTherefore, one can choose $K$ where the entropy contribution of the next eigenvalues becomes negligible.\n\nApart from introducing and listing the relevant methods, we have also experimentally applied these strategies on a subset of the $\\texttt{high-school}$ node classification dataset in the following table, accompanied by the accuracy performance. Note the supervised methodology picked $\\sim5.1\\\\%$ of eigenvalues for $\\textbf{L}_1$, which is consistent with the unsupervised methods ($\\sim5.5\\\\%$).\n\n**Table: Selected optimal values of eigenvalue-eigenvector $(K)$ pairs from $\\sim$ 5800 ones for $\\mathbf{L}_1$ by different relevant methods.**\n| |Eigengap | Energy-based | Knee detection | Spectral entropy|Cross-Validation|\n|---|---|---|---|---|---|\n|K|322 | 312 | 289 |312|300|\n|Acc|91.7 | 92.0 | 91.6 |92.0|91.9|\n\nRegarding the effect of varying $K$ on the performance, we provide comprehensive experimental results on a subset of the $\\texttt{high-school}$ node classification dataset in the following table. From these results, we can conclude that choosing the best $K$ heavily relies on the specific data distribution, but generally, only a small fraction of the eigenvalues is enough to reach acceptable performance. We also observe that increasing $K$ does not necessarily lead to better performance, as high-frequency components might contain more noisy or non-relevant content [5].\n\n**Table: Performance (accuracy) results across a selected set of eigenvalue-eigenvector $(K)$ pairs from $\\sim$ 300 ones for $\\mathbf{L}_0$.**\n|$1\\\\%$| $3\\\\%$ | $10\\\\%$ | $30\\\\%$ | $50\\\\%$ | $70\\\\%$|\n|---|---|---|---|---|---|\n|88.0$\\pm$5.2 | 90.6$\\pm$2.1 | **93.4$\\pm$3.4** | $91.4\\pm1.9$ | 90.2$\\pm$3.2 | 90.4$\\pm$1.6|\n\n---\nReferences:\n\n[1] J. Shi, et al, “Normalized cuts and image segmentation,” IEEE TPAMI, 2000\n\n[2] M. Belkin, et al, “Laplacian eigenmaps for dimensionality reduction and data representation,” Neural computation, 2003\n\n[3] I. M. Johnstone, “On the distribution of the largest eigenvalue in principal components analysis,” The Annals of statistics, 2001\n\n[4] M. De Domenico, et al, “Spectral entropies as information-theoretic tools for complex network comparison,” Physical Review X, 2016\n\n[5] Dong, X., et al. Learning Laplacian matrix in smooth graph signal representations. IEEE TSP 2016."}, {"review_id": "srqrPTiLr4", "reviewer": "Reviewer_z9BE", "summary": "The 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.", "questions": "- 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.", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\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)", "was_revised": true, "final_justification": "The authors have addressed all my comments and provided additional experiments confirming the theoretical findings.", "rebuttal": "We appreciate the reviewer for providing constructive comments on our paper. We'll include the feedback in the camera-ready paper. In the following, we have addressed the reviewer's concerns:\n\n**Q1:** Firstly, our framework and its theoretical claims are also valid and straightforwardly transferable to the case of weighted structures, not just binarized. Therefore, the incidence matrices can be non-integer in general. Also, the proof of Theorem 4.3 does not need the error matrices to have small norm. Theorem 4.3 holds in general, and having a small norm is only needed in the proof of Corollary 4.4. We will clarify these points in the camera-ready.