Original: I SOMETRIC T RANSFORMATION I NVARIANT AND E QUIVARIANT G RAPH C ONVOLUTIONAL N ETWORKS
Revised: I SOMETRIC T RANSFORMATION I NVARIANT AND E QUIVARIANT G RAPH C ONVOLUTIONAL N ETWORKS

Original: Anonymous authors

Original: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Graphs are one of the most important data structures for representing pairwise relations between objects.
Revised: Graphs are one of the most important data structures for representing pairwise relations between objects.

Original: Speciﬁcally, a graph embedded in a Euclidean space is essential to solving real problems, such as physical simulations.
Revised: Speciﬁcally, a graph embedded in a Euclidean space is essential to solving real problems, such as physical simulations.

Original: A crucial requirement for applying graphs in Euclidean spaces to physical simulations is learning and inferring the isometric transformation invariant and equivariant features in a computationally efﬁcient manner.
Revised: A crucial requirement for applying graphs in Euclidean spaces to physical simulations is learning and inferring the isometric transformation invariant and equivariant features in a computationally efﬁcient manner.

Original: In this paper, we propose a set of transformation invariant and equivariant models based on graph convolutional networks, called IsoGCNs.
Revised: In this paper, we propose a set of transformation invariant and equivariant models based on graph convolutional networks, called IsoGCNs.

Original: We demonstrate that the proposed model has a competitive performance compared to state-of-the-art methods on tasks related to geometrical and physical simulation data.
Revised: We demonstrate that the proposed model has a competitive performance compared to state-of-the-art methods on tasks related to geometrical and physical simulation data.

Original: Moreover, the proposed model can scale up to graphs with 1M vertices and conduct an inference faster than a conventional ﬁnite element analysis, which the existing equivariant models cannot achieve.
Revised: Moreover, the proposed model can scale up to graphs with 1M vertices and conduct an inference faster than a conventional ﬁnite element analysis, which the existing equivariant models cannot achieve.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Graph-structured data embedded in Euclidean spaces can be utilized in many different ﬁelds such as object detection, structural chemistry analysis, and physical simulations.
Revised: Graph-structured data embedded in Euclidean spaces can be utilized in many different ﬁelds such as object detection, structural chemistry analysis, and physical simulations.

Original: Graph neural networks (GNNs) have been introduced to deal with such data.
Revised: Graph neural networks (GNNs) have been introduced to deal with such data.

Original: The crucial properties of GNNs include permutation invariance and equivariance.
Revised: The crucial properties of GNNs include permutation invariance and equivariance.

Original: Besides permutations, isometric transformation invariance and equivariance must be addressed when considering graphs in Euclidean spaces because many properties of objects in the Euclidean space do not change under translation and rotation.
Revised: Besides permutations, isometric transformation invariance and equivariance must be addressed when considering graphs in Euclidean spaces because many properties of objects in the Euclidean space do not change under translation and rotation.

Original: Due to such invariance and equivariance, 1) the interpretation of the model is facilitated; 2) the output of the model is stabilized and predictable; and 3) the training is rendered efﬁcient by eliminating the necessity of data augmentation as discussed in the literature (Thomas et al., 2018; Weiler et al., 2018;
Revised: Due to such invariance and equivariance, 1) the interpretation of the model is facilitated; 2) the output of the model is stabilized and predictable; and 3) the training is rendered efﬁcient by eliminating the necessity of data augmentation as discussed in the literature (Thomas et al., 2018; Weiler et al., 2018;

Original: Fuchs et al., 2020).
Revised: Fuchs et al., 2020).

Original: Isometric transformation invariance and equivariance are inevitable, especially when applied to physical simulations, because every physical quantity and physical law is either invariant or equivariant to such a transformation.
Revised: Isometric transformation invariance and equivariance are inevitable, especially when applied to physical simulations, because every physical quantity and physical law is either invariant or equivariant to such a transformation.

Original: Another essential requirement for such applications is computational efﬁciency because the primary objective of learning a physical simulation is to replace a computationally expensive simulation method with a faster machine learning model.
Revised: Another essential requirement for such applications is computational efﬁciency because the primary objective of learning a physical simulation is to replace a computationally expensive simulation method with a faster machine learning model.

Original: In the present paper, we propose IsoGCNs , a set of simple yet powerful models that provide computationally-efﬁcient isometric transformation invariance and equivariance based on graph convolutional networks (GCNs) (Kipf & Welling, 2017).
Revised: In the present paper, we propose IsoGCNs , a set of simple yet powerful models that provide computationally-efﬁcient isometric transformation invariance and equivariance based on graph convolutional networks (GCNs) (Kipf & Welling, 2017).

Original: Speciﬁcally, by simply tweaking the deﬁnition of an adjacency matrix, the proposed model can realize isometric transformation invariance.
Revised: Speciﬁcally, by simply tweaking the deﬁnition of an adjacency matrix, the proposed model can realize isometric transformation invariance.

Original: Because the proposed approach relies on graphs, it can deal with the complex shapes that are usually presented using mesh or point cloud data structures.
Revised: Because the proposed approach relies on graphs, it can deal with the complex shapes that are usually presented using mesh or point cloud data structures.

Original: Besides, a speciﬁc form of the IsoGCN layer can be regarded as a spatial differential operator that is essential for describing physical laws.
Revised: Besides, a speciﬁc form of the IsoGCN layer can be regarded as a spatial differential operator that is essential for describing physical laws.

Original: In addition, we have shown that the proposed approach is computationally efﬁcient in terms of processing graphs with up to 1M vertices that are often presented inphysical simulations that correspond to real problems.
Revised: In addition, we have shown that the proposed approach is computationally efﬁcient in terms of processing graphs with up to 1M vertices that are often presented in real physical simulations.

Original: Moreover, the proposed model exhibited faster inference compared to a conventional ﬁnite element analysis approach at the same level of accuracy.
Revised: Moreover, the proposed model exhibited faster inference compared to a conventional ﬁnite element analysis approach at the same level of accuracy.

Original: Therefore, an IsoGCN can suitably replace physical simulations regarding its power to express physical laws and faster, scalable computation.
Revised: Therefore, an IsoGCN can suitably replace physical simulations regarding its power to express physical laws and faster, scalable computation.

Revised: The corresponding implementation and the dataset are available online 1 .

Original: The main contributions of the present paper can be summarized as follows:
Revised: The main contributions of the present paper can be summarized as follows:

Original: • We construct isometric invariant and equivariant GCNs, called IsoGCNs for the speciﬁed input and output tensor ranks.
Revised: • We construct isometric invariant and equivariant GCNs, called IsoGCNs for the speciﬁed input and output tensor ranks.

Original: • We demonstrate that an IsoGCN model enjoys competitive performance against state-ofthe-art baseline models on the considered tasks related to physical simulations.
Revised: • We demonstrate that an IsoGCN model enjoys competitive performance against state-ofthe-art baseline models on the considered tasks related to physical simulations.

Original: • We conﬁrm that IsoGCNs arecomputationally efﬁcient and scalable to graphs with 1M vertices and achieves inference considerably faster than conventional ﬁnite element analysis.
Revised: • We conﬁrm that IsoGCNs are scalable to graphs with 1M vertices and achieve inference considerably faster than conventional ﬁnite element analysis.

Original:  R ELATED WORK
Revised:  R ELATED WORK

Original: Graph neural networks.
Revised: Graph neural networks.

Original: The concept of a GNN was ﬁrst proposed by Baskin et al.
Revised: The concept of a GNN was ﬁrst proposed by Baskin et al.

Original: ; Sperduti & Starita (1997) and then improved by (Gori et al., 2005; Scarselli et al., 2008).
Revised: ; Sperduti & Starita (1997) and then improved by (Gori et al., 2005; Scarselli et al., 2008).

Original: Although many variants of GNNs have been proposed, these models have been uniﬁed under the concept of message passing neural networks (Gilmer et al., 2017).
Revised: Although many variants of GNNs have been proposed, these models have been uniﬁed under the concept of message passing neural networks (Gilmer et al., 2017).

Original: Generally, message passing is computed with nonlinear neural networks, which can incur a tremendous computational cost.
Revised: Generally, message passing is computed with nonlinear neural networks, which can incur a tremendous computational cost.

Original: In contrast, the GCN developed by Kipf & Welling (2017) is a considerable simpliﬁcation of a GNN, that uses a linear message passing scheme expressed as
Revised: In contrast, the GCN developed by Kipf & Welling (2017) is a considerable simpliﬁcation of a GNN, that uses a linear message passing scheme expressed as

Original: [Equation]
Revised: [Equation]

Original: where H in ( H out ) is an input (output) feature of the l th layer, ˆ A is a renormalized adjacency matrix with self-loops, and W is a trainable weight.
Revised: where H in ( H out ) is an input (output) feature of the l th layer, ˆ A is a renormalized adjacency matrix with self-loops, and W is a trainable weight.

Original: A GCN, among the variants of GNNs, is essential to the present study because the proposed model is based on GCNs for computational efﬁciency.
Revised: A GCN, among the variants of GNNs, is essential to the present study because the proposed model is based on GCNs for computational efﬁciency.

Original: Invariant and equivariant neural networks.
Revised: Invariant and equivariant neural networks.

Original: A function f : X → Y is said to be equivariant to a group G when f ( g · x ) =
Revised: A function f : X → Y is said to be equivariant to a group G when f ( g · x ) =

Original: g · f ( x ) , for all g ∈ G and x ∈ X , assuming that group G acts on both X and Y .
Revised: g · f ( x ) , for all g ∈ G and x ∈ X , assuming that group G acts on both X and Y .

Original: In particular, when f ( g · x ) = f ( x ) , f is said to be invariant to the group G .
Revised: In particular, when f ( g · x ) = f ( x ) , f is said to be invariant to the group G .

Original: Group equivariant convolutional neural networks were ﬁrst proposed by Cohen & Welling (2016) for discrete groups.
Revised: Group equivariant convolutional neural networks were ﬁrst proposed by Cohen & Welling (2016) for discrete groups.

Original: Subsequent studies have categorized such networks into continuous groups (Cohen et al., 2018), three-dimensional data (Weiler et al., 2018), and general manifolds (Cohen et al., 2019).
Revised: Subsequent studies have categorized such networks into continuous groups (Cohen et al., 2018), three-dimensional data (Weiler et al., 2018), and general manifolds (Cohen et al., 2019).

