PIORF: Physics-Informed Ollivier-Ricci Flow for Long–Range Interactions in Mesh Graph Neural Networks
Authors: Youn-Yeol Yu
Jeongwhan Choi
Jaehyeon Park
Kookjin Lee
Noseong Park
Venue: ICLR 2025 Poster (2025)
Keywords: graph neural network
fluid dynamics
simulation
mesh
physics
over-squashing
rewiring
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: PIORF is a physics-informed graph rewiring method that integrates Ollivier–Ricci curvature with velocity gradients to improve long-range interactions in mesh-based GNNs, significantly mitigating over-squashing in fluid dynamics simulations.

Abstract: Recently, data-driven simulators based on graph neural networks have gained attention in modeling physical systems on unstructured meshes. However, they struggle with long-range dependencies in fluid flows, particularly in refined mesh regions. This challenge, known as the 'over-squashing' problem, hinders information propagation. While existing graph rewiring methods address this issue to some extent, they only consider graph topology, overlooking the underlying physical phenomena. We propose Physics-Informed Ollivier--Ricci Flow (PIORF), a novel rewiring method that combines physical correlations with graph topology. PIORF uses Ollivier--Ricci curvature (ORC) to identify bottleneck regions and connects these areas with nodes in high-velocity gradient nodes, enabling long-range interactions and mitigating over-squashing. Our approach is computationally efficient in rewiring edges and can scale to larger simulations. Experimental results on 3 fluid dynamics benchmark datasets show that PIORF consistently outperforms baseline models and existing rewiring methods, achieving up to 26.2\% improvement.

Published as a conference paper at ICLR 2025
PIORF: PHYSICS-INFORMED OLLIVIER-RICC, FLOW FOR LONG-RANGE INTERACTIONS IN MESH GRAPH NEURAL NETWORKS
Youn-Yeol Yu Jeongwhan Choi Jachyeon Park2 Kookjin Lee Noseong Park2 2
Yonsei University KAAIST
AArizona State University
ABSTRACT
Recently, data-driven simulators based on graph neural networks have gained at- tention modeling physical systems on unstructured meshes. However, they struggle with long-range dependencies in fluid flows, particularly in refined mesh regions. This challenge, known as the 'over-squashing' problem, hinders infor- mation propagation. While existing graph rewiring methods address this issue to some extent, they only consider graph topology, overlooking the underlying phys- ical phenomena. We propose Physies-Informed Oliivier-Ricci Flow (PIORF), a novel rewiring method that combines physical correlations with graph topology. PIORF uses Ollivier-Ricct curvature (ORC) to identify bottleneck regions and connects these areas with nodes in high-velocity gradient nodes, enabling long- range interactions and mitigating over-squashing. Our approach is computation- ally efficient in rewiring edges and can scale to larger simulations. Experimental results 3 fluid dynamics benchmark datasets show that PIORF consistently out- performs baseline models and existing rewiring methods, achieving to 26.2% improvement.
AirFoil
(Velocity) 24.5%
(Pressure)
11.3%
AirFoil (Pressure)
DNS 20%
0.04
0.06
0.10
-206
CylinderFlow B (Velocity)
(a) ORC distribution
MGN
AirFoil (Density)
23.5%
+ DIGL
13.3%
FoSR
EAGLE (Pressure)
BORF
EAGLE (Velocity)
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
(b) Velocity contour
The radar plot shows the per- centage improvement over MGN for each method on datasets. The radial dis- tance indicates the magnitude of improve- ment. PIORF consistently outperforms other methods with substantial gains partic- ularly AIRFOIL (24,5% for Velocity) and CYLINDER (21.3% for Pressure).
Figure 2:
Figure 1: Visualization of PIORF rewiring in CYLINDERFLOW-TINS (a) Blue areas indicate po- tential bottlenecks. Red circles (O) denote critical bottleneck nodes. (b) The black circle (O) denotes the highest velocity node. PIORF connects bottle- neck nodes (O) with high-velocity nodes (O).
INTRODUCTION
Solving the Havier-Stokes equations that govern fluid dynamics remains an open problem. In the absence of an analytical solution, most studies use numerical methods, representatively, finite ele-
Equaal contribution.
Corresponding author.

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Published as a conference paper at ICLR 2025
ment methods (FEMs) (Madenci & Guven, 2015; Stolarski et al., 2018; Abaqus, 2011; Dhatt al., 2012) to discretize differential equations spatially and temporally to account for complex physics. To optimize computational resources while maintaining accuracy in simulations involving unstructured surfaces, mesh refinement techniques are commonly used. These methods allocate higher resolu- tion regions of interest that require more detailed analysis, such as areas with steep gradients or complex geometries. While this approach balances computational cost with simulation accuracy, it results in a complex and irregular mesh structure (Löhner, 1995; Liu al., 2022).
The high computational cost of traditional numerical solutions has sparked interest in data-driven simulators based on graph neural networks (GNNs). Graph machine learning approaches, par- ticularly MeshGraphNets (MGNs) (Pfaff et al., 2020), have shown promising results in modeling physical systems on unstructured meshes. So far, studies using MGNs have shown accurate predic- tions for various physical systems Jenchez-Gonzaler et al., 2020; Fortunato et al., 2022; Yu et al., 2024). However, these methods face the challenge of capturing the long-range dependence of fluid flows, which is essential for accurately simulating complex phenomena such as turbulence (Benzi & Toschi, 2023).
Mesh refinement and over-squa problem. The core problem in using GNNs for fluid dy- namics simulations lies in balancing mesh refinement and information propagation. To achieve accurate simulations, it is essential to use finer meshes, especially in regions with significant ve- locity gradients, such as in boundary conditions (e.g. walls, holes, inlets, and outlets) (Katz & Sankaran, 2011; Baker, 2005). However, this refinement introduces two critical issues: as infor- mation propagates through the graph, it is repeatedly compressed, leading to an 'over-squashing' problem (Alon & Yahav, 2021; Topping et al., 2021). The over-squashing occurs in areas of local mesh refinement (Imai & Aoki, 2006) and near boundary conditions where the mesh is non-uniform, resulting in some nodes having few neighbors. ii) As the mesh becomes finer, MGNs need to per- form more message-passing steps to propagate information over the same physical distance. This leads to 'under-reaching" problems (Fortunato et al., 2022), where the model struggles to capture interactions beyond a certain range. These issues are particularly pronounced in fluid dynamics simulations. As the mesh becomes finer, the challenges increase, creating a trade-off between the demand for high-resolution simulations and the capacity of GNNs to efficiently process the graphs.
Limitations of existing solutions. While several graph rewiring methods have been proposed to address over-squashing (Topping al., 2021; Karhadkar et al., 2022: Nguyen al., 2023; Black et al., 2023; Arnaiz-Rodrigeez et al., 2022), they typically consider only the graph topology. This approach is insufficient for fluid dynamics simulations, where the underlying physical phenomena play crucial role in determining important long-range interactions.
Main idea. To address these challenges, we propose Physics-Infommn Informed Ollivier-Riie Ricci Flow (PI- ORF)!, a novel method that incorporates physical quantities such as flow velocity into graph rewiring. PIORF uses graph topology and physical phenomena to reduce over-squashing and en- hance information flow. We use the Ollivier-Ricci curvature (ORC) (Ollivier, 2009) to identify bottleneck regions in the graph structure. Fig. 1 depicts the key idea behind our PIORF using a CYLINDERFFLOW-TINY simulation. The ORC distribution (Fig. I(a)) reveals potential bottleneck areas (blue regions), with red circles (O) marking nodes of minimum curvature. The velocity mag- nitude contour (Fig. 1(b)) shows areas of rapid fluid velocity changes, with the black circle (O) indicating the highest velocity node. Our approach connects these bottleneck nodes with nodes in high-velocity gradient regions, enabling long-range interactions and mitigating over-squashing.
Contributions.
Our contributions are summarized as follows:
To the best of our knowledge, we are the first introduce a rewiring method that considers both graph topology and physical phenomena for fluid dynamics simulations.
Our PIORF method shows excellent computational efficiency by adding multiple edges with a single calculation compared to existing rewiring methods.
We extend PIORF to handle temporal mesh graphs and apply it to dynamic simulation environ- ments such as the EAGLE dataset, demonstrating the scalability of PIORF larger mesh graphs.