\n\nWe'd also like to clarify the meaning of having a small norm for $\\mathbf{E}_k$. From the proof of Corollary 4.4, we observe that being small is *w.r.t* the scale of the maximum eigenvalue of the Hodge Laplacians, *i.e.*, $\\lambda\\_{max}(\\mathbf{L}\\_{k,d})$. Therefore, it is not necessarily considered as being small in an absolute manner.\n\nFinally, regarding the nature of error matrices $\\mathbf{E}_k$, we clarify that the removal, addition, and weight changing of edges can be modeled by the additive model $\\hat{\\mathbf{B}}_1=\\mathbf{B}_1+\\mathbf{E}_1$.\n\nTo simplify the discussion, we start with a simple toy example. Let's consider an undirected three-node graph in which $\\mathcal{V}=${$1,2,3$} and $\\mathcal{E}=${$(1,2),(1,3)$}. In this case, we have $\\mathbf{B}_1=[[1 \\\\:\\\\:\\\\: 1][-1 \\\\:\\\\:\\\\: 0][0 \\\\:\\\\:\\\\: -1]]\\in\\mathbb{R}^{3\\times2}$. Then, one special form of an incidence error matrix corresponding to changes in the current edge weights can be stated as $\\mathbf{E}\\_1=[[-\\epsilon\\_{12} \\\\:\\\\:\\\\: -\\epsilon\\_{13}][\\epsilon\\_{12} \\\\:\\\\:\\\\: 0][0 \\\\:\\\\:\\\\: \\epsilon\\_{13}]]\\in\\mathbb{R}^{3\\times2}$. Therefore, after structural perturbation, we have $\\hat{\\mathbf{B}}_1=\\mathbf{B}_1+\\mathbf{E}_1=[[1-\\epsilon\\_{12} \\\\:\\\\:\\\\: 1-\\epsilon\\_{13}][-1+\\epsilon\\_{12} \\\\:\\\\:\\\\: 0][0 \\\\:\\\\:\\\\: -1+\\epsilon\\_{13}]]\\in\\mathbb{R}^{3\\times2}$. So, we have $\\\\|\\mathbf{E}\\_1\\\\|=\\sqrt{2}\\sqrt{\\epsilon^2\\_{12}+\\epsilon^2\\_{13}}$, and, given the scale of $\\epsilon\\_{12}$ and $\\epsilon\\_{13}$, this norm can be small or not. As a conclusion, since our definition of simplicial perturbation is not limited to only the addition or removal of edges (or higher-order connections), the norm of the error matrices can take different scales. On the other hand, and in the case of weighted graphs and in connection with the previously mentioned examples, assume that we have the edge weight $e\\_{12}=0.0625$ (instead of 1). This implies that $\\mathbf{B}_1=[[0.25 \\\\:\\\\:\\\\: 1][-0.25 \\\\:\\\\:\\\\: 0][0 \\\\:\\\\:\\\\: -1]]\\in\\mathbb{R}^{3\\times2}$. Now, for only modeling the removal of this edge, we just need to have $\\epsilon\\_{12}=0.25$ in $\\mathbf{E}_1$, leading to a small norm for $\\mathbf{E}\\_1$. Therefore, the scale of the norm for $\\mathbf{E}$ heavily depends on the scale of edge weights.\n\n**Q2:** In connection with the response to the previous question, the perturbation of $\\mathbf{B}_k$ could model potential erroneous structures where we have the addition of non-existing simplicial connections, the removal of existing simplices, or changes in simplicial connection weights in a principled manner. For further clarification, one can state the toy graph from the response to **Q1** as $\\mathcal{V}=${$1,2,3$} and $\\mathcal{E}=${$(1,2),(1,3),(2,3)$} in which edge $(2,3)$ takes zero value because of disconnectivity. Then, the incidence matrix is stated as $\\mathbf{B}_1=[[1 \\\\:\\\\:\\\\: 1 \\\\:\\\\:\\\\: 0][-1 \\\\:\\\\:\\\\: 0 \\\\:\\\\:\\\\: 0][0 \\\\:\\\\:\\\\: -1 \\\\:\\\\:\\\\: 0]]\\in\\mathbb{R}^{3\\times3}$. Now, as an example, for modeling removing edge $(1,2)$ and adding edge $(2,3)$, the error matrix can take the form of $\\mathbf{E}_1=[[-1\\\\:\\\\:\\\\: 0 \\\\:\\\\:\\\\: 0][1 \\\\:\\\\:\\\\: 0 \\\\:\\\\:\\\\: 1][0 \\\\:\\\\:\\\\: 0 \\\\:\\\\:\\\\: -1]]\\in\\mathbb{R}^{3\\times3}$. Again, if we consider the case of changing edge weights (not removal or addition) or a weighted graph, the norm of the error matrix can take a very small value. Therefore, the stability of COSIMO against perturbations shows its robustness to such cases.