Original: These methods are based on CNNs; thus, they cannot handle mesh or point cloud data structures as is.
Revised: These methods are based on CNNs; thus, they cannot handle mesh or point cloud data structures as is.

Original: Speciﬁcally, 3D steerable CNNs (Weiler et al., 2018) uses voxels (regular grids), which though relatively easy to handle, are not efﬁcient because they represent both occupied and non-occupied parts of an object (Ahmed et al., 2018).
Revised: Speciﬁcally, 3D steerable CNNs (Weiler et al., 2018) uses voxels (regular grids), which though relatively easy to handle, are not efﬁcient because they represent both occupied and non-occupied parts of an object (Ahmed et al., 2018).

Original: In addition, a voxelized object tends to lose the smoothness of its shape, which can lead to drastically different behavior in a physical simulation, as typically observed in structural analysis and computational ﬂuid dynamics.
Revised: In addition, a voxelized object tends to lose the smoothness of its shape, which can lead to drastically different behavior in a physical simulation, as typically observed in structural analysis and computational ﬂuid dynamics.

Original: Thomas et al.
Revised: Thomas et al.

Original: ; Kondor (2018) discussed how to provide rotation equivariance to point clouds.
Revised: ; Kondor (2018) discussed how to provide rotation equivariance to point clouds.

Original: Speciﬁcally, the tensor ﬁeld network (TFN) (Thomas et al., 2018) is a point cloud based rotation and translation equivariant neural network the layer of which can be written as
Revised: Speciﬁcally, the tensor ﬁeld network (TFN) (Thomas et al., 2018) is a point cloud based rotation and translation equivariant neural network the layer of which can be written as

Original: [Equation]
Revised: [Equation]

Original: where ˜ H ( l ) in ,i ( ˜ H ( l ) out ,i ) is a type- l input (output) feature at the i th vertex, φ lkJ : R ≥ 

Revised: where ˜ H ( l ) in ,i ( ˜ H ( l ) out ,i ) is a type- l input (output) feature at the i th vertex, φ lk J : R ≥

Original: → R is a trainable function, Y Jm is the m th component of the J th spherical harmonics, and Q lkJm is the ClebschCordan coefﬁcient.
Revised: → R is a trainable function, Y Jm is the m th component of the J th spherical harmonics, and Q lkJm is the ClebschCordan coefﬁcient.

Original: The SE(3)-Transformer (Fuchs et al., 2020) is a variant of the TFN with selfattention.
Revised: The SE(3)-Transformer (Fuchs et al., 2020) is a variant of the TFN with selfattention.

Original: These models achieve high expressibility based on spherical harmonics and message passing with nonlinear neural networks.
Revised: These models achieve high expressibility based on spherical harmonics and message passing with nonlinear neural networks.

Original: However, for this reason, considerable computational resources are required.
Revised: However, for this reason, considerable computational resources are required.

Original: In contrast, the present study allows a signiﬁcant reduction in the computational costs because it eliminates spherical harmonics and nonlinear message passing.
Revised: In contrast, the present study allows a signiﬁcant reduction in the computational costs because it eliminates spherical harmonics and nonlinear message passing.

Original: From this perspective,
Revised: From this perspective,

Original: IsoGCNs are also regarded as a simpliﬁcation of the TFN, as seen in equation 14.
Revised: IsoGCNs are also regarded as a simpliﬁcation of the TFN, as seen in equation 14.

Original: Physical simulations using GNNs.
Revised: Physical simulations using GNNs.

Original: Several related studies, including those by Sanchez-Gonzalez et al.
Revised: Several related studies, including those by Sanchez-Gonzalez et al.

Original: (2018; 2019); Alet et al.
Revised: (2018; 2019); Alet et al.

Original: ; Chang & Cheng (2020) focused on applying GNNs to learn physical simulations.
Revised: ; Chang & Cheng (2020) focused on applying GNNs to learn physical simulations.

Original: These approaches allowed the physical information to be introduced to GNNs; however, addressing isometric transformationinvariance and equivariance was out of the scope of their research.
Revised: These approaches allowed the physical information to be introduced to GNNs; however, addressing isometric transformation equivariance was out of the scope of their research.

Original: In the present study, we incorporate isometric transformation invariance and equivariance into
Revised: In the present study, we incorporate isometric transformation invariance and equivariance into

Original: GCNs, thereby, ensuring the stability of the training and inference under isometric transformation.
Revised: GCNs, thereby, ensuring the stability of the training and inference under isometric transformation.

Original: Moreover, the proposed approach is efﬁcient in processing large graphs with up to 1M vertices that have a sufﬁcient number of degrees of freedom to express complex shapes.
Revised: Moreover, the proposed approach is efﬁcient in processing large graphs with up to 1M vertices that have a sufﬁcient number of degrees of freedom to express complex shapes.

Original:  I SOMETRIC T RANSFORMATION INVARIANT AND EQUIVARIANT GRAPH CONVOLUTIONAL LAYERS
Revised:  I SOMETRIC T RANSFORMATION INVARIANT AND EQUIVARIANT GRAPH CONVOLUTIONAL LAYERS

Original: In this section, we discuss how to construct IsoGCN layers that correspond to the isometric invariant and equivariant GCN layers.
Revised: In this section, we discuss how to construct IsoGCN layers that correspond to the isometric invariant and equivariant GCN layers.

Original: To formulate a model, we assume that: 1) only attributes associated with vertices and not edges; and 2) graphs do not contain self-loops.
Revised: To formulate a model, we assume that: 1) only attributes associated with vertices and not edges; and 2) graphs do not contain self-loops.

Original: Here, G =
Revised: Here, G =

Original: ( V , E ) denotes a graph and d denotes the dimension of a Euclidean space.
Revised: ( V , E ) denotes a graph and d denotes the dimension of a Euclidean space.

Original: In this paper, we refer to tensor as geometric tensors, and we consider a (discrete) rank-
Revised: In this paper, we refer to tensor as geometric tensors, and we consider a (discrete) rank-

Original: p tensor ﬁeld H ( p ) ∈ R |V|× f × d p , where |V| denotes the number of vertices and f ∈ Z
Revised: p tensor ﬁeld H ( p ) ∈ R |V|× f × d p , where |V| denotes the number of vertices and f ∈ Z

Original: + ( Z + denotes the positive integers).
Revised: + ( Z + denotes the positive integers).

Original: Here, f denotes the number of features (channels) of H ( p ) , as shown in Figure 1 (a).
Revised: Here, f denotes the number of features (channels) of H ( p ) , as shown in Figure 1 (a).

Original: With the indices, we denote H ( p ) i ; g ; k 1 k 2 ...
Revised: With the indices, we denote H ( p ) i ; g ; k 1 k 2 ...

Original: k p , where i permutes under the permutation of vertices and k 1 , . . .
Revised: k p , where i permutes under the permutation of vertices and k 1 , . . .

Original: , k p refers to the Euclidean representation.
Revised: , k p refers to the Euclidean representation.

Original: Thus, under the permutation, π : H ( p ) i ; g ; k 1 k 2 ...
Revised: Thus, under the permutation, π : H ( p ) i ; g ; k 1 k 2 ...

Original: k p (cid:55)→ H ( p ) π ( i ); g ; k 1 k 2 ...k p , and under orthogonal transformation, U : H ( p ) i ; g ; k 1 k 2 ...
Revised: k p (cid:55)→ H ( p ) π ( i ); g ; k 1 k 2 ...k p , and under orthogonal transformation, U : H ( p ) i ; g ; k 1 k 2 ...

Original: k p (cid:55)→ (cid:80) l 1 ,l 2 ,...,l p U k 1 l 1 U k 2 l 2 . . .
Revised: k p (cid:55)→ (cid:80) l 1 ,l 2 ,...,l p U k 1 l 1 U k 2 l 2 . . .

Original: U k
Revised: U k

Original: p l p H ( p ) i 
Revised: p l p H ( p ) i ; g ; l 1 l 2 ...l p .

Revised: 3. C ONSTRUCTION OF AN ISOMETRIC ADJACENCY MATRIX

Revised: <

Original: [Table] [Table]
Revised: [Table] [Table]

Revised: /3vEeIdUv8Wmq2DrlLXWr5Dms8jkTFMUxwpWjXuZi3EO2t19fBcXMI6Dtc4ryHmZJvrh/bwefP9y3kPmGWhuvY9T96bfqz6u0EzRwxgtDL4fxtlEPXZDpwj03yngLp8GOlguAM9aGUQg4kxkeVPQK3hYE1vTXOrlyXUBGE7FNn72zeA3Uozy/G5t2Xucc1Lo0zkv/wE1XkZmAPh7uKqvvtVQn5O2IV71uzRqxWbm/ydaXKKgLvUbfgmoYTDCOEvy+1EJwd2n71bWu3/

Original: /fdrL8/erPxxs/kzn18s/e3S3y39dml96R+X/ri0v3S6dLUULhVL/7H0n0v/9ebojXzz9c0PtulPf9L0+Zsl7+fNv/8veR5hWg==</latexit> G ij ;;:[Equation]
Revised: Ks3hx/Wpt/fdrL8/erPxxs/ kzn18s/e3S3y39dml96R+X/ri0v3S6dLUULhVL/7H0n0v/9ebojXzz9c0PtulPf9L0+Zsl7+fNv/8veR5hWg==</latexit> G ij ;;:

Original: [Equation]
Revised: [Equation]

Original: Before constructing an IsoGCN, an isometric adjacency matrix (IsoAM), which is at the core of the IsoGCN concept must be deﬁned.
Revised: Before constructing an IsoGCN, an isometric adjacency matrix (IsoAM), which is at the core of the IsoGCN concept must be deﬁned.

Original: The proof of each proposition can be found in
Revised: The proof of each proposition can be found in

Original: Appendix B.
Revised: Appendix B.

Original: An IsoAM G ∈ R |V|×|V|× d is deﬁned as:
Revised: An IsoAM G ∈ R |V|×|V|× d is deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: where G ij ;;: is a slice in the spatial index of G , x i ∈ R
Revised: where G ij ;;: is a slice in the spatial index of G , x i ∈ R

Original: d is the position of the i th vertex (rank1 tensor), and T ijkl ∈
Revised: d is the position of the i th vertex (rank1 tensor), and T ijkl ∈

Original: R
Revised: R

Original: d × d is an untrainable transformation invariant and orthogonal transformation equivariant rank-2 tensor.
Revised: d × d is an untrainable transformation invariant and orthogonal transformation equivariant rank-2 tensor.