OOr code is available here: https: / //g///ttttt/ thub. www.b..wo../eeeoooeeeot..ee..eee wwt.:/co//yyyoeee honp/yoydeeepppooff
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As shown in Fig. 2, PIORF consistently outperforms MGN model and other rewiring methods across 3 benchmark datasets, achieving up to 26.2% improvement.
RELATED WORK
2.1
MESH-BASED SIMULATION MODELS
Using GNNs to predict the results of complex physical systems is a popular area of scientific machine learning (SciML) (Li al., 2020; Michalowska et al., 2023; Belbute- et al., 2020; Mrowca et al., 2018; Li et al., 2019; 2018; Pfaff et al., 2020). Among them, MGN performs local message passing by re-expressing as graph from a mesh. The strength of MGN lies in its ability to use mesh-based representations commonly used in many commercial simulation tools to numeri- cally solve partial differential equations (PDEs). Instead of solving the PDEs directly, MGN learns variety of systems while incorporat-
the underlying dynamics from data and can be applied ing boundary conditions. However, in order to obtain a more accurate solution approximate, MGN often requires finer meshes. A larger number of nodes causes the GNN's under-reaching problem and requires more layers for effective long-range interactions, which reduces learning efficiency. To address this, recent studies have investigated methods enable long-range interaction by forming hierarchical structure (Fortunato et al., 2022: Cao et al., 2023) or using Transformer (Janny et al., 2023: Yu al., 2024). Fortunato et al. (2022) introduce a dual-layer structure designed to propa- gate messages at two different resolutions. Janny et al. (2023) proposes a clustering-bassee pooling method and performs global self-attention. Cao al. (2023) reviews the shortcomings of current pooling methods and proposes Bi-Stride Multi-Scale (BSMS), hierarchical GNN using bi-stride pooling. Yu et al. (2024) use hierarchical mesh graphs and has an ability to capture long-range interactions between spatially distant locations within an object.
2.2
-QURSSQUASHING AND GRAPH REWIRING METHODS
The issue of over-squa was initially identified by Alon & Yahav (2021) and has since emerged as significant challenge in GNNs when dealing with long-range dependencies. This phenomenon occurs when the information aggregated from a large number of neighbors is compressed into fixed-sized node feature vector, resulting in considerable loss of information (Alon & Ya- hav, 2021). Several approaches have been studied to address the over-squashing problem in GNNs (Finkelshtein et al., 2023; Shi et al., 2023; Errica et al., 2023; Choi et al., 2024; Fesser & Weber, 2024; Choi et al., 2025). While alternative message-passing strategies, such as expanded width-aware message passing (Choi et al., 2024), have gained attention, graph rewiring - adding or removing edges has been the most actively proposed (Gasteiger et al., 2019; Topping et al., 2021: Nguyen al., 2023; Rodrigue al., 2022; Karhadkar al., 2022: Black et al., 2023; Banerjee et al., 2022: Attali et al., 2024). Gasteiger et al. (2019) propose DIGL rewiring method that computes kernel evaluation and sparsification of the adjacency matrix. DIGL smooths the ad- jacency of the graph. which makes it tend to connect nodes at short distances (Coifman & Lafon, 2006). However. this makes it not suitable for tasks that require longer diffusion distances. Top- ping al. (2021) propose curvature-based graph rewiring strategy. This method identifies edges with minimal negative curvature and adds new edges around them. First-order spectral rewiring (FoSR) proposed by Karhadkar et (2022) calculates the change in spectral gap due to edge addi- tion and selects the edge that maximizes the gap. Nguyen et al. (2023) propose batch Ollivier-Rica flow (BORF) using ORC simultaneously solve the over-smoothing and over-squashing problems. BORF works in batches and calculates the curvature with minimum and maximum in each batch. Then, connections are added to the set with the minimum edge value to uniformly weaken the graph bottleneck. BORF does not recalculate the graph curvature within each batch, but rather reuses the already computed optimal transfer plan between sets to determine which edges should be added. Recently, Attali et al. (2024) alleviate over-squahhig by using Delaunay triangulation, but this is not appropriate because mesh-based simulations are already constructed by the triangulation.
Despite interest in over-squashing in GNNs, over-squashing in mesh-based GNNs such MGN remains unexplored (See Table 5). Since mesh structures have different characteristics from graph structures used in existing research, and existing rewiring methods define bottlenecks for graph topologies from a geometric perspective, it is necessary to verify that existing rewiring methods are suitable for mesh graphs with certain number of distributed edges.
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3 PRELIMINARIES
3.1 MESHGRAPHNETS (MGN)
MGNs (Pfaff al., 2020) are class of GNNs designed for mesh-bases simulation, using an framework. The encoder encodes as multigraph, the nodes of the mesh converted to graph nodes, and the mesh edges become bidirectional mesh-edges. The proces- sor updates all node and edge embeddings by performing multiple message passing along the mesh edges through multiple GraphNet blocks (Sanchez- Gonzalez et al., 2020). Finally, the decoder pre- dicts the subsequent state by using the updated latent node representations.
Prooder-roocessor-DDe
Encoder. The mesh M at time is transformed into a graph G = (V,8), where the mesh nodes become graph nodes v; E V. and the mesh edges become bidirectional edges (ž,j) E E. For each edge, we define the mesh edge feature mij, which encodes connectivity information. The edge features derived from the relative displacement vector Xij x; x, and its norm |x,yl- Node features include the velocity w, and the node type n, which indicates the boundary conditions. The input and output characteristics for each dataset detailed in Appendix C.2.
Processor.
The processor consists of several GraphNet blocks. Each block sequentially updates node and edge embeddings through message passing operations. v! and ei, denote the node and edge embeddings at layer I, respectively. The update equations are:
f v v)),, 9
1-1
vtt fv Σ Ee
(I)
JEN,
where JE and fv are learnalee functions parameterized multi-layer perceptrons (MLPs), and W, denotes the set of neighbors of node i.
Decoder and updater. To predict the next time state from the current time, an MLP decoder is
used predict one or more output features oj, such as the velocity gradient w, density gradient p; and the next pressure p;- The velocity gradient is used to calculate the next velocity w 4+1 through an updater, which performs first-order integration (W++1 w w().
Training loss.
Following the MGN approach, the training loss uses the mean squared error (MSE):
M
C=
Σ
( ++1
w
22
+
Σ
fot
21t1)2
(2)
1=1
where |V] is the number of nodes.
3.2 OLLIVIER -ICCII CURVATURE ON GRAPHS
Ricci curvature, fundamental concept in differential geometry, describes the average dispersion of geodesics in the local region of Riemannian manifold. In the context of graphs, ORC (Ollivier, 2009) extends these concepts to graphs and considers random walks between nearby points using Wasserstein distances between Markov chains.
Given graph G (V,E) and a pair of nodes i, E V, ORC x(i,)) of edge (i.j) E E is defined as:
<(i, = Wi (m, my)
(3)
d(i,j)
where d(ū,j) is the shortest-path distance between nodes i and m; is probability distribution of I-step random walk from node i, and W1 is the Wasserstein distance order 1. For a node E V, (p) represents the probability that random walker starting at i will reach in one step. The Wasserstein distance W, (m, my) between probability distributions m, and m, is defined as:
(im;;,m)) inf =e)(mm..m))
Σ x(D.0)((p.0)
(4)
(n-g)Ev2
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where [[(,; my) is the set of joint probability distributions with marginals m, and my.
ORC quantifies the variance of a geodesic and has positive, negative, and zero values. When is O (x(i.j) 0), the geodesics tend to remain parallel, when it is negative value (x(i, < 0), they diverge, and when a positive value (m((), > 0), they converge. ORC on edges with high negative values is known to cause over-squashing (Topping et al., 2021). Equation (4) requires defining the probability distribution for function neighbor nodes. Since the radius of the neighbor nodes in the graph is I, given one-step random walk m from node node p is defined as:
p No
mx(p)
dog(s)
(5)
0
otherwise,
where dig(1) is the degree of node which means the number of element in MG.
4 PIORF: PHYSICS- OLLIVIER-RICC FL.w
In this section, we introduce PIORF, a novel rewiring method to improve long-range dependencies.
Design Goals. Our proposed method is designed with the following 3 goals:
(Physical Context) The method should incorporate physical quantities (e.g., velocity) with topol- ogy (e.g., ORC) to improve long-range interactions.
(Efficiency) The computational cost of adding new edges should be lower than that of existing rewiring methods.
(Accuracy) The prediction error should be lower than that of other rewiring methods.