\n\n**Q3:** We'll clarify in the answer to **Q4** that $t$ is indeed a parameter of the model, not a hyperparameter. Previous studies typically analyze the over-smoothing phenomenon by stacking several graph or simplicial layers. Therefore, we provide comprehensive results on the $\\texttt{high-school}$ node classification dataset averaged over three random seeds in the following table regarding the tradeoff between over-smoothing and accuracy. We observe that the proposed COSIMO can maintain the performance even with an increasing number of layers up to 16 in terms of accuracy and Dirichlet energy. We also observe that COSIMO is more robust against over-smoothing than SCCNN, where extremely smoothness happens after $n_l=4$ and leads to severely reduced performance. The key observation and trend here are also fairly consistent with the results in Figure 3.\n\n**Acc:**\n|**Method** | $n_l$=1 | $n_l$=2 | $n_l$=4 | $n_l$=8 | $n_l$=16 | $n_l$=32|\n|---|---|---|---|---|---|---|\n|SCCNN | 57 | 81 | 43 | 16 | 12 | 10 |\n|COSIMO | 79.33 | 86.33 | 91.33 | 91.33 | 88.00 | 53.67|\n\n**Dirichlet Energy:**\n|**Method** | $n_l$=1 | $n_l$=2 | $n_l$=4 | $n_l$=8 | $n_l$=16 | $n_l$=32|\n|---|---|---|---|---|---|---|\n|SCCNN | 0.88| 0.80| 0.75| 0.53 | 0.42 | 0.07|\n|COSIMO | 0.94 | 0.92 | 0.92 | 0.92 | 0.88 | 0.19|\n\n**Q4:** As stated in Remark 5.6, and due to the differentiability of our framework *w.r.t.* the simplicial receptive fields, in all the experimental results (except the over-smoothing analysis in Section 6.2), the values of $t$ are learned during the training process, *i.e.*, $t$ are parameters, not hyperparameters. We considered $t$ as a hyperparameter only in the over-smoothing analysis in Section 6.2 to illustrate the effect of different values for $t$ on the over-smoothing phenomenon.\n\n**Q5:** Eq. (8) implies a definition. In fact, it integrates joint and independent dynamics that act as source terms; this has been shown to improve robustness to over-smoothing and stability as outlined in [1]. The analytical form of the solution to the set of PDEs in eqs. (5), (6), and (7) take the form of eq. (9), which can be stated in a **general form** as eq. (8). Note that $\\mathbf{x}\\_{k,d}(t\\_d) + \\mathbf{x}\\_{k,u}(t\\_u)$ in eq.(8) is actually the general symbolic form of $\\mathbf{x}\\_{k}(t\\_d,t\\_u)$ in eq. (9). We will modify and clarify these points in the camera-ready version. \n\n---\nReferences:\n\n[1] A. Han, et al, “From continuous dynamics to graph neural networks: Neural diffusion and beyond,” TMLR 2025"}]

Author rebuttal:

Meta-review: This paper proposes COSIMO, a novel simplicial neural network to process data on simplicial complexes. The method design seems effective and novel, and the paper demonstrates that their approach works well empirically and provides rigorous theoretical analysis. Following the reviewers' feedback, the authors can refine the paper to be more clear.

With positive scores from all reviewers, I vote for acceptance.
Decision: Accept (poster)