Original: Note that we denote G ij ;; k to be consistent with the notation of H ( p ) i ; g ; k 1 k 2 ...
Revised: Note that we denote G ij ;; k to be consistent with the notation of H ( p ) i ; g ; k 1 k 2 ...

Original: k p because i and j permutes under the vertex permutation and k represents the spatial index while the number of features is always 1.
Revised: k p because i and j permutes under the vertex permutation and k represents the spatial index while the number of features is always 1.

Original: The IsoAM can be viewed as a weighted adjacency matrix for each direction and reﬂects spatial information while the usual weighted adjacency matrix cannot because a graph has only one adjacency matrix.
Revised: The IsoAM can be viewed as a weighted adjacency matrix for each direction and reﬂects spatial information while the usual weighted adjacency matrix cannot because a graph has only one adjacency matrix.

Original: If the size of the set { G ij ;;: (cid:54) = 0 } j is greater than or equal to d , then it can be deemed to be a frame, which is a generalization of a basis.
Revised: If the size of the set { G ij ;;: (cid:54) = 0 } j is greater than or equal to d , then it can be deemed to be a frame, which is a generalization of a basis.

Original: For the simplest case, one can deﬁne T ijkl =
Revised: For the simplest case, one can deﬁne T ijkl =

Original: δ il δ jk
Revised: δ il δ jk

Original: A ij I
Revised: A ij I

Original: (Figure 1 (b)), where δ ij is the Kronecker delta, A is the adjacency matrix of the graph, and I is the identity matrix that is the simplest rank-2 tensor.
Revised: (Figure 1 (b)), where δ ij is the Kronecker delta, A is the adjacency matrix of the graph, and I is the identity matrix that is the simplest rank-2 tensor.

Original: In another case, T ijkl can be determined from the geometry of a graph, as deﬁned inequation 16.
Revised: In another case, T ijkl can be determined from the geometry of a graph, as deﬁned in equation 16.

Original: Nevertheless, in the bulk of this section, we retain T ijkl abstract to cover various forms of interaction, such as position-aware GNNs (You et al., 2019).
Revised: Nevertheless, in the bulk of this section, we retain T ijkl abstract to cover various forms of interaction, such as position-aware GNNs (You et al., 2019).

Original: Here, G is composed of only untrainable parameters and thus can be determined before training.
Revised: Here, G is composed of only untrainable parameters and thus can be determined before training.

Original: Proposition 3.1.
Revised: Proposition 3.1.

Original: IsoAM deﬁned in equation 4 is both translation invariant and orthogonal transformation equivariant, i.e., for any isometric transformation ∀ t ∈ R 3 , U ∈
Revised: IsoAM deﬁned in equation 4 is both translation invariant and orthogonal transformation equivariant, i.e., for any isometric transformation ∀ t ∈ R 3 , U ∈

Original: O( d ) , T : x (cid:55)→
Revised: O( d [Equation]

Original: Ux + t , 
Revised: , T

Original: [Equation]
Revised: [Equation]

Original: Based on the deﬁnition of the GCN layer in the equation 1, let
Revised: Based on the deﬁnition of the GCN layer in the equation 1, let

Original: G ∗ H (0) ∈
Revised: G ∗ H (0) ∈

Original: R |V|× f × d denote the convolution between G and the rank-0 tensor ﬁeld H (0)
Revised: R |V|× f × d denote the convolution between G and the rank-0 tensor ﬁeld H (0)

Original: ∈ R |V|× f ( f ∈ Z + ) as follows:
Revised: ∈ R |V|× f ( f ∈ Z + ) as follows:

Original: [Equation]
Revised: [Equation]

Original: With a rank-1 tensor ﬁeld H (1) ∈ R |V|× f × d , let G (cid:12) H (1) ∈ R |V|× f and G (cid:12) G ∈ R |V|×|V| denote the contractions which are deﬁned as follows:
Revised: With a rank-1 tensor ﬁeld H (1) ∈ R |V|× f × d , let G (cid:12) H (1) ∈ R |V|× f and G (cid:12) G ∈ R |V|×|V| denote the contractions which are deﬁned as follows:

Original: [Equation]
Revised: [Equation]

Original: The contraction of IsoAMs G (cid:12) G can be interpreted as the inner product of each component in the
Revised: The contraction of IsoAMs G (cid:12) G can be interpreted as the inner product of each component in the

Original: IsoAMs.
Revised: IsoAMs.

Original: Thus, the subsequent proposition follows.
Revised: Thus, the subsequent proposition follows.

Original: Proposition 3.2.
Revised: Proposition 3.2.

Original: The contraction of IsoAMs G (cid:12) G is isometric transformation invariant, i.e., for any isometric transformation
Revised: The contraction of IsoAMs G (cid:12) G is isometric transformation invariant, i.e., for any isometric transformation

Original: ∀ t ∈ R 3 , U ∈
Revised: ∀ t ∈ R 3 , U ∈

Original: O( d ) , T : x (cid:55)→
Revised: O( d ) , T : x (cid:55)→

Original: Ux + t , G (cid:12) G
Revised: Ux + t , G (cid:12) G

Original: (cid:55)→ G (cid:12) G .
Revised: (cid:55)→ G (cid:12) G .

Original: With a rank- p tensor ﬁeld H ( p ) ∈ R |V|× f × d p , let G ⊗ H ( p ) ∈ R |V|× f × d 1+ p  and G ⊗ G ∈ R
Revised: With a rank- p tensor ﬁeld H ( p ) ∈ R |V|× f × d p , let G ⊗ H ( p ) ∈ R |V|× f × d 1+ p . and G ⊗ G ∈ R

Original: |V|×|V|× ddenote the tensor products deﬁned as follows:
Revised: |V|×|V|× ddenote the tensor products deﬁned as follows:

Original: [Equation]
Revised: [Equation]

Original: The tensor product of IsoAMs G ⊗ G can be interpreted as the tensor product of each of the IsoAMs components.
Revised: The tensor product of IsoAMs G ⊗ G can be interpreted as the tensor product of each of the IsoAMs components.

Original: Thus, the subsequent proposition follows:
Revised: Thus, the subsequent proposition follows:

Original: Proposition 3.3.
Revised: Proposition 3.3.

Original: The tensor product of the IsoAMs G ⊗ G is isometric transformation equivariant in terms of the rank-2 tensor, i.e., for any isometric transformation
Revised: The tensor product of the IsoAMs G ⊗ G is isometric transformation equivariant in terms of the rank-2 tensor, i.e., for any isometric transformation

Original: ∀ t ∈ R 3 , U ∈
Revised: ∀ t ∈ R 3 , U ∈

Original: O( d ) , T : x (cid:55)→
Revised: O( d ) , T : x (cid:55)→

Original: Ux 
Revised: Ux + t , and ∀ i, j ∈ 1 , . . . , |V| , ( G ⊗ G ) ij ;; k 1 k

Original: [Equation]
Revised: [Equation]

Original: t , and ∀ i, j ∈ 1 , . .
Revised: U k

Original: [Equation]
Revised: [Equation]

Original: |
Revised: U

Original: [Equation]
Revised: [Equation]

Original: This proposition is easily generalized to the tensors of higher ranks by deﬁning the p th tensor power of G as follows: (cid:78) 

Revised: This proposition is easily generalized to the tensors of higher ranks by deﬁning the p th tensor power of G as follows: (cid:78)

Original: G = 1 , (cid:78) 1 G = G , and (cid:78) p G = (cid:78) p − 1 G ⊗ G .
Revised: G = 1 , (cid:78) 1 G = G , and (cid:78) p G = (cid:78) p − 1 G ⊗ G .

Original: Namely, (cid:78) p G is isometric transformation equivariant in terms of rank- p tensor.
Revised: Namely, (cid:78) p G is isometric transformation equivariant in terms of rank- p tensor.

Original: Therefore, one can see that ( (cid:78) p G ) ⊗ H ( q )
Revised: Therefore, one can see that ( (cid:78) p G ) ⊗ H ( q )

Original: = ( (cid:78) p − 1 G ) ⊗ ( G ⊗ H ( q ) ) .
Revised: = ( (cid:78) p − 1 G ) ⊗ ( G ⊗ H ( q ) ) .

Original: Moreover, the convolution can be generalized for (cid:78) p G and the ranktensor ﬁeld H (0)
Revised: Moreover, the convolution can be generalized for (cid:78) p G and the ranktensor ﬁeld H (0)

Original: ∈ R |V|× f as follows:
Revised: ∈ R |V|× f as follows:

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: The contraction can be generalized for (cid:78) p G and the rank- q tensor ﬁeld H ( q ) ∈ R |V|× f × d q ( p ≥ q ) as speciﬁed below:
Revised: The contraction can be generalized for (cid:78) p G and the rank- q tensor ﬁeld H ( q ) ∈ R |V|× f × d q ( p ≥ q ) as speciﬁed below:

Original: [Equation]
Revised: [Equation]

Original: For the case p <
Revised: For the case p <

Original: q , the contraction can be deﬁned similarly.
Revised: q , the contraction can be deﬁned similarly.

Original: 3. C ONSTRUCTION OF I SO GCN
Revised: 3. C ONSTRUCTION OF I SO GCN

Original: An IsoGCN utilizes an IsoAM to realize the isometric transformation invariant and equivariant layers.

Original: Using the operations deﬁned above, we can construct IsoGCN layers, which take the tensor ﬁeld of any rank as input, and output the tensor ﬁeld of any rank, which can differ from those of the input.
Revised: Using the operations deﬁned above, we can construct IsoGCN layers, which take the tensor ﬁeld of any rank as input, and output the tensor ﬁeld of any rank, which can differ from those of the input.

Original: In addition, one can show that these layers are also equivariant under the vertex permutation, as discussed in Maron et al.
Revised: In addition, one can show that these layers are also equivariant under the vertex permutation, as discussed in Maron et al.

Original: 3.2. I SOMETRIC TRANSFORMATION INVARIANT LAYER
Revised: 3.2. I SOMETRIC TRANSFORMATION INVARIANT LAYER

Original: As can be seen in Proposition 3.1, the contraction of IsoAMs is isometric transformation invariant.
Revised: As can be seen in Proposition 3.1, the contraction of IsoAMs is isometric transformation invariant.