To achieve these goals, PIORF selects nodes hased on their topological properties and physical quantities. We extend the ORC to node-level curvature, denoted % for node i. This node curvature "/ is computed as:
Rewiring with PIORE.
k(i,j). JENN
(6)
L
The rewiring proceeds as Algorithm 1, of which the key steps are described as follows:
i) PIORF selects [8|V]] nodes with the lowest "/ from V in order to form set S, that |S| |8|| where 8 E (0, 1) is the pooling ratio. is our sole hyperparameter.
ii) For each E S, PIORF computes the Euclidean distance d(w,, w/) between velocities w. and Wi for all nodes i E V/ s.
iii) For each E S, PIORF identifies nodes arg max d(ws, ws) with the largest veloeity differences and defines their set as Rgg
iv) PIORF adds bidirectional edges (s, r) and (r, 8) to the graph & for all s E S and r E R.-
For the sake of convenience in explanation, the physical quantity used in PIORF is described based on the use of velocity from the node features. By integrating both physical and topological proper- ties, PIORF enhances long-range interactions and mitigates over-squashing. The detailed description of the notations used in the formulas written so far summarized in Appendix E.
Algorithm Physics- Informed Ollivier-Rieci Flow (PIORF)
1: Input: A graph 9 (V, pooling ratio 8 E (0, velocity w; of node i
2: Output: Rewired graph g' (V,E*)
3: Calculate the ORC, 7/ for all nodes in V using Equation (6)
4: Selects | nodes with the lowest ? in order to form a set S, where |S| = [
5: For each node
S, calculate the Euclidean distance d(w., ws) between velocities of and all other nodes E V s.
6: Find the node r arg max (ww,, wi) with the largest velocity differences. Add bidirectional edges (s, and (r,, s) g'.
8: return G' (V,E")
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Physical interpretation. In fluid dynamics, the distinction between laminar (Schubauer & Skram- stad, 1947) and turbulent (Mathieu & Scott, 2000) flows, as quantified by velocity and the Reynolds numbers (Lissaman, 1983), is important for understanding system behavior. The relationship be- tween velocity and pressure is described through the rate of change and is explained by the Navier- Stokes equations (Temam, 2001). The velocity refers to the speed which fluid moves specific point in space. The pressure is the force exerted by a fluid per unit area on the surfaces. PIORF in- tegrates this physical context by adding edges between nodes with significant velocity differences. This allows the model to help with long-rang interactions to better simulate real-world phenomena such fluid turbulence. Physically, connecting nodes with large velocity differences indicates re- gions of instability. Unlike existing rewiring methods, PIORF ensures that the rewiring process takes on the actual physical context of the system, leading to physically meaningful signal propagation. With this physical insight, PIORF can improve long-range interactions and prediction performance in physics-basl based simulations.
Computational efficiency. Unlike existing rewiring methods, which rely greedy algorithms to iteratively add edges based on their objective functions (Karhadkar et al., 2022; Black et al., 2023), PIORF introduces
efficient approach. PIORF identifies nodes with significant differences in physical quantities and adds new edges in a single pass. This avoids the high computational cost of iterative edge addition and, thus, improves scalability. We show that PIORF is more efficient than other rewiring methods in rewiring new edges in Section 6.3.
5 DISCUSSION
In this section, we analyze mesh graphs and distinguish our method from graph pooling techniques.
75
Cyllnder Flow
WALL
Airfoil
pegres
Percent
50
25
-0.2
0.0
0.2
0.4
3
4
5
6
7
8
Curvature
Degree
(a)
(b)
(c)
Figure 3: Structural analyses mesh graphs: (a) Correlation between ORC and node degree in training dataset of CYLINDERFLOW, revealing potential information bottlenecks. (b) Node degree distribution across datasets, showing the prevalence of degree-6 nodes in uniform regions. (c) Non- uniform mesh refinement near boundary conditions.
Analysis of mesh graphs. We use ORC analyze the topology of mesh graphs of fluid dynamics benchmark datasets. This analysis reveals several key insights:
Fig. 3(a) shows strong negative correlation between ORC and node degree. This relationship identifies potential information bottlenecks in the mesh graph, particularly in high-degree nodes.
Fig. 3(b) indicates a prevalence of degree-6 nodes uniform regions, typical of Delaunay trian- gulation (Weatherill, 1992). However, boundary condition nodes (e.g., holes, walls, inlets, and outlets) show lower degrees due to their sparse distribution, as shown in Fig. 3(c).
In computational fluid dynamics (CFD) (Anderson Wendt, 1995). local mesh refinement often applied to enhance accuracy in specific areas. This process leads to gradual transition from fine meshes near boundaries to coarser meshes, resulting non-uniform structures (Fig. 3())..
These findings emphasize the relationship between the mesh configuration, boundary conditions, and the risk of information bottlenecks GNNs used for fluid dynamics simulations. Fig. 8 in Appendix B shows the ORC distribution obtained through this information for each dataset.
Pooling and rewiring methods in mesh graphs. Due to mesh graphs with more than thousands of nodes, node pooling techniques (Fortunato al., 2022; Cao al., 2023; Yu et al., 2024) are widely

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used to reduce computational complexity and enhance the capture of long-range interactions. We extend the application of our PIORF beyond MGN to hierarchical models such as BSMS (Cao et al., 2023) and HMT (Yu et al., 2024). While these models already incorporate pooling effectively reduce the number nodes, we hypothesize that applying PIORF to the pooled structures could further optimize edge connections. This integration of pooling and rewiring aims refine capacity of the model to represent complex physical relationships across different scales. In Section 6, we explore whether this combination can yield additional improvements in fluid dynamics benchmarks.
6 EXPERIMENTS
6.1 EXPERIMENTS ON FLUID DYNAMICS BENCHMARK DATASETS
Datasets. We evaluate our method on two publicly available datasets: CYLONDERFLOW and AIR- Foir.. Both datasets follow the Navies-Stee Stookss equations (Temam, 2001), but differ in their flow characteristics. CYLINDERFLOW shows laminar flow behavior. In contrast, AIRFOIL represents turbulent flow model with high velocity, where fluid particles move irregularly in time and space. (See Appendix B for detailed description of datasets.)
Setting. We compare our PIORF against rewiring methods: DIGL (Gasteiger et al., 2019), FoSR (Karhadkar et al., 2022), SDRF (Topping et al., 2021), and BORF (Nguyen et al., 2023) These are apllied to 4 different models architectures: MGN (Pfaff et al., 2020). BSMS (Cao al., 2023), Graph Transformer (GT) (Dwivedi & Bresson, 2020) and HMT (Yu et al.. 2024). BSMS is hierarchical GNN and HMT a hierarchical Transformer. For MGN, we use 15 blocks. For optimal performance, BSMS is set to level 7 for CYLINDER WELINEERFF.OO FL. and level 9 for FOR Detailed hyper- parameters for all baselines are provided in Appendix C. Each experiment repeated 5 times with different random seeds. All experiments are performed on NVIDIA 3090 and Intel Core-i9 CPUs.
Results. Table shows comprehensive performance comparison of different rewiring methods across the model architectures. PIORF consistently outperforms other rewiring baselines when applied to MGN, BSMS, GT, and HMT models. For CYLINDERFLOW, PIORF achieves the lowest RMSE in both velocity and pressure when applied to MGN. This improvement is especially sig- nificant compared to MGN and other rewiring methods. For AIRFOIL, PIORF achieves the best performance in all cases. Fig. 4 shows the superiority of PIORF by showing velocity magnitude contours at the final timestep. Our PIORF results closely align with the ground truth, especially in regions marked by black boxes.
Sensitivity to pooling ratio. We analyze sensitivity to pooling ratio 8, which is our sole hyper- parameter and determines the number of new edge connections. Fig. 5 shows how rollout RMSE varies with 5 for both datasets. Velocity RMSE of CYI.NDERFFLOW is optimal at 3%, while pres- sure RMSE generally improves with higher ratios. For AIRFOIL, velocity RMSE is best at 7%, and pressure RMSE at 7%. Across all cases, 1% pooling ratio often performs worse than MGN, while 9% increases standard deviations. The results show the need to tune å specfically for each dataset. Fig. (a) shows the Velocity RMSE of CYLINDERFLOW, and it can be seen that the average and standard deviation of RMSE increase at 9% where a large number of edges are connected.
6.2 SCALING TO LARGER FLUID DYNAMICS
Table 2: Comparison of fluid dynamics datasets
Datasets. To evaluate scalability and effi ciency of PIORF, we use EAGLE (Janny et al,, 2023), which simulates turbulent flows created by drones in various scenes. As shown in Table 2. EAGLE significantly sur- passes FYLWNDERFLOW and AIRFOIL in scale and complexity. EAGLE has dynamic meshes (Malcevic & Ghattas, 2002: Jasak, 2009), where the mesh positions and boundary conditions change at each time step. This dynamic nature requires temporal graph rewiring, presenting a more challenging and realistic scenario compared to the static meshes of CYLINDERFILON and FoII.
Dataset
Size
Dynamic Scene Dynamic Mesh
CYLINDERFLOW 15GB
AIRFOIL EAGLE
56GB
X
270GB
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Table I: RMSE (rollout-all, x10*) for our PIORF and other rewiring methods.
CYLinDERFF.Ow
AIRFon.
Method
Velocity
Pressure
Velocity
Pressure
Density
MGN
48.8 +: 5.6
36.7 + 2.4
10,261 + 832
3,043,186
282.514
29,4 + 2.7
DIGL
62.0
46.0 + 0.4
11,534 +
3,495,260
252,832
33.6
2.2
SDRF
43.0 + 3.0
35.5 1 1.0
10,714 +: 11,068 + 377
3,238,730 183,094
31.1
1 1.9
FoSR
43.7 = 3.2
35.0 + 1.2
3,314,506
164,026
31.9 = 1.5
BORF
48.5 = 7.8
36,9 = 2.2
10,029 + 410
2,884,555
186,003
28.1 = 1.8
PIORF
41.6 1: 3.9
28,9 1. 1.5
7,743 584
2,245,858
142,452
22.5 1: 1.4
BSMS
78.7 + 2.8
50.7 + 2.2
10,883 + 460
2,640,398 158,480
26.5 + 2.1
DIGL
237.8 1: 7.3
163.6 1: 8.5
40,312 +: 3,936
8,218,660 +: 1,2811200
81.3 d: 11.5
SDRF
78.0 +
50.7 d: 1,9
36,539 3,980
7,426,023 642,555
74.2 +: 8.1
FOSR
82.2 +
52.3 d:
41,831
8,490,283 352,622
84.0 +:
BORF
84.9 +
54.2 +: 1.6
10,750
2,632,487 126,177
25.8 1.2
PIORF
76.9
50.6 + 2.4
10,482 ::
2,584,690 163,680
25.4
GT
54.3 + 7.3
40.0 + 2.0
10,002 +:
2,979.573
29.0 0.9
DIGL
68.4 3.6
49.5 + 1.0
11,004 +
3,331,160 192,098
32.7 + 1.9
SDRF
52.1 9.1
39.0 + 1.3
10,354 + 610
3,120,743 179,587
30.1
- 1.9
FoSR
50.7 8.7
39.3 + 1.6
11,211 = 868
3,415,094 312,517
33.6 = 3.1
BORF
58.9 9.7
40.6 + 2.4
9,830
2,883,648
136,064
28.6 1.3
PIORF
48.5 = 4.5
31,3 + 2.3
7,429 778
2,124,920
130,279
21.4 = 1.2
HMT
71.0 1: 1.2
51.1 +: 1.5
5,303 1: 414
1,251,955 1: 79,764
12.8 1: 0.8
DIGL
76.3 1.7
54.0 1 0.7
5,176 409
1,232,486 1: 79,250
12.5 1: 0.8
SDRF
71.0
51.3
32,695
7,579.699
247,004
74.4 1:
FOSR
72.1 =
52.3 +
35,474
1,011
8,137,115 231,038
79.4 - 2.0
BORF
74.2 3.1
53.6
5,591 + 416
1,306,555
+ 82,509
13.3 = 0.8
PIORF
70.9 + 1.6
50.9 + 0.8
4,961
378
1,182,495 + 67,499
12.1 - 0.7
(a) Ground Truth
(b) MGN PIORF
(c) MGN BORF
(d) MGN
(c) Ground Truth
(f) MGN + PIORF
(g) MGN + BORF
(h) MGN
Figure 4: Comparison of 2D cross-sectional velocity magnitude contours for CYLINDERFLOW (a)- (d) and AIRFOIL (e)-(h) the last time step with the largest cumulative error. It is most similar to ground truth when PIORF is applied. The closer the color is to red, the faster the velocity. The black boxes (D) highlight regions where PIORF shows particular accuracy in predicting complex flow structures. PIORF consistently achieves the closest match to ground truth on both datasets. More rollout images can be found Appendix D.
Setting. We use MGN with 15 layers and maintain the same baseline rewiring methods, adjusting only dataset-specific hyperparameters. We set the velocity noise standard deviation to 0.02 in all methods. DIGL is set with a alpha to 0.01 and eps at 0.4. For SDRF, set maximum of 10
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Ie3
le9
RMSE (Velocity)
RMSE (Pressure)
RMSE (Pressure)
RMSE (Density)
50
35-
30
45-
30
2.5
25
35
Meen MGN
25
2.0
20
Pooling Ratio (%)
Pooling Ratio (%)
Pooling Ratio (%)
Pooling Ratio (%)
(a) CYLINDERFLOW
(b) CYLINDERFLOW
(c) AIRFOIL
(d) AIRFOIL
Figure 5: Sensitivity pooling ratio 8. The dashed lines represent RMSE of MGN without rewiring.
iterations and edge removal, and for FoSR, iterations. To ensure statistical significance, we repeat each experiment 5 times with different seeds.
initial power of and maximum of 20
Table 3: Rollout-all RMSE (x 1x10))
Results. The timeout of BORF highlights the computational challenges in applying rewiring to the saneesccae dataset. As shown in Table 3, our PIORF outperforms all baselines, achieving 14.5% improvement in velocity RMSE over MGN. While other rewiring methods such as SDRF and FoSR show some improvements, they are significantly smaller compared to PIORF. Fig. 1l in Appendix D shows the result of the last step with different rewiring methods applied.
EAGLE
Model
Velocity
Pressure
MGN
2,280 + 135 10,893 + 632
DIGL
2,623 +:2 114 12,688 1 698
SDRF
2,186 :: 10,504 1: 297
FoSR
2,254
10,755 1.
BORF Time-out
Time-out
PIORF 1,950
9,449 167
6.3 COMPUTATIONAL EFFICIENCY
Given the large scale of mesh graphs, with thousands of nodes and tens of thousands edges (see Table Appendix B), we need add a large number of edges to alleviate over-squashing. However, existing rewiring methods require multiple iterations to add or delete edges, leading to increased computational overhead. Fig. 6 shows the computation time required to add varying numbers of edges when rewiring one trajectory in the CYLINDERFLOW, AIRFOIL, and EAGLE datasets. PIORF maintains the lowest computation time in all datasets and edge counts. This due to the ability of PIORF to compute all the necessary rewiring in single pass, avoiding an it- erative process. In contrast, BORF shows steep increase computation time as the number of added edges grows, particularly evident in EAGLE. Although SDRF and FoSR more efficient than BORF, they still show a trend of increasing computational time, emphasizing the scalability advantage of PIORF in handling large-scale fluid dynamics simulations.
10
10c
SDRP
SDRF
10²
PoSit
Computation Time
Computation Time
1o
BORP
202
PIORE
20
PIORF
PIORF
Computation
201
10a
202
10a
10a
30
100
150
200
50
100
150
200
20
40
60
Number of added edges
Numbar of added edges
Number of added edges
(a) CYLINDERFLOW
(b) AIRFOIL
(c) EAGLE
Figure 6: Comparison of computation time as the number of edges added increases.
6.4 ABLATION STUDIES
We conduct ablation studies to evaluate components of PIORF and Table summarizes findings.
Choice of physical value for rewiring. We analyze the impact of using velocity or pressure to identify nodes for edge rewiring in PIORE, For CYLInDERFI.OWw incompressible flow (Panton,

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Table 4: Rollout-all RMSE (x10*) for PIORE and the ablations.