Original: Therefore, for the isometric transformation invariant layer with a rank-0 input tensor ﬁeld f : R |V|× f in (cid:51) H (0)in (cid:55)→ H (0)out ∈ R |V|× f out ( f in , f out ∈ Z + ), the activation function σ , and the trainable parameter matrix W ∈ R f in × f out can be constructed as H (0)out = σ (cid:16) ( G (cid:12) G )
Revised: Therefore, for the isometric transformation invariant layer with a rank-0 input tensor ﬁeld f : R |V|× f in (cid:51) H (0)in (cid:55)→ H (0)out ∈ R |V|× f out ( f in , f out ∈ Z + ), the activation function σ , and the trainable parameter matrix W ∈ R f in × f out can be constructed as H (0)out = σ (cid:16) ( G (cid:12) G )

Original: H (0)in W (cid:17) .
Revised: H (0)in W (cid:17) .

Original: By deﬁning L := G (cid:12) G ∈ R |V|×|V| , it can be simpliﬁed as H (0)out
Revised: By deﬁning L := G (cid:12) G ∈ R |V|×|V| , it can be simpliﬁed as H (0)out

Original: = σ (cid:16) LH (0)in W (cid:17) , which has the same form as a GCN (equation 1), with the exception that ˆ A is replaced with L .
Revised: = σ (cid:16) LH (0)in W (cid:17) , which has the same form as a GCN (equation 1), with the exception that ˆ A is replaced with L .

Original: An isometric transformation invariant layer with the rank- p input tensor ﬁeld H ( p ) in ∈ R |V|× f in × d p can be formulated as H (0)out
Revised: An isometric transformation invariant layer with the rank- p input tensor ﬁeld H ( p ) in ∈ R |V|× f in × d p can be formulated as H (0)out

Original: = F p → 0 ( H ( p ) in ) =
Revised: = F p → 0 ( H ( p ) in ) =

Original: σ (cid:16)(cid:104)(cid:78) p G (cid:12) H ( p ) in (cid:105) W (cid:17) .
Revised: σ (cid:16)(cid:104)(cid:78) p G (cid:12) H ( p ) in (cid:105) W (cid:17) .

Original: If p = 1 , such approaches utilize the inner products of the vectors in R d , these operations correspond to the extractions of a relative distance and an angle of each pair of vertices, which are employed in Klicpera et al.
Revised: If p = 1 , such approaches utilize the inner products of the vectors in R d , these operations correspond to the extractions of a relative distance and an angle of each pair of vertices, which are employed in Klicpera et al.

Original: 3.2. I SOMETRIC TRANSFORMATION EQUIVARIANT LAYER
Revised: 3.2. I SOMETRIC TRANSFORMATION EQUIVARIANT LAYER

Original: To construct an isometric transformation equivariant layer, one can use linear transformation, convolution and tensor product to the input tensors.
Revised: To construct an isometric transformation equivariant layer, one can use linear transformation, convolution and tensor product to the input tensors.

Original: If both the input and the output tensor ranks are greater than 0, one can apply neither nonlinear activation nor bias addition because these operations will cause an inappropriate distortion of the isometry because isometric transformation does not commute with them in general.
Revised: If both the input and the output tensor ranks are greater than 0, one can apply neither nonlinear activation nor bias addition because these operations will cause an inappropriate distortion of the isometry because isometric transformation does not commute with them in general.

Original: However, a conversion that uses only a linear transformation, convolution, and tensor product does not have nonlinearity, which limits the predictive performance of the model.
Revised: However, a conversion that uses only a linear transformation, convolution, and tensor product does not have nonlinearity, which limits the predictive performance of the model.

Original: To add nonlinearity to such a conversion, we can ﬁrst convert the input tensors to rank-0 ones, apply nonlinear activations, and then multiply them to the higher rank tensors.
Revised: To add nonlinearity to such a conversion, we can ﬁrst convert the input tensors to rank-0 ones, apply nonlinear activations, and then multiply them to the higher rank tensors.

Original: For instance, the nonlinear isometric transformation equivariant layer with the rank- m input tensor ﬁeld H ( m ) in ∈ R |V|× f in × d m and the rank- l ( m ≤ l ) output tensor H ( l ) out ∈ R |V|× f out ×
Revised: The nonlinear isometric transformation equivariant layer with the rank- m input tensor ﬁeld H ( m ) in ∈ R |V|× f in × d m and the rank- l ( m ≤ l ) output tensor H ( l ) out ∈ R |V|× f out ×

Original: d l can be deﬁned as:
Revised: d l can be deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: where × denotes multiplication with broadcasting and W ml ∈ R f in × f out are trainable weight matrices multiplied in the feature direction.
Revised: where × denotes multiplication with broadcasting and W ml ∈ R f in × f out are trainable weight matrices multiplied in the feature direction.

Original: If m = 0 , we regard G ⊗ H (0) as G ∗ H (0) .
Revised: If m = 0 , we regard G ⊗ H (0) as G ∗ H (0) .

Original: If m = l , one can add the residual connection (He et al., 2016) in equation 12.
Revised: If m = l , one can add the residual connection (He et al., 2016) in equation 12.

Original: If m > l ,
Revised: If m > l ,

Original: [Equation]
Revised: [Equation]

Original: In general, the nonlinear isometric transformation equivariant layer with the rank-0 to rank- M input tensor ﬁeld { H ( m ) in } Mm =0 and the rank- l output tensor ﬁeld H ( l ) out can be deﬁned as:
Revised: In general, the nonlinear isometric transformation equivariant layer with the rank-0 to rank- M input tensor ﬁeld { H ( m ) in } Mm =0 and the rank- l output tensor ﬁeld H ( l ) out can be deﬁned as

Original: [Equation]
Revised: [Equation]

Original: where f gather denotes a function such as summation, product and concatenation in the feature direction.
Revised: where f gather denotes a function such as summation, product and concatenation in the feature direction.

Original: One can see that this layer is similar to that in the TFN (equation 2), while there are no spherical harmonics and trainable message passing.
Revised: One can see that this layer is similar to that in the TFN (equation 2), while there are no spherical harmonics and trainable message passing.

Original: To be exact, the output of the layer deﬁned above is translation invariant.
Revised: To be exact, the output of the layer deﬁned above is translation invariant.

Original: To output translation equivariant variables such as the vertex positions after deformation (which change accordingly with the translation of the input graph), one can ﬁrst deﬁne the reference vertex position x ref for each graph, then compute the translation invariant output using equation 14, and ﬁnally, add x ref to the output.
Revised: To output translation equivariant variables such as the vertex positions after deformation (which change accordingly with the translation of the input graph), one can ﬁrst deﬁne the reference vertex position x ref for each graph, then compute the translation invariant output using equation 14, and ﬁnally, add x ref to the output.

Original: For more detailed information on IsoGCN modeling, see Appendix D.
Revised: For more detailed information on IsoGCN modeling, see Appendix D.

Original: 3. E XAMPLE OF I SO AM
Revised: 3. E XAMPLE OF I SO AM

Original: The IsoGCN G is deﬁned in a general form for the propositions to work with various classes of graph.
Revised: The IsoGCN G is deﬁned in a general form for the propositions to work with various classes of graph.

Original: In this section, we concretize the concept of the IsoAM to apply an IsoGCN to mesh structured data.
Revised: In this section, we concretize the concept of the IsoAM to apply an IsoGCN to mesh structured data.

Original: Here, a mesh is regarded as a graph regarding the points in the mesh as vertices of the graph and assuming two vertices are connected when they share the same cell.
Revised: Here, a mesh is regarded as a graph regarding the points in the mesh as vertices of the graph and assuming two vertices are connected when they share the same cell.

Original: A concrete instance of IsoAM ˜ D , D ∈ R
Revised: A concrete instance of IsoAM ˜ D , D ∈ R

Original: |V|×|V|× d is deﬁned as follows:
Revised: |V|×|V|× d is deﬁned as follows:

Original: Expression [Table]
Revised: [Table] [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: where R |V|×|V| (cid:51) A ( m ) :
Revised: where R |V|×|V| (cid:51) A ( m ) :

Original: = min ( (cid:80) mk =

Revised: = min ( (cid:80) mk =

Original: A k , 1) is an adjacency matrix up to m hops and w ij ∈ R is an untrainable weight between the i th and j th vertices that is determined depending on the tasks 1 .
Revised: A k , 1) is an adjacency matrix up to m hops and w ij ∈ R is an untrainable weight between the i th and j th vertices that is determined depending on the tasks 2 .

Original: By regarding
Revised: By regarding

Original: Tijkl
Revised: T

Original: [Equation]
Revised: [Equation]

Original: ) / (cid:107) x
Revised: / (cid:107) x

Original: j − x i (cid:107) 2 in equation 4, one can see that D is qualiﬁed as an
Revised: j − x i (cid:107) 2 in equation 4, one can see that D is qualiﬁed as an

Original: IsoAM.
Revised: IsoAM.

Original: Because a linear combination of IsoAMs is also an IsoAM, ˜ D is an IsoAM.
Revised: Because a linear combination of IsoAMs is also an IsoAM, ˜ D is an IsoAM.

Original: Thus, they provide both translation invariance and orthogonal transformation equivariance.
Revised: Thus, they provide both translation invariance and orthogonal transformation equivariance.

Original: ˜ D can be obtained only from the mesh geometry information, thus can be computed in the preprocessing step.
Revised: ˜ D can be obtained only from the mesh geometry information, thus can be computed in the preprocessing step.

Original: Here, ˜ D is designed such that it corresponds to the gradient operator model used in physical simulations (Tamai & Koshizuka, 2014; Swartz & Wendroff, 1969).
Revised: Here, ˜ D is designed such that it corresponds to the gradient operator model used in physical simulations (Tamai & Koshizuka, 2014; Swartz & Wendroff, 1969).

Original: As presented in Table 1 and Appendix C, ˜ D is closely related to many differential operators, such as the gradient, divergence, Laplacian, Jacobian, and Hessian.
Revised: As presented in Table 1 and Appendix C, ˜ D is closely related to many differential operators, such as the gradient, divergence, Laplacian, Jacobian, and Hessian.

Original: Therefore, the considered IsoAM plays an essential role in constructing neural network models that are capable of learning differential equations.
Revised: Therefore, the considered IsoAM plays an essential role in constructing neural network models that are capable of learning differential equations.

Original:  E XPERIMENT
Revised:  E XPERIMENT

Original: To test the applicability of the proposed model, we composed the following two datasets: 1) a differential operator dataset of grid meshes; and 2) an anisotropic nonlinear heat equation dataset of meshes generated from CAD data.
Revised: To test the applicability of the proposed model, we composed the following two datasets: 1) a differential operator dataset of grid meshes; and 2) an anisotropic nonlinear heat equation dataset of meshes generated from CAD data.