Actions
CYLINDERFL.OW
AIRFoIL
Ablation Model
Add Remove
Velocity
Pressure
Velocity
Pressure
Density
MGN
48.8 + 5.6
36.7 = 24
10,261 + 832
3,043,186 = 282,514
29.4 = 2.7
Velocity
41.6 +:
28.9 + 1.5
7,743
2,245,858
142,452
22.5
Pressure
80.9 13.4
75.8 15.3
7,768 288
2,293,481
108,098
23.1 +
Random
43.4 +: 2.4
32.3 4: 1.2
10,317 +: 771
3,115,406 1: 230,796
30.2 1: 2.2
Only Removal
'
42.0 +: 2.3
36.8 + 0.5
10,890 +
3,289.964
31.7 + 0.8
Both
49.0 d: 7.5
31.1 + 2.9
7,813 551
2,334,583 = 182,600
23.4 1: 1.9
Weighted Edges
53.2 1: 8.6
44.1 1: 4.5
11,214 1: 3,486,655 1: 203,277
32.9 1: 1.6
To Senders
53.4 + 1.2
35.8 + 1.0
10,358 + 866
3,099,548 :: 320,553
29.7 :: 2.8
To Receivers
47.9 4.9
35.0 +:
10,421 + 704 3,132,703 = 119,582
30.3 = 1.2
2024), velocity-based rewiring significantly outperforms pressure-based rewiring. This aligns with Bernoulli's 's principle for incompressible flows, where velocity changes more indicate key flow dy- namics. For FOR,, compressible (Saad, 1985) and turbulent flow (Mathieu & Scott, 2000), pressure-based and velocity- -ased rewiring performs well and outperforms other rewiring methods.
Effect of physical-informed node selection. PIORF selects the nodes based on ORC-identified bottlenecks and nodes with high physical changes. To assess the impact of using physical values in this selection process, we compare our approach ("Velocity") with a modified version (" ("Random") where nodes are chosen based on ORC bottlenecks but the second node is selected randomly. ignor- ing physical values. Results show that physics-ifformed informed selection outperforms random selection.
Effect of edge addition/removal. We analyze the effects of edge addition ("Velocityy"), removal ("Only Removal"), and both ("Both"). Removal ("Only Removal") removes the edge with the high- est positive curvature. Interestingly, edge addition alone yields the best performance for all datasets, suggesting that adding new edges is beneficial than removing existing ones.
Weighted edges. We explore the impact of weighted edges by the L2 distance of velocity when calculating ORC in Equation (3) and Equation (4). The "Weighted Edges" results indicate that this approach does not improve performance. It means that binary edge existence might sufficient for capturing relevant physical relationships.
Directionality rewiring. We dissect the effect of directional rewiring by adding one-way edge sets. 'To Senders" is when aggregation is performed from receivers to senders, while "To Receivers" the opposite. The results show that bidirectional rewiring outperforms unidirectional approaches.
CONCLUSIONS
We introduce PIORF as new rewiring method that simultaneously considers the topology and physical correlation of the mesh graph and experimentally demonstrate best performance in the field of physics mesh simulation. Moreover, we show for the first time that applying our rewiring method to hierarchical GNNs and Transformer also improves model performance in mesh graph.
Limitations and future work. One limitation of PIORF is its dependence on the choice of phys- ical values for rewiring. Future research could focus on developing adaptive mechanisms for se- lecting the most relevant physical quantities automatically. Another important direction to extend PIORF to handle dynamic adaptive mesh refinement (Bangerth & Rannacher, 2003; Cerveny et al., 2019), which could include integrating PIORF with error estimation techniques that enable more targeted refinement in areas with large solution errors. Additionally, extending PIORF to appli- cations in multi-body dynamics (Choi et al., 2013), equivariant graphs (Satorras et al., 2021), 1), and particle-based simulations (Li et al., 2018) is important of future work.
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ETHICS STATEMENT
Our proposed PIORF is rewiring method designed for modeling physical systems on unstructured meshes, and thus poses clear negative societal or ethical implications. However, potential misuse or application of the algorithm in unintended areas could result in unintended consequences.
Additionally, this paper may have implications regarding the carbon footprint and accessibility of learning algorithms. Recently, as the computational demands in machine learning research have grown, they have led to an increasing carbon footprint. Our proposed method contributes re- ducing this carbon footprint by not only improving performance but also enhancing computational efficiency in such contexts.
REPRODUCIBILITY
We provide the source code for our experimental environments and proposed method. In future, we intend to make this source code available for the benefit of the community. PIORF source code can be found in the following: https: //t::///iii /gitttb.......... com/ yuyudeep/piors
PIORF has single hyperparameter, the pooling ratio 8. The best hyperparameter option for repro- duction in each dataset is described in Section 5, along with sensitivity analysis. Additionally, the experimental settings for the proposed method and baseline can be found in Section 6.1, Section 6.2, and Appendix C.
ACKNOWLEDGEMENTS
This work was supported by the LG Display and an IITP grant funded by the Korean government (MSIT) (No. RS-2020-11201361, Artificial Intelligence Graduate School Program (Yonsei Univer- sity)). K. Lee acknowledges support from the U.S. National Science Foundation under grant IIS 2338909.
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Supplementary Materials for "PIORF"
Table of Contents
A Comparison of Rewiring Methods and Complexcity
15
B Datasets Details
16
Baseline Details
17
C.I. Rewiring Methods
17
C.2 Models
17
D Other Variable Contour and Rollout Figures
19
E Notations
22
F Additional Ablation Studies
23
G Additional Discussion
23
Graph topology changes.
23
G.2 Effective Resistance on Graphs.
24
G.3 Relationship Between Accumulated Error and Velocity Gradient.
24
G.4 Discussion on ORC in PIORF
25
A COMPARISON OF REWIRING METHODS AND COMPLEXCITY
We further compare existing graph rewiring methods with our proposed method. As shown in Ta- ble 5, method, PIORF, takes the physical context into account, which other rewiring methods do not. This is key characteristic of our approach, which aims to overcome the limitations of existing methods for learning fluid dynamics simulations that are not designed for this purpose.
The complexity of PIORF ((E)) where [}] the number of edges and dmax is the maximal degree. In particular, simulation datasets in fluid dynamics use thousands nodes and edges to ensure solution accuracy, so the PIORF method is advantageous in applying more edges to these datasets. One of the biggest differences is the computational cost. Existing methods such as DIGL, SDRF, FoSR, and BORF incur significant computational cost in the process of selecting which edges to rewiring to optimize their own defined objective function (See Fig. 6). Our method, on the other hand, performs the rewiring without any objective function optimization, which beneficial in terms of computational cost. Another important difference is the number of hyperparameters. Existing rewiring methods typically require two more hyperparameters, while our PIORF uses only one hyperparameter, the pooling ratio. This has the advantage of reducing the hyperparameter search space.
of thousands of
Table 5: Comparison of different rewiring methods and our PIORE.
Methods
Indicator
Complexity
Geometry
Physics
SDRF
Balanced Forman curvature
O(E (1E)
FoSR
Spectral gap
O(V2)
x
BORF
Ollivier-Ricci curvature
OPE (
/
x
PIORF
Ollivier-Ricii curvature with physical context
O OIE d
/
d
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B DATASETS DETAILS
SYLINDERFLOWW-TE dataset used in Fig. is used to illustrate the concept PIORF and is not desinged for evaluation. We three public datasets for evaluation, and Table 6 shows information such as the number of cases, number of steps, number nodes and number of edges for each dataset. AIRFOIL and EAGLE datasets turbulent flow models, and CYLINDERFLOW is laminar flow model. Fig. 7 shows the velocity magnitude contour image of each datasets. In all datasets, the velocity is high in areas near boundary conditions such as walls. Fig. 8 shows the distribution of ORC by each dataset. When creating a mesh, nodes with high degrees occur due to local fine mesh and boundary conditions. Red circles are nodes where the degree is or higher and bottlenecks occur.
Table 6: Dataset description: Fluid dynamics behavior, number of trajectories for each data set, time step, and average number of nodes, edges, and cells in the training data set. A cell refers to an element and is small unit that makes up mesh. In the case of a triangular mesh, cell consists of three nodes.
Datasets
Behavior Cases (Train) Cases (Test) Steps Nodes (avg) Edges (avg) Cells (avg)
CYLINDERFLOW
Laminar
1,000
100
600
1,886
10,848
3,538
AIRFOIL
Turbulent
1,000
100
600
5,233
30,898
10,216
EAGLE
Turbulent
947
118
990
3,389
20,023
6,623
100
200
300
(a) CYLINDERFLOW
(b)
AIRFOIL
(c) EAGLE
Figure 7: Velocity magnitude contour image for each dataset. In all cases, changes in velocity occur in walls where fluid cannot flow.
CYL.NDERF1.OW-TIN CYL.NDERFFLOW the dataset used illustrate the concept of our PIORF for understanding the flow of fluid in narrow passages around a cylinder. To create CYLINDERFLOW-TIN dataset, we consider performing simulation modelling in an environment similar to that of CYLINDERFLOW. We use the Ansys Fluent solver (Stolarski et al., 2018) to generate the dataset. The number nodes is approximately 300 and the fluid input is air.
CYLINDERF1.OW CYL.INEEFFI.ON is important in many industrial applications, such the cooling of cylindrical pipes, by analyzing the flow of fluid around a cylinder. The flow can exhibit laminar or turbulent flow behavior depending on factors such flow rate, fluid density, and cylinder size. CYLINDERFLOW dataset (Pfaff et al., 2020) consists of 1,000 analysis results, with each case containing 600 time steps. The dataset contains single cylinder, but includes variety of Reynolds numbers, sizes, and positions.
FORF.. AIRFOIL an application of aerodynamics and the most basic CFD modeling. AIR- FOIr., also known as wings, is utilized in the design of airfoils and various other aerodynamic ap- plications such as aircraft, helicopters, and spacecraft. AIRFOIL plays a central role designing an airplane's wings to generate lift, control flight, and move through airflow. Moreover, it is very important to design an aerodynamic design that is effective in a specific range of flow conditions. AIRFOIL dataset (Pfaff et al., 2020) consists of 1,000 analysis results, with each case containing 600 time steps. The data set contains one AIRFOIL and various input conditions, such as velocity and pressure, with the fluid density changing every step.
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EAGLE. EAGLE is large-scale dataset for learning non-steady fluid dynamics. This is sim- ulation of the airflow generated by drone moving in 2D environment with various boundary shapes. It is much more difficult than other datasets such CYLINDERFLOW models the complex ground effect turbulence created by the drone's airflow according to its control laws. Different scene geometries produce completely different results, resulting in highly turbulent and non non-perionn periodic eddies and high flow diversity. In the field of learned simulators, EAGLE is the first apply a dynamic mesh effect in which the shape and position of the mesh change every time step. It accurately simulates fluid behavior by changing the drone's position over time.
AIRFOIL as
-0. 8.030-0.025 0.000 0.025 8.059 0.075
0.300
0.125
-0.04
-0.02
0.00
0.02
0.04
-0.04
-0.02
0.00
0.02
0.04
0.06
(a) CYLINDERFLOW
(b) AIRFOIL
EAGLE
Figure 8: ORC distribution image for each dataset. Red circles (O) are the nodes where the degree high and a bottleneck occurs.
Representative physical quantity. The velocity refers to the speed at which fluid moves at a specific point in space. The pressure is the force exerted by fluid per unit area on the surfaces. The density is a measure of how much mass is contained within a given volume of the substance. is defined as mass per unit volume. The density of fluid depends temperature and pressure. These three physical quantities are related by Bernoulli's equation. When density is constant, increasing velocity causes pressure decrease.
C BASELINE DETAILS
We compare four competitive rewiring methodologies and four models. For models, MGN (Pfaff et al., 2020), BSMS, GT, and HMT are used, along with rewiring methods such as DIGL (Gasteiger al., 2019), SDRF (Topping al., 2021). FoSR (Karhadkar al., 2022), and BORF (Nguyen al., 2023). For all datasets. training steps are set to 10,000,000. Velocity noise standard deviation is 0.02 and for CYL.NDERFI.Ow and FORF datasets, respectively.
C.I REWIRING METHODS
For DIGL, set alpha to 0.01 and 0.4 for eps. For SDRF, max number of iterations is 10. Edge removal is not used. For FOSR, initial power and max number iterations are set and 20, respectively. In the case of BORF, the deletion for each batch are set to 4 and 2, respectively. We use the official implementation released by the authors on GitHub for all rewiring baselines:
number of iterations is set to 10, and edge addition and
SDRF: o00.:tttttee//ttttooooo//////eee FoSR: http://///g/////a//////////////a
DIGL: https::pp//o
BORF: httptttttttt///t.t.ooo///aaaaaaa
C.2 MODELS
MGN. To align with the MGN methodology, apply 15 iterations of message passing in all datasets. All MLPs have a hidden vector size 128. Table indicates the input, edge, and output features used for each dataset.
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Table 7: Details of features for each dataset. P; is the fluid density and w, is the velocity of the fluid. w, is the gradient of velocity, is the node type, and X; is the position of the node.
Datasets
Inputs maj
Inputs V.
Outputs Oz
CYLINDERFL.OW
Xijo Xij
ni, Wi
Wi.Pi
AIRFOIL
Xijs Xij
By Wa Pi
Pi Pi
EAGLE
Xijo Xij
Dí Wi
Wi.Pi
We implement the BSMS model with Tensorflow. And according the best hyperpa- rameters of BSMS, levels 7 and 9 are used for CYLInDERFLOOw and AIRFOIL, respectively. Noise standard deviation is set the same as MGN. All MLPs have hidden vector size of 128. The En- coder/decoder are set to those in MGN.
BSMS.
The hidden dimension size inside its Transformer is set to 128. FFN used three linear layers and two ReLU activations. To ensure numerical stability, the results obtained with the exponential term within the softmax function are constrained to fall in the range of [-2, 2]. We use the FFN with- using positional encoding. There are 15 transformer blocks with 4 heads. The encoder/decoder are set to those in MGN.
GT.
HMT. Because contact edges are not used fluid datasets, we only use HMT among the mod- ules of the HCMT model. The hidden dimension size of HMT is set to 128, and both FFN and numerical stability are set to the same as GT. There are 15 transformer blocks with heads. The encoder/decoder are set to those in MGN.
We use the official implementation released by authors on GitHub for all baselines models:
MGN: https:://gttubb.ccmmmm
deleee-deeiinnen nd/ delepmindddeeemiieeeete dewpmindddnnddeeee research
troe/nasterrrmashqraapntsss
BSMS: htSSaa/tttt//tttto...oo////////p
GT: httt:////.tthhoooooooeeeeeeeeeee
HMT: http:/psr//ggtttut..o/o///dddeee
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D OTHER VARIABLE CONTOUR AND ROLLOUT FIGURES
Figs. to 11 are rollout images of CYLINDERFLOW, AIRFon. and EAGLE, from the last time step with the highest cumulative error.
Models
Traj.
Traj. 2
Traj.
Ground Truth
MGN
+ DIGL
FOSR
SDRF
BORF
PIORF
0.0
0.5
1.0
1.5
2.0
2.5
Figure 9: The velocity magnitude contours of various rewiring methods compared to the ground truth CYLWNDERFLLOW
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Models
Traj.
Traj. 2
Traj.
Ground Truth
MGN
DIGL
FoSR
+ SDRF
BORF
PIORF
50
100
150
200
250
300
350
Figure 10: The velocity magnitude contours of various rewiring methods compared to the ground truth at AIR FOIL.
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Models
Traj.
Traj. 2
Traj.
Ground Truth
MGN
+ DIGL
SDRF
FoSR
PIORF
2
6
10
12
Figure 11: The velocity magnitude contours of various rewiring methods compared to the ground truth EAGLE.
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E NOTATIONS
Table 8 outlines the key notations used the paper.
Table Notation summary.
Name
index
senders
i
receivers
j
ORC(edge)
<((,.))
ORC(node)
nodes
u,
the shortest distance
d(u, v)
distribution of 1-step random walk from node
ma
LI Wassertein transport distance
W1 Wi H4))
sets nodes sets of edges
V
8
ORC Pooling Ratio
5
Inputs(edge features)
m,y
Inputs(node features)
V;
Outputs
Edge Hidden Features
Updated Edge Hidden Features
Node Hidden Features
Vi
Updated Node Hidden Features
Node MLP
fV
Edge MLP
Number of nodes
V
Number of edges
[
Mesh Positions
X,
Relative Mesh Positions
X:j
Norm Relative Mesh Positions
Xis
Node Type
Velocity
Velocity Gradient
W
Predicted Velocity Gradient
Pressure
Pe
Predicted Pressure
P
Density
Density Gradient
Pi
Predicted Density Gradient
P
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Published as a conference paper at ICLR 2025
F ADDITIONAL ABLATION STUDIES
Our proposed rewiring method has node selection steps that depend on ingredients such as degree, ORC, and physical context. We conduct additional ablation studies evaluate performance across different ingredient selections. The pooling ratio for all experiments is 3%.
Table shows performance based on ingredient selection. The first step is select nodes based on curvature("Fommer" Algorithm lines 3-4), and the second is to select nodes based on physical context("Latter", Algorithm lines 5-6). We define the following four rewiring methods for ablation studies: i) "Ablation I", where the former refers to high degree and the latter to physics, ii) "Ablation 2". where the former refers to random and the latter to physics, iii) "Ablation 3", where the former refers to random and the latter to random, and iv) "Ablation 4", where the former refers to ORC and the latter to random.
In CylinderFlow, "Ablation 1" shows results with high-degree selection, achieving improved per- formance compared to MGN. However, underperforms relative to PIORF, as exhibits varying curvature values for the same degree. "Ablation 2" and "Ablation 4" show performance based on the choice of physical context and ORC, respectively. Both outperform MGN, and Physical Context provides slightly better performance than ORC. "Ablation 3" is the result of randomly selecting both the former and the latter and adding edges, and is similar to the performance of MGN.
Table 9: Rollout-all RMSE (x (x10²) for PIORF and the ablations.
Ingredient
CYLINDERFLOW
AIRFOIL
Model
Former
Latter
Velocity
Pressure
Velocity
Pressure
Density
MGN PIORF
48.8 + 5.6
36.7 1:2 24
10,261 +: 832
3,043,186 +: 282,514
29.4 +: 2.7
ORC
Physies
41.6 + 199
28.9 : 1.5
7,743 +: 584
2,245,858 +: 142,452
22.5 + 1.4
Ablation
Degree
Physics
44.9 5.7
33.3 111
10,379 + 607
3,065,807 + 276.373
29.6 + 2.6
Ablation 2
Random
Physics
44.6 + 0.8
31.1 1.0
10,150 + 505
2,936,397 + 177,037
28.4 + 188
Ablation 3
Random
Random
48.4 1.6
35.3
11,220 +: 538
3,305,957 + 176,642
31.9 +
Ablation 4
ORC
Random
43.4
32
10,317
3,115,406 + 230,796
30.2
G ADDITIONAL DISCUSSION
G.1 GRAPH TOPOLOGY CHANGES.
We analyze changes in graph topology in each dataset. Fig. shows a comparison of curvature distributions between the original graph and the graph using PIORF The graph constructed after applying PIORF shows the removal highly negative curvatures that cause bottlenecks (Topping al., 2021).
0.8
Original
Original
0.6
SDRF
Density
BORF PIORF
Density
0.6
BORF
0.4
PIORF
0.4
0.2
0.2
0.0
0.%
0.1
0.0
0.1
0.2
-0.1
0.0
0.1
0.2
Ollivier curvature
Ollivier curvature
(a) CYLINDERFLOW
(b) AIRFOIL
Figure 12: Comparison of curvature distributions between the original graph and the graph using PIORF. The æ-axis represents the Olivier curvature of the edges, and the plots show a kernel density estimate of the curvature distribution.
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G.2 EFFECTIVE RESISTANCE ON GRAPHS.
The effective resistance (Black et al., 2023) provides a metric for measuring over-squashing. We randomly pick up 10,000 sample graphs from each dataset and analyze the total resistance (the of the effective resistance between pairs nodes). Table 10 shows the total effective resistance results in the original graph and the graph after applying the PIORF method in each dataset. The total effective resistance is significantly reduced, which indicates that the bottleneck is alleviated and enables long-rand raagge propagation.
Table 10: Total resistance for our PIORF and baselines.
Methods
CylinderFlow
Airfoil
MGN
2,491,084
15,644.891
SDRF
2,487,198
15,628,620
BORF 2,398,661
15,403,
PIORF
1,653,709
10,140,834
G.3 RELATIONSIIIP BETWEEN ACCUMULATED ERROR AND VELOCITY GRADIENT.
In the field of dynamics learning simulations, such as MGN, the model iteratively predicts the next step. The longer the simulation steps, the more accumulated emro occurs during inference. Fig. 13 shows the change in accumulated error and the gradient of velocity for each step after applying PIORE. Areas with significant accumulated errors depend on the velocity gradient. In PIORE, which reflects this physical quantity, the overall accumulated error is reduced compared to the original.
0.35
6.04
6.30
6.65
6.00
0.00
0.0
(a) MGN step 200
(b) MGN step 300
(c) MGN step 400
6.12
6.25
6.00
0.3
8.04
62
(d) PIORF at step
(c) PIORF at step 300
(f) PIORF at step 400
0.75
0.29e
0359
0.075
0.89
0.00
0.099
0.000
0.00
(g) Gradient step 100
(h) Gradient step 200
Gradient at step 300
Figure 13: Comparison of accumulated error distribution and gradient magnitude velocity distri- bution according to PIORF application.
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G.4 DISCUSSION ON ORC IN PIORF
In studies proposing rewiring methods in the field of graph machine learning, metrics with various curvature concepts are typically used to optimize edge addition removal by maximizing their values (Topping et al., 2021; Nguyen et al., 2023). In physics simulation, alternatives to our ORC- based approach could include methods such as SDRF (Topping et al.. 2021) and BORF (Nguyen et al., 2023) that use different metries for optimization. For example, SDRF uses a balanced Forman curvature, which provides more conservative estimate compared to ORC (Nguyen et al., 2023, Lemma 4.1). While BORF similarly uses ORC for rewiring, our experiments in Table demonstrate why less suitable for physics simulation.
Metrics that can be presented as alternatives with similar role to ORC are Forman curvature (Sree- jith et al., 2016) and betweenness centrality (Barthelemy, 2004). However, these metrics not near the boundary conditions effectively. This region where fluid flow changes
capture the and is also crucial from a domain knowledge. While Forman curvature, based on the graph Lapla- cian, is casier and faster to compute than Ollivier-Rice curvature, it is less geometric (Ni et al., 2019). We choose ORC specifically because better captures geometric characteristic, particularly around boundary conditions where fluid flow changes dramatically. Betweenness centrality could be used for source node selection, its high complexity O([VIE)) and need for global information make it impractical for mesh graphs with thousands of nodes and edges.