Original: In this section, we discuss our machine learning model, the deﬁnition of the problem, and the results for each dataset.
Revised: In this section, we discuss our machine learning model, the deﬁnition of the problem, and the results for each dataset.

Original: Using ˜ D deﬁned in Section 3.3, we constructed a neural network model considering an encodeprocess-decode conﬁguration (Battaglia et al., 2018).
Revised: Using ˜ D deﬁned in Section 3.3, we constructed a neural network model considering an encodeprocess-decode conﬁguration (Battaglia et al., 2018).

Original: The encoder and decoder were comprised of component-wise MLPs and tensor operations.
Revised: The encoder and decoder were comprised of component-wise MLPs and tensor operations.

Original: For each task, we tested m = 2 , 5 in equation 16 to investigate the effect of the number of hops considered.
Revised: For each task, we tested m = 2 , 5 in equation 16 to investigate the effect of the number of hops considered.

Original: In addition to the GCN (Kipf & Welling, 2017), we chose GIN (Xu et al., 2018), SGCN (Wu et al., 2019), Cluster-GCN (Chiang et al., 2019), and GCNII (Chen et al., 2020) as GCN variant baseline models.
Revised: In addition to the GCN (Kipf & Welling, 2017), we chose GIN (Xu et al., 2018), SGCN (Wu et al., 2019), Cluster-GCN (Chiang et al., 2019), and GCNII (Chen et al., 2020) as GCN variant baseline models.

Original: For the equivariant models, we chose the TFN (Thomas et al., 2018) and SE(3)-Transformer (Fuchs et al., 2020) as the baseline.
Revised: For the equivariant models, we chose the TFN (Thomas et al., 2018) and SE(3)-Transformer (Fuchs et al., 2020) as the baseline.

Original: We implemented these models using PyTorch 1.6.0 (Paszke et al., 2019) and PyTorch Geometric 1.6.1 (Fey & Lenssen, 2019).
Revised: We implemented these models using PyTorch 1.6.0 (Paszke et al., 2019) and PyTorch Geometric 1.6.1 (Fey & Lenssen, 2019).

Original: For both the TFN and SE(3)-Transformer, we used implementation of Fuchs et al.
Revised: For both the TFN and SE(3)-Transformer, we used implementation of Fuchs et al.

Original: (2020) 2 because the computation of the TFN is considerably faster than the original implementation, as claimed in Fuchs et al.
Revised: (2020) 3 because the computation of the TFN is considerably faster than the original implementation, as claimed in Fuchs et al.

Original: For each experiment, we minimized the mean squared loss using the Adam optimizer (Kingma & Ba, 2014).
Revised: For each experiment, we minimized the mean squared loss using the Adam optimizer (Kingma & Ba, 2014).

Original: The corresponding implementation and the dataset will be made available online.
Revised: The corresponding implementation and the dataset will be made available online.

Original: The details of the experiments can be found in
Revised: The details of the experiments can be found in

Original: Appendix E and F.
Revised: Appendix E and F.

Original: 4. D IFFERENTIAL OPERATOR DATASET
Revised: 4. D IFFERENTIAL OPERATOR DATASET

Original: To demonstrate the expressive power of IsoGCNs, we created a dataset to learn the differential operators.
Revised: To demonstrate the expressive power of IsoGCNs, we created a dataset to learn the differential operators.

Original: We ﬁrst generated a pseudo-2D grid mesh randomly with only one cell in the Z direction and 10 to 100 cells in the X and Y directions.
Revised: We ﬁrst generated a pseudo-2D grid mesh randomly with only one cell in the Z direction and 10 to 100 cells in the X and Y directions.

Original: We then generated scalar ﬁelds on the grid meshes
Revised: We then generated scalar ﬁelds on the grid meshes

Original: [Figure]
Revised: [Figure]

Original: show only the best setting for for models.

Original: a full table

Original: [Table]
Revised: [Table]

Original: and analytically calculated the gradient, Laplacian, and Hessian ﬁelds.
Revised: and analytically calculated the gradient, Laplacian, and Hessian ﬁelds.

Original: We generated 100 samples for each train, validation, and test dataset.
Revised: We generated 100 samples for each train, validation, and test dataset.

Original: For simplicity, we set w ij = 1 in equation 16 for all ( i, j ) ∈ E .
Revised: For simplicity, we set w ij = 1 in equation 16 for all ( i, j ) ∈ E .

Original: To compare the performance with the GCN models, we simply replaced an IsoGCN layer with a GCN or its variant layers while keeping the number of hops m the same to enable a fair comparison.
Revised: To compare the performance with the GCN models, we simply replaced an IsoGCN layer with a GCN or its variant layers while keeping the number of hops m the same to enable a fair comparison.

Original: We adjusted the hyperparameters for the equivariant models to ensure that the number of parameters in each was almost the same as that in the IsoGCN model.
Revised: We adjusted the hyperparameters for the equivariant models to ensure that the number of parameters in each was almost the same as that in the IsoGCN model.

Original: For more details regarding the model architecture, see Appendix E. We conducted the experiments using the following settings
Revised: For more details regarding the model architecture, see Appendix E. We conducted the experiments using the following settings:

Original: : 1) inputting the scalar ﬁeld and predicting the gradient ﬁeld (rank-0 → rank-1 tensor); 2) inputting the scalar ﬁeld and predicting the Hessian ﬁeld (rank-0 → rank-2 tensor); 3) inputting the gradient ﬁeld and predicting the Laplacian ﬁeld (rank-1 → rank-0 tensor); and 4) inputting the gradient ﬁeld and predicting the Hessian ﬁeld (rank-1 → rank-2 tensor).
Revised: 1) inputting the scalar ﬁeld and predicting the gradient ﬁeld (rank-0 → rank-1 tensor); 2) inputting the scalar ﬁeld and predicting the Hessian ﬁeld (rank-0 → rank-2 tensor); 3) inputting the gradient ﬁeld and predicting the Laplacian ﬁeld (rank-1 → rank-0 tensor); and 4) inputting the gradient ﬁeld and predicting the Hessian ﬁeld (rank-1 → rank-2 tensor).

Original: Figure 2 and Table 2 present a visualization and comparison of predictive performance, respectively.
Revised: Figure 2 and Table 2 present a visualization and comparison of predictive performance, respectively.

Original: The results show that an IsoGCN outperforms other GCN models for all settings.
Revised: The results show that an IsoGCN outperforms other GCN models for all settings.

Original: This is because the IsoGCN model has information on the relative position of the adjacency vertices, and thus understands the direction of the gradient, whereas the other GCN models cannot distinguish where the adjacencies are, making it nearly impossible to predict the gradient directions.
Revised: This is because the IsoGCN model has information on the relative position of the adjacency vertices, and thus understands the direction of the gradient, whereas the other GCN models cannot distinguish where the adjacencies are, making it nearly impossible to predict the gradient directions.

Original: Adding the vertex positions to the input feature to other GCN models exhibited a performance improvement, however as the vertex position is not a translation invariant feature, it could degrade the predictive performance of the models.
Revised: Adding the vertex positions to the input feature to other GCN models exhibited a performance improvement, however as the vertex position is not a translation invariant feature, it could degrade the predictive performance of the models.

Original: Thus, we did not input x as a vertex feature to the IsoGCN model or other equivariant models to retain their isometric transformation invariant and equivariant natures.
Revised: Thus, we did not input x as a vertex feature to the IsoGCN model or other equivariant models to retain their isometric transformation invariant and equivariant natures.

Original: IsoGCNs perform competitively against other equivariant models with shorter inference time as shown in Ta-  ble 7.
Revised: IsoGCNs perform competitively against other equivariant models with shorter inference time as shown in Ta- [Figure] ble 7.

Original: As mentioned in Section 3.3, ˜ D corresponds to the gradient operator, which is now conﬁrmed in practice.
Revised: As mentioned in Section 3.3, ˜ D corresponds to the gradient operator, which is now conﬁrmed in practice.

Original: 4. A NISOTROPIC NONLINEAR HEAT EQUATION DATASET
Revised: 4. A NISOTROPIC NONLINEAR HEAT EQUATION DATASET

Original: To apply the proposed model to a real problem, we adopted the anisotropic nonlinear heat equation.
Revised: To apply the proposed model to a real problem, we adopted the anisotropic nonlinear heat equation.

Original: We considered the task of predicting the time evolution of the temperature ﬁeld based on the initial temperature ﬁeld, material property, and mesh geometry information as inputs.
Revised: We considered the task of predicting the time evolution of the temperature ﬁeld based on the initial temperature ﬁeld, material property, and mesh geometry information as inputs.

Original: We randomly selected 82 CAD shapes from the ﬁrst 200 shapes of the ABC dataset (Koch et al., 2019), generate ﬁrst-order tetrahedral meshes using a mesh generator program, Gmsh (Geuzaine & Remacle, 2009), randomly set the initial temperature and anisotropic thermal conductivity, and ﬁnally conducted a ﬁnite element analysis (FEA) using the FEA program FrontISTR 3 (Morita et al., 2016; Ihara et al., 2017).
Revised: We randomly selected 82 CAD shapes from the ﬁrst 200 shapes of the ABC dataset (Koch et al., 2019), generate ﬁrst-order tetrahedral meshes using a mesh generator program, Gmsh (Geuzaine & Remacle, 2009), randomly set the initial temperature and anisotropic thermal conductivity, and ﬁnally conducted a ﬁnite element analysis (FEA) using the FEA program FrontISTR 4 (Morita et al., 2016; Ihara et al., 2017).

Original: For this task, we set w ij =
Revised: For this task, we set w ij =

Original: V eﬀective j /V eﬀective i , where V eﬀective i denotes the effective volume of the i th vertex (equation 46.)
Revised: V eﬀective j /V eﬀective i , where V eﬀective i denotes the effective volume of the i th vertex (equation 46.)

Original: Similarly to the differential operator dataset, we tested the number of hops m = 2 , 5 .
Revised: Similarly to the differential operator dataset, we tested the number of hops m = 2 , 5 .

Original: However because we put four IsoAM operations in one model, the number of hops visible from the model is 8 ( m = 2 ) or 20 ( m = 5 ).
Revised: However because we put four IsoAM operations in one model, the number of hops visible from the model is 8 ( m = 2 ) or 20 ( m = 5 ).

Original: As is the case with the differential operator dataset, we replaced an IsoGCN layer accordingly for GCN or its variant models.
Revised: As is the case with the differential operator dataset, we replaced an IsoGCN layer accordingly for GCN or its variant models.

Original: In the case of k = 2 , we reduced the number of parameters for each of the equivariant models to fewer than the
Revised: In the case of k = 2 , we reduced the number of parameters for each of the equivariant models to fewer than the

Original: IsoGCN model because they exceeded the memory of the GPU (NVIDIA Tesla V100 with 32 GiB memory) with the same number of parameters.
Revised: IsoGCN model because they exceeded the memory of the GPU (NVIDIA Tesla V100 with 32 GiB memory) with the same number of parameters.

Original: In the case of k = 5 , neither the TFN nor the SE(3)-
Revised: In the case of k = 5 , neither the TFN nor the SE(3)-

Original: Transformer ﬁts into the memory of the GPU even with the number of parameters equal to 10.
Revised: Transformer ﬁts into the memory of the GPU even with the number of parameters equal to 10.

Original: For more details about the dataset and the model, see Appendix F.
Revised: For more details about the dataset and the model, see Appendix F.

Original: Figure 3 and Table 3 present the results of the qualitative and quantitative comparisons for the test dataset.
Revised: Figure 3 and Table 3 present the results of the qualitative and quantitative comparisons for the test dataset.

Original: The IsoGCN demonstrably outperforms all other baseline models.
Revised: The IsoGCN demonstrably outperforms all other baseline models.

Original: Moreover, owing to the computationally efﬁcient isometric transformation invariant nature of IsoGCNs, it also achieved a high prediction performance for the meshes that had a signiﬁcantly larger graph than those considered in the training dataset.
Revised: Moreover, owing to the computationally efﬁcient isometric transformation invariant nature of IsoGCNs, it also achieved a high prediction performance for the meshes that had a signiﬁcantly larger graph than those considered in the training dataset.

Original: The IsoGCN can scale up to 1M vertices, which is practical and is considerably greater than that reported in Sanchez-Gonzalez et al.
Revised: The IsoGCN can scale up to 1M vertices, which is practical and is considerably greater than that reported in Sanchez-Gonzalez et al.

Original: Therefore, we conclude that IsoGCN models can be trained on relatively smaller meshes 4 to save the training time and then used to apply the inference to larger meshes without observing signiﬁcant performance deterioration.[Figure]
Revised: Therefore, we conclude that IsoGCN models can be trained on relatively smaller meshes 5 to save the training time and then used to apply the inference to larger meshes without observing signiﬁcant performance deterioration.

Original: Table 4 reports the preprocessing and inference computation time using the equivariant models with m = 2 as the number of hops and FEA using FrontISTR 5.0.0.
Revised: Table 4 reports the preprocessing and inference computation time using the equivariant models with m = 2 as the number of hops and FEA using FrontISTR 5.0.0.

Original: We varied the time step ( ∆ t = 1 . 0 , 0 . 5 ) for the FEA computation to compute the t = 1 . 0 time evolution thus, resulting in different computation times and errors compared to an FEA with ∆ t = 0 .
Revised: We varied the time step ( ∆ t = 1 . 0 , 0 . 5 ) for the FEA computation to compute the t = 1 . 0 time evolution thus, resulting in different computation times and errors compared to an FEA with ∆ t = 0 .

Original: 01 , which was considered as the ground truth.
Revised: 01 , which was considered as the ground truth.

Original: Clearly, the IsoGCN is 3 to 5- times faster than the FEA with the same level of accuracy, while other equivariant models have almost the same speed as FrontISTR with ∆ t = 0 . 
Revised: Clearly, the IsoGCN is 3- to 5- times faster than the FEA with the same level of accuracy, while other equivariant models have almost the same speed as FrontISTR with ∆ t = 0 . 5 .

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: [Table]
Revised: [Table]

Original: In this study [Table] we proposed the GCNbased isometric transformation invariant and equivariant models called
Revised: In this study, we proposed the GCNbased isometric transformation invariant and equivariant models called

Original: IsoGCN .
Revised: IsoGCN .

Original: We discussed an example of an isometric adjacency matrix (IsoAM) that was closely related to the essential differential operators.
Revised: We discussed an example of an isometric adjacency matrix (IsoAM) that was closely related to the essential differential operators.

Original: The experiment results conﬁrmed that the proposed model leveraged the spatial structures and can deal with large-scale graphs.
Revised: The experiment results conﬁrmed that the proposed model leveraged the spatial structures and can deal with large-scale graphs.

Original: The computation time of the IsoGCN model is signiﬁcantly shorter than the
Revised: The computation time of the IsoGCN model is signiﬁcantly shorter than the

Original: FEA, which other equivariant models cannot achieve.
Revised: FEA, which other equivariant models cannot achieve.

Original: Therefore, IsoGCN must be the ﬁrst choice to learn physical simulations because of its computational efﬁciency as well as isometric transformation invariance and equivariance.
Revised: Therefore, IsoGCN must be the ﬁrst choice to learn physical simulations because of its computational efﬁciency as well as isometric transformation invariance and equivariance.

Original: Our demonstrations were conducted on the mesh structured dataset based on the FEA results.
Revised: Our demonstrations were conducted on the mesh structured dataset based on the FEA results.

Original: However, we expect IsoGCNs to be applied to various domains, such as object detection, molecular property prediction, and physical simulations using particles.
Revised: However, we expect IsoGCNs to be applied to various domains, such as object detection, molecular property prediction, and physical simulations using particles.

Original: [Table]
Revised: A CKNOWLEDGMENTS

Original: | = [Table] |V| = [Table]
Revised: The authors gratefully acknowledge Takanori Maehara for his helpful advice and NVIDIA for hardware donations.

Original: |V| = [Table]
Revised: This work was supported by JSPS KAKENHI Grant Number 19H01098.

Original: [Table]
Revised: [Table]

Original: A N OTATION
Revised: A N OTATION

Original: G
Revised: G

Original: A graph
Revised: A graph

Original: V
Revised: [Equation]

Original: A vertex set |V|
Revised: A vertex set [Equation]

Original: The number of vertices
Revised: The number of vertices

Original: E
Revised: E

Original: An edge set
Revised: An edge set

Original: Z +
Revised: [Equation]

Original: The positive integers d
Revised: The positive integers d

Original: The dimension of the Euclidean space
Revised: The dimension of the Euclidean space

Original: [Equation]
Revised: [Equation]

Original: The position of the i th vertex
Revised: The position of the i th vertex

Original: [Equation]
Revised: [Equation]

Original: Element k of x i
Revised: Element k of x i

Original: [Equation]
Revised: [Equation]

Original: The isometric adjacency matrix (IsoAM) (equation 4)
Revised: The isometric adjacency matrix (IsoAM) (equation 4)

Original: G
Revised: G

Original: [Equation]
Revised: [Equation]

Original: Slice of G in the spatial index (equation 4) 
Revised: Slice of G in the spatial index (equation 4) G

Original: [Equation]
Revised: [Equation]

Original: Element ( i, j, k ) of G
Revised: Element ( i, j, k ) of G

Original: [Equation]
Revised: [Equation]

Original: A rank- p tensor ﬁeld tensor ( f, p ∈ Z+ )
Revised: A rank- p tensor ﬁeld tensor ( f, p ∈ Z

Original: [Equation]
Revised: [Equation]

Revised: [Equation]

Original: Element ( i ; g ; k 1 , k 2 , . . . , k p ) of H ( p ) .
Revised: Element ( i ; g ; k 1 , k 2 , . . . , k p ) of H ( p ) .

Original: i refers to the permutation representation, k 1 , . . .
Revised: i refers to the permutation representation, k 1 , . . .

Original: k p refer to the Euclidean representation, and g denotes the feature index (See section 3).
Revised: k p refer to the Euclidean representation, and g denotes the feature index (See section 3).

Original: (cid:34) p (cid:79) G (cid:35)
Revised: (cid:34) [Equation] (cid:79) G (cid:35)

Original: [Equation]
Revised: [Equation]

Original: Convolution of the p th power of G and rank-0 tensor ﬁeld H (0) (equation 6, equation 10)
Revised: Convolution of the p th power of G and rank-0 tensor ﬁeld H (0) (equation 6, equation 10)

Original: [Table]
Revised: [Table]

Original: Contraction of the p th power of G and rank- q tensor ﬁelds (equation 7, equation 
Revised: Contraction of the p th power of G and rank- q tensor ﬁelds (equation 7, equation 11)

Original: [Table]
Revised: [Table]

Original: Tensor product of the p th power of G and
Revised: Tensor product of the p th power of G and

Original: rank- q tensor ﬁelds H ( q ) (equation 8)
Revised: rank- q tensor ﬁelds H ( q ) (equation 8)

Original: [Table]
Revised: [Table]

Original: The rank- p input tensor ﬁeld of the considered layer
Revised: The rank- p input tensor ﬁeld of the considered layer

Original: [Table]
Revised: [Table]

Original: The
Revised: The

Original: [Table]
Revised: [Table]

Original: The activation function
Revised: The activation function

Original: W
Revised: W

Original: The trainable parameter matrix
Revised: The trainable parameter matrix

Original: [Table]
Revised: [Table]

Original: An [Table] matrix
Revised: An adjacency matrix

Original: [Table]
Revised: [Table]

Original: The Kronecker delta
Revised: The Kronecker delta

Original: [Table]
Revised: [Table]

Original: The effective volume of the i th vertex (equation 46)
Revised: The effective volume of the i th vertex (equation 46)

Original: [Table]
Revised: [Table]

Original: The mean volume of the i th vertex (equation 47)
Revised: The mean volume of the i th vertex (equation 47)

Original: [Table]
Revised: [Table]

Original: A concrete instance of IsoAM (equation 15)
Revised: A concrete instance of IsoAM (equation 15)

Original: B P ROOFS OF PROPOSITIONS
Revised: B P ROOFS OF PROPOSITIONS

Original: In this section, we present the proofs of the propositions described in Section 3.
Revised: In this section, we present the proofs of the propositions described in Section 3.

Original: Let R(cid:51)g
Revised: Let R(cid:51)

Original: ( x l , x k ) =
Revised: [Equation] [Equation]

Original: ( x k − x l ) .
Revised: k − [Equation]

Original: Note that G is expressed using g ( x i , x j ) as G ij ;;: = (cid:80) k,l ∈V ,k (cid:54) =
Revised: Note that G is expressed using g ( x i , x j ) as G ij ;;: = (cid:80) k,l ∈V ,k (cid:54) =

Original: l T ijkl g ( x l , x k ) .
Revised: l T ijkl g ( x l , x k ) .

Original: B. P ROOF OF P ROPOSITION 3. 
Revised: B. P ROOF OF P ROPOSITION 3.

Original: Proof.
Revised: Proof.

Original: First, we demonstrate the invariance with respect to the translation with ∀ t ∈ R d .
Revised: First, we demonstrate the invariance with respect to the translation with ∀ t ∈ R d .

Original: g ( x i , x j ) is transformed invariantly as follows under translation:
Revised: g ( x i , x j ) is transformed invariantly as follows under translation:

Original: [Equation]
Revised: [Equation]

Original: By deﬁnition, T ijkl is also translation invariant.
Revised: By deﬁnition, T ijkl is also translation invariant.

Original: Thus,
Revised: Thus,

Original: [Equation]
Revised: [Equation]

Original: We then show an equivariance regarding the orthogonal transformation with ∀ U ∈
Revised: We then show an equivariance regarding the orthogonal transformation with ∀ U ∈

Original: O( d ) .
Revised: O( d ) .

Original: g ( x i , x j ) is transformed as follows by orthogonal transformation:
Revised: g ( x i , x j ) is transformed as follows by orthogonal transformation:

Original: [Equation]
Revised: [Equation]

Original: By deﬁnition, T ijkl is transformed to UT ijkl
Revised: By deﬁnition, T ijkl is transformed to UT ijkl

Original: U − 1 by orthogonal transformation.
Revised: U − 1 by orthogonal transformation.

Original: Thus,
Revised: Thus,

Original: [Equation]
Revised: [Equation]

Original: Therefore, G is both translation invariant and an orthogonal transformation equivariant.
Revised: Therefore, G is both translation invariant and an orthogonal transformation equivariant

Revised: [Equation]

Original: B. P ROOF OF P ROPOSITION 3. 
Revised: B. P ROOF OF P ROPOSITION 3.

Original: Proof.
Revised: Proof.

Original: Here, G (cid:12) G is translation invariant because G is translation invariant.
Revised: Here, G (cid:12) G is translation invariant because G is translation invariant.

Original: We prove rotation invariance under an orthogonal transformation ∀
Revised: We prove rotation invariance under an orthogonal transformation ∀

Original: U ∈
Revised: U ∈

Original: O( n ) .
Revised: O( n ) .

Original: In addition, G (cid:12) G is transformed under
Revised: In addition, G (cid:12) G is transformed under

Original: U as follows:
Revised: U as follows:

Original: [Equation]
Revised: [Equation]

Original: the dummy

Original: [Equation]

Original: Therefore
Revised: Therefore

Original: [Equation]
Revised: [Equation]

Original: G (cid:12) G is isometric transformation invariant
Revised: G (cid:12) G is isometric transformation invariant

Original: [Equation]
Revised: [Equation]

Original: B. P ROOF OF P ROPOSITION 3. 
Revised: B. P ROOF OF P ROPOSITION 3.

Original: Proof.
Revised: Proof.

Original: G ⊗ G is transformed under ∀ U ∈ O( n ) as follows:
Revised: G ⊗ G is transformed under ∀ U ∈ O( n ) as follows:

Original: [Equation]
Revised: [Equation]

Original: By regarding G ij ;; n G jl ;; o as one matrix H
Revised: By regarding G ij ;; n G jl ;; o as one matrix H

Original: no , it follows the coordinate transformation of ranktensor UHU T for each i , j , and l .
Revised: no , it follows the coordinate transformation of ranktensor UHU T for each i , j , and l .

Original: C P HYSICAL INTUITION OF ˜ D
Revised: C P HYSICAL INTUITION OF ˜ D

Original: In this section, we discuss the connection between the concrete IsoAM example ˜ D and the differential operators such as the gradient, divergence, the Laplacian, the Jacobian, and the Hessian operators.
Revised: In this section, we discuss the connection between the concrete IsoAM example ˜ D and the differential operators such as the gradient, divergence, the Laplacian, the Jacobian, and the Hessian operators.

Original: Let φ i ∈ R denote a rank-0 tensor (scalar) at the i th vertex.
Revised: Let φ i ∈ R denote a rank-0 tensor (scalar) at the i th vertex.

Original: Let us assume a partial derivative model of a rank-0 tensor φ at the i th vertex regarding the k th axis ( ∂φ/∂x k )
Revised: Let us assume a partial derivative model of a rank-0 tensor φ at the i th vertex regarding the k th axis ( ∂φ/∂x k )

Original: i ∈ R ( k ∈ { 1 , . . .
Revised: i ∈ R ( k ∈ { 1 , . . .

Original: , d } ) , that is based on the gradient model in the least squares moving particle semi-implicit method (Tamai &
Revised: , d } ) , that is based on the gradient model in the least squares moving particle semi-implicit method (Tamai &

Original: Koshizuka, 2014).
Revised: Koshizuka, 2014).

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Although one could deﬁne w ij as a function of the distance (cid:107) x j − x i (cid:107) , w ij was kept constant with respect to the distance required to maintain the simplicity of the model with fewer hyperparameters.
Revised: Although one could deﬁne w ij as a function of the distance (cid:107) x j − x i (cid:107) , w ij was kept constant with respect to the distance required to maintain the simplicity of the model with fewer hyperparameters.

Original: C. G RADIENT
Revised: C. G RADIENT

Original: ˜ D can be viewed as a Laplacian matrix based on D ; however, ˜ D ∗ H (0) can be interpreted as the gradient within the Euclidean space.
Revised: ˜ D can be viewed as a Laplacian matrix based on D ; however, ˜ D ∗ H (0) can be interpreted as the gradient within the Euclidean space.

Original: Let ∇ H (0)
Revised: Let ∇ H (0)

Original: ∈
Revised: ∈

Original: R |V|× f × d be an approximation of the gradient of H (0) .
Revised: R |V|× f × d be an approximation of the gradient of H (0) .

Original: Using equation 25, the gradient model can be expressed as follows:
Revised: Using equation 25, the gradient model can be expressed as follows:

Original: [Equation]
Revised: [Equation]

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Using this gradient model, we can conﬁrm that (˜ D ∗ H (0) )
Revised: Using this gradient model, we can conﬁrm that (˜ D ∗ H (0) )

Original: i ; g ; k = ( ∇ H (0) )
Revised: i ; g ; k = ( ∇ H (0) )

Original: i ; glk because
Revised: i ; glk because

Original: [Equation]
Revised: [Equation]

Original: the dummy indexl
Revised: dummy index

Original: [Equation]
Revised: [Equation]

Original: j [Equation] Therefore, ˜ D ∗ can be interpreted as the gradient operator within a Euclidean space.
Revised: Therefore, ˜ D ∗ can be interpreted as the gradient operator within a Euclidean space.

Original: C. D IVERGENCE
Revised: C. D IVERGENCE

Original: We show that ˜ D (cid:12) H (1) corresponds to the divergence.
Revised: We show that ˜ D (cid:12) H (1) corresponds to the divergence.

Original: Using D , the divergence model ∇ · H (1) ∈
Revised: Using D , the divergence model ∇ · H (1) ∈

Original: R |V|× f is expressed as follows:
Revised: R |V|× f is expressed as follows:

Original: [Equation]
Revised: [Equation]

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original: ˜ D (cid:12) H [Equation] 1) is
Revised: ˜ D (cid:12) H (1) is

Original: [Equation]
Revised: [Equation]

Original: C. L APLACIAN OPERATOR
Revised: C. L APLACIAN OPERATOR

Original: We prove that ˜ D (cid:12) ˜
Revised: We prove that ˜ D (cid:12) ˜

Original: D corresponds to the Laplacian operator within a Euclidean space
Revised: D corresponds to the Laplacian operator within a Euclidean space

Original: [Equation]
Revised: [Equation]

Original: Using equation
Revised: Using equation

Original: [Equation]
Revised: [Equation]

Original: the Laplacian model ∇ 2 H (0)
Revised: the Laplacian model ∇ 2 H (0)

Original: ∈ R |V|× f can be expressed as follows:
Revised: ∈ R |V|× f can be expressed as follows:

Original: [Equation]
Revised: [Equation]

Original: Then, (˜ D (cid:12) ˜ D ) H (0) is
Revised: Then, (˜ D (cid:12) ˜ D ) H (0) is

Original: [Equation]
Revised: [Equation]

Original: ( ∵ Change the dummy index m → j for the third and fourth terms )
Revised: ( ∵ Change the dummy index m → j for the third and fourth terms )

Original: [Equation]
Revised: [Equation]

Original: ( ∵ Change the dummy index n → l for the second and fourth terms )
Revised: ( ∵ Change the dummy index n → l for the second and fourth terms )

Original: [Equation]
Revised: [Equation]

Original: C. J ACOBIAN AND H ESSIAN OPERATORS
Revised: C. J ACOBIAN AND H ESSIAN OPERATORS

Original: Considering a similar discussion
Revised: Considering a similar discussion

Original: [Equation]
Revised: [Equation]

Original: we can show the following dependencies.
Revised: we can show the following dependencies.

Original: For the Jacobian model,
Revised: For the Jacobian model,

Original: [Equation]
Revised: [Equation]

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: the 
Revised: the Hessian

Original: [Equation]
Revised: [Equation]

Original: D I SO GCN MODELING DETAILS
Revised: D I SO GCN MODELING DETAILS

Original: To achieve isometric transformation invariance and equivariance, there are several rules to follow.
Revised: To achieve isometric transformation invariance and equivariance, there are several rules to follow.

Original: Here, we describe the desired focus when constructing an IsoGCN model.
Revised: Here, we describe the desired focus when constructing an IsoGCN model.

Original: In this section, a rank- p tensor denotes a tensor the rank of which is p ≥ 1 and σ denotes a nonlinear activation function.
Revised: In this section, a rank- p tensor denotes a tensor the rank of which is p ≥ 1 and σ denotes a nonlinear activation function.

Original: W is a trainable weight matrix and b is a trainable bias.
Revised: W is a trainable weight matrix and b is a trainable bias.

Original: D. A CTIVATION AND BIAS
Revised: D. A CTIVATION AND BIAS

Original: As the nonlinear activation function is not isometric transformation equivariant, nonlinear activation to rank- p tensors cannot be applied, while one can apply any activation to rank-0 tensors.
Revised: As the nonlinear activation function is not isometric transformation equivariant, nonlinear activation to rank- p tensors cannot be applied, while one can apply any activation to rank-0 tensors.

Original: In addition, adding bias is also not isometric transformation equivariant, one cannot add bias when performing an afﬁne transformation to rank- p tensors.
Revised: In addition, adding bias is also not isometric transformation equivariant, one cannot add bias when performing an afﬁne transformation to rank- p tensors.

Original: Again, one can add bias to rank-0 tensors.
Revised: Again, one can add bias to rank-0 tensors.

Original: Thus, for instance, if one converts from rank-0 tensors H (0) to rank-1 tensors using IsoAM G ,
Revised: Thus, for instance, if one converts from rank-0 tensors H (0) to rank-1 tensors using IsoAM G ,

Original: G ∗ σ ( H (0) W + b ) and ( G ∗ σ ( H (0) ))
Revised: G ∗ σ ( H (0) W + b ) and ( G ∗ σ ( H (0) ))

Original: W are isometric equivariant functions, however ( G ∗ H (0) )
Revised: W are isometric equivariant functions, however ( G ∗ H (0) )

Original: W + b and σ (cid:16) ( G ∗ σ ( H (0) ))
Revised: W + b and σ (cid:16) ( G ∗ σ ( H (0) ))

Original: W (cid:17) are not due to the bias and the nonlinear activation, respectively.
Revised: W (cid:17) are not due to the bias and the nonlinear activation, respectively.

Original: Likewise, regarding a conversion from rank-1 tensors H (1) to rank-0 tensors, σ (cid:16) ( G (cid:12) H (1) ) W + b (cid:17) and σ (cid:16) G (cid:12) ( H (1) W ) (cid:17) are isometric transformation invariant functions; however, G (cid:12) ( H (1) W + b ) and ( G (cid:12) σ ( H (1) ))
Revised: Likewise, regarding a conversion from rank-1 tensors H (1) to rank-0 tensors, σ (cid:16) ( G (cid:12) H (1) ) W + b (cid:17) and σ (cid:16) G (cid:12) ( H (1) W ) (cid:17) are isometric transformation invariant functions; however, G (cid:12) ( H (1) W + b ) and ( G (cid:12) σ ( H (1) ))

Original: W + b are not.
Revised: W + b are not.

Original: To convert rank- p tensors to rank- q tensors ( q ≥ 1 ), one can apply neither bias nor nonlinear activation.
Revised: To convert rank- p tensors to rank- q tensors ( q ≥ 1 ), one can apply neither bias nor nonlinear activation.

Original: To add nonlinearity to such a conversion, we can multiply the converted rank-0 tensors σ (( (cid:78) p G (cid:12) H ( p ) )
Revised: To add nonlinearity to such a conversion, we can multiply the converted rank-0 tensors σ (( (cid:78) p G (cid:12) H ( p ) )

Original: W + b ) with the input tensors H ( p ) or the output tensors H ( q ) .
Revised: W + b ) with the input tensors H ( p ) or the output tensors H ( q ) .

Original: D. P REPROCESSING OF INPUT FEATURE
Revised: D. P REPROCESSING OF INPUT FEATURE

Original: Similarly to the discussion regarding the biases, we have to take care of the preprocessing of rank- p tensors to retain isometric transformation invariance because adding a constant array and component-wise scaling could distort the tensors, resulting in broken isometric transformation equivariance.
Revised: Similarly to the discussion regarding the biases, we have to take care of the preprocessing of rank- p tensors to retain isometric transformation invariance because adding a constant array and component-wise scaling could distort the tensors, resulting in broken isometric transformation equivariance.

Original: For instance, H ( p ) / Std all (cid:104) H ( p ) (cid:105) is a valid transformation to retain isometric transformation equivariance, assuming Std all (cid:104) H ( p )
Revised: For instance, H ( p ) / Std all (cid:104) H ( p ) (cid:105) is a valid transformation to retain isometric transformation equivariance, assuming Std all (cid:104) H ( p )

Original: (cid:105) ∈ R is a standard deviation of all components of H ( p ) .
Revised: (cid:105) ∈ R is a standard deviation of all components of H ( p ) .

Original: However, conversions such as H ( p ) /
Revised: However, conversions such as H ( p ) /

Original: Std component (cid:104) H ( p ) (cid:105) and H ( p ) −
Revised: Std component (cid:104) H ( p ) (cid:105) and H ( p ) −

Original: Mean (cid:104) H ( p ) (cid:105) are not isometric transformation equivariant, assuming that Std component (cid:104) H ( p )
Revised: Mean (cid:104) H ( p ) (cid:105) are not isometric transformation equivariant, assuming that Std component (cid:104) H ( p )

Original: (cid:105) ∈
Revised: (cid:105) ∈

Original: R d p is a component-wise standard deviation.
Revised: R d p is a component-wise standard deviation.

Original: D. S CALING
Revised: D. S CALING

Original: Because the concrete instance of IsoAM ˜ D corresponds to the differential operator, the scale of the output after operations regarding ˜ D can be huge.Thus, we rescale ˜ D using the scaling factor (cid:104) Mean sample ,
Revised: Because the concrete instance of IsoAM ˜ D corresponds to the differential operator, the scale of the output after operations regarding ˜ D can be huge.

Original: i

Original: 2 ii 1 + D 2 ii ;; 1 / 2 , where Mean sample ,i denotes the mean over the samples and vertices.
Revised: Thus, we rescale ˜ D using the scaling factor where Mean sample , denotes the mean over the samples and vertices.

Original: D. I MPLEMENTATION
Revised: D. I MPLEMENTATION

Original: Because an adjacency matrix A is usually a sparse matrix for a regular mesh, A ( m ) in equationis also a sparse matrix for a sufﬁciently small m .
Revised: Because an adjacency matrix A is usually a sparse matrix for a regular mesh, A ( m ) in equationis also a sparse matrix for a sufﬁciently small m .

Original: Thus, we can leverage sparse matrix multiplication in the IsoGCN computation.
Revised: Thus, we can leverage sparse matrix multiplication in the IsoGCN computation.

Original: This is one major reason why IsoGCNs can compute rapidly.
Revised: This is one major reason why IsoGCNs can compute rapidly.

Original: If the multiplication (tensor product or contraction) of IsoAMs must be computed multiple times the associative property of the IsoAM can be utilized.
Revised: If the multiplication (tensor product or contraction) of IsoAMs must be computed multiple times the associative property of the IsoAM can be utilized.

Original: For instance, it is apparent that (cid:104)(cid:78) k G (cid:105) ∗ H (0)
Revised: For instance, it is apparent that (cid:104)(cid:78) k G (cid:105) ∗ H (0)

Original: =
Revised: =

Original: G ⊗ ( G ⊗ . . .
Revised: G ⊗ ( G ⊗ . . .

Original: ( G ∗ H (0) )) .
Revised: ( G ∗ H (0) )) .

Original: Assuming that the number of nonzero elements in A ( m ) equals n and H (0)
Revised: Assuming that the number of nonzero elements in A ( m ) equals n and H (0)

Original: ∈ R |V|× f , then the computational complexity of the right-hand side is O ( n |V| fd k ) .
Revised: ∈ R |V|× f , then the computational complexity of the right-hand side is O ( n |V| fd k ) .

Original: This is an exponential order regarding d .
Revised: This is an exponential order regarding d .

Original: However, d and k are usually small numbers (typically d = 3 and k ≤ 4 ).
Revised: However, d and k are usually small numbers (typically d = 3 and k ≤ 4 ).

Original: Therefore one can compute an IsoGCN layer with a realistic spatial dimension d and tensor rank k fast and memory efﬁciently.
Revised: Therefore one can compute an IsoGCN layer with a realistic spatial dimension d and tensor rank k fast and memory efﬁciently.

Original: In our implementation, both a sparse matrix operation and associative property are utilized to realize fast computation.
Revised: In our implementation, both a sparse matrix operation and associative property are utilized to realize fast computation.

Original: E E XPERIMENT DETAILS : DIFFERENTIAL OPERATOR DATASET E. M ODEL ARCHITECTURES
Revised: E E XPERIMENT DETAILS : DIFFERENTIAL OPERATOR DATASET E. M ODEL ARCHITECTURES

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]

Original: [Table] [Equation] [Table] [Equation]
Revised: <latexit

Original: [Table]
Revised: [Table]

Original: [Equation] [Table] YeUfV36zsrnyu5U/ rhyu9FeuVsKVx5X/
Revised: +

Original: [Table] [Table]
Revised: [Table] [Table]

Original: PmP/
Revised: +ZeV368crvRXrlbCla8r/

Original: [Table]
Revised: [Table]

Original: = =</latexit [Table] [Identity] <latexit sha
Revised: /fnPzZvHm39/

Original: [Table]
Revised: [Table]

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Original: [Table]
Revised: [Table]

Original: [Table] [Table] [Table] /yLF9cv9rY/OeNl+dv1n6/Xf+Zz89W/n7l1yu/Xdlc+ZeV368crvRXrlbCla8r/7Xyp5X/fnPzZvHm39/8h2364x/Vff5uxft585//C8rqZeY=</latexit> Propagation <latexit sha1_base64="+u4ErTcb29kkW7mQY4fORWw9Ars=">AAA0yniclVtbc9zKcabtXBzGiY/tx7xMhVLZSVGMKCkVP6RS5p0Ul/erpNVRYbG9WIi4CTMLkoI3T/4Lfk2e/ZPyb9Izg0H3YEEdm1US0d8305jL19O9WHBUJLFUL
Revised: /erpNVRYbG9WIi4CTMLkoI3T/

Original: [Table]
Revised: [Table]

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Original: [Figure]
Revised: [Figure]

Original: [Table]
Revised: [Table]

Original: Figure 4 represents the IsoGCN model used for the differential operator dataset.
Revised: Figure 4 represents the IsoGCN model used for the differential operator dataset.

Original: We used the tanh activation function as a nonlinear activation function because we expect the target temperature ﬁeld to be smooth.
Revised: We used the tanh activation function as a nonlinear activation function because we expect the target temperature ﬁeld to be smooth.

Original: Therefore, we avoid using non-differentiable activation functions such as the rectiﬁed linear unit (ReLU) (Nair & Hinton, 2010).
Revised: Therefore, we avoid