25


Reviews:
[{"review_id": "7RxLQIg1Dq", "reviewer": "Reviewer_5Uy1", "summary": "In this paper, authors propose Physics-Informed Ollivier–Ricci Flow (PIORF) which builds on the Ollivier–Ricci Flow. This innovative rewiring method integrates physical correlations with graph topology to address the over-squashing problem, which traditional approaches often overlook by focusing solely on graph topology without considering underlying physical phenomena. The proposed method is designed with the following 3 goals: 1) physical context, 2) efficiency, 3) accuracy. Furthermore, authors extend the ORC to node-level curvature, which is the core of the RIOFR. Experimental results on 3 fluid dynamics benchmark datasets show that PIORF consistently outperforms baseline models and existing rewiring methods.", "questions": "1(W1). Inadequate theory compared to previous work(BORF)[1]. And authors complain that 'BORF works in batches and calculates the curvature with a minimum and maximum in each batch. Then, connections are added to the set with the minimum edge value to uniformly weaken the graph bottleneck. To save computation time, BORF does not recalculate the graph curvature within each batch, but rather reuses the already computed optimal transfer plan between sets to determine which edges should be added.', but there is no indication in this paper that there is an increase in efficiency compared to BORF\n\n2(W1). The addition of edges also changes the topology, which may lead to negative consequences that are not addressed in the paper.\n\n3(W1). Computational efficiency is demonstrated only through experimental illustrations of efficiency gains, with no accompanying theoretical analysis.\n\n4(W1). Contrary to the existing theory. Topping's work [2] suggests that a fully connected graph does not have over-squashing (maximum curvature). However, as shown in Figure 3(a), the authors claim that the degree of a node is negatively correlated with curvature. These two theories are clearly contradictory. The authors should clarify whether this discrepancy arises from specific conditions discussed in the context of fluid dynamics. The degree may not be correlated to the level of curvature.\n\n5(W2). There is no clear rationale for why adding edges between $s$ and $r$ would address the underlying physical phenomena. The authors only provide some intuitive insights from physics and experimental simulations, but these edges could potentially have a negative impact.\n\n6(W3). Legends should be provided, and the meanings of the colors should be clarified to help readers better understand the impact of color changes in the figures (Figure 1, 4, 7, 8, 9,10 11).\n\n[1] Khang Nguyen, Nong Minh Hieu, Vinh Duc Nguyen, Nhat Ho, Stanley Osher, and Tan Minh Nguyen. Revisiting over-smoothing and over-squashing using ollivier-ricci curvature. In In- ternational Conference on Machine Learning, pp. 25956–25979. PMLR, 2023\n\n[2] Jake Topping, Francesco Di Giovanni, Benjamin Paul Chamberlain, Xiaowen Dong, an Michael M Bronstein. Understanding over-squashing and bottlenecks on graphs via curvature. arXiv preprint arXiv:2111.14522, 2021", "limitations": "", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The authors introduce a novel Ollivier–Ricci Flow, termed PIORF, to address the over-squashing problem that neglects physical phenomena.\n\n2. The authors extend the ORC to node-level curvature.\n\n3. The effectiveness of the method is validated through experimental verification.", "weaknesses": "1. There is a lack of sufficient theoretical analysis compared with former works.\n\n2. The paper does not provide an explanation for how the added edges address the underlying physical phenomena.\n\n3. The legends in the figures are not clear.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "XS5gcHJBdm", "reviewer": "Reviewer_Jfxr", "summary": "The paper proposes a novel graph rewiring method when using GNNs to solve PDEs. The basic idea is to combine regions based on both graph topology (Ollivier-Ricci curvature (ORC)) and physical features defined on each graph node (e.g. velocity). The proposed method first identifies bottleneck regions, i.e., with low ORC at the nodes, and then connects them with nodes having a maximum difference in physical properties. It seems that the first step identifies well-connected graph clusters typically far away on the graph and then connects them through basically one or two other nodes, namely the one(s) having the lowest and/or highest chosen physical feature. This way, information can propagate quickly between distant regions. The presented experiments seem promising.", "questions": "See weaknesses.", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- Combining topological and physical features. ORC seems meaningful for identifying clusters. Combining it with the physical features also makes sense, as these are quantities of interest.\n- Performance on the presented benchmarks is good.\n- Overall, the manuscript is well written and has many ablations/baselines.", "weaknesses": "- **W1: Why ORC?** I get the point of wanting to connect clusters (according to the used library for computing the ORC, low ORC reveals \"bridges within clusters\": https://pypi.org/project/GraphRicciCurvature/), but I'm not convinced that this is the best way. Note that I don't say that it is not the right way, I just say that there is no discussion of alternatives. \n- **W2: Why not node degree?** If the ORC and the degree of a node are strongly correlated, there should not be much difference between using the degree of a node versus a low ORC. I strongly recommend ablating this design choice (**Ablation 1**). If both perform comparably, there is no reason to use ORC, as the degree of a node is a much simpler concept. Thus, Occam's razor would favor it, and talking about adaptation by other researchers, one would not have to use a separate library to compute curvatures.\n- **W3: 6.4 Ablations.** I was not expecting this amount of ablations regarding the second half of your algorithm (lines 5-6 of Algorithm 1, physical features), but I was definitely expecting more ablations regarding the first part (lines 3-4 of Algorithm 1, ORC). In particular, how about randomly picking $S$ nodes instead of the current lines 5-6 of the algorithm relying on the ORC (**Ablation 2**)? This would show the importance of using physical features for rewiring, which seems to be a major part of the approach.\n- **W4: Simplest baseline.** I appreciate your effort in the model comparison part! However, I'm missing the simplest possible rewiring technique: randomly pick the same number of bidirectional edges as you use ($S$ if I'm not missing something), and see how this model performs (**Ablation 3**). From what I know, the chosen baselines (DIGL, SDRF, etc.) are designed for very different downstream tasks, and I'm not surprised that they underperform.\n\nI'll be more than happy to increase my score upon adding the suggested Ablations 1-3.\n\n**Minor/Typos:**\n- L. 376: \"changes\" -> \"change\"?\n- L. 425: \"boxe\" -> \"boxes\"?\n- L. 462:  \"PIORF maintains the lowest computation time in all datasets and edge counts.\" Is this true? The orange FoSR line is nearly identical; by the way, it is also pretty hard to see the orange line, probably even impossible if I print the manuscript on paper -> please think of a better visualization (e.g. log y-axis?). Please fix the whole paragraph regarding which approach is \"the fastest\", as there seem to be two of them.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "v0o34adkzn", "reviewer": "Reviewer_isvf", "summary": "This paper introduces an enhancement in mesh-based graph network simulators by addressing the over-squashing problem using a novel physics-informed rewiring approach. Notably, the method is able to scale to large scale fluid simulations due to the controlled complexity in performing rewiring. Experimentally, the proposed approach achieves notable improvement over existing mesh-based graph networks across three benchmarks, showcasing its efficacy.", "questions": "Q1. Is the approach also applicable to particle-based simulations? e.g., [1].\n\nQ2. Could the approach be combined with other physical inductive biases like equivariance [2], [3], [4]? It would be interesting to have some of these discussions in the paper or as related literature.\n\nQ3. Could the approach be applied together with constraint-aware graph simulators like [5]?\n\nQ4. Are there any real-world dataset on which the model can be evaluated? Practitioners would be more exciting to see how the method performs in actual real-world scenarios.\n\n[1] Li et al. Learning particle dynamics for manipulating rigid bodies, deformable objects, and fluids.\n\n[2] Satorras et al. E(n)-equivariant Graph Neural Networks.\n\n[3] Huang et al. Equivariant Graph Mechanics Networks with Constraints.\n\n[4] Han et al. Learning Physical Dynamics with Subequivariant Graph Neural Networks.\n\n[5] Rubanova et al. Constraint-based graph network simulator.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The paper is well written and the proposed approach is easy to follow.\n\n2. The motivation is very clear with an elegant solution based on ORC.\n\n3. The discussions on the improved efficiency of the proposed approach over existing rewiring methods should be appreciated.\n\n4. Experiments are thorough and the results are well presented.", "weaknesses": "These are some minor weaknesses which do not substantially hurt the paper but would be good to add more discussions.\n\n1. The proposed approach is mainly developed upon mesh-based representations. Adding more discussions to whether this could be generalized to ubiquitous graph network simulators (e.g., with particle representations or rigid bodies) would enhance the contribution.\n\n2. Some metrics are not clear to readers, e.g., Pressure, Density. It would be helpful to provide some explanations on the actual physical interpretation of these quantities, in the context of the considered simulation environments.\n\n3. All datasets seem to be artificial and adding more experiments (if applicable) on some real-world datasets would significantly enhance the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "HZ0l72RA9d", "reviewer": "Reviewer_SBAf", "summary": "The paper introduces a method called PIORF that enhances long-range interactions in mesh graph neural networks for fluid dynamics simulations. It uses Ollivier–Ricci curvature (ORC) to identify bottleneck regions in the graph and connects these areas with nodes having high velocity gradients. The authors evaluate PIORF on three fluid dynamics datasets and choose multiple backbone GNNs to demonstrate its effectiveness. The results show that PIORF outperforms existing graph rewiring methods in fluid simulation.", "questions": "Overall, my concerns are as follows:\n\n1. Whether over-squashing exist in or significantly affect fluid dynamic learning?\n2. How does PIORF alleviate over-squashing? more insights into model designs.\n3. Does the performance improvement result from that PIORF alleviate over-squashing?", "limitations": "", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The paper studies the over-squashing problem in fluid simulation.\n2. The proposed methods consider both the graph topology and physical quantity.\n3. The authors conduct experiments on various backbones.", "weaknesses": "I appreciate the authors' effort in exploring the over-squashing problem in fluid dynamic learning. However, multiple issues are still underexplored from my point of view. The details are as follows:\n\n1. Most graph rewriting methods study the classification problem in graphs like citation networks and social networks. The significance of over-squashing in fluid simulation is underexplored.  Whether and how does it affect learning performance? One observation that raises such a question is that in Table 1, existing graph rewriting methods enhance errors in most cases, indicating that over-squashing might not be the key issue in learning fluid dynamics.\n\n2. For the method, connecting the nodes with the highest velocity gradient is straightforward. How does the author avoid connecting nodes with low influence but high-velocity gradients? \n\n3. What is the ratio of negative ORC? Do you consider it to determine the pooling ratio?\n\n4. The authors do not provide time complexity.\n\n5. From Table 4, Random and Only Removal also significantly reduces model errors. Do they alleviate the over-squashing problem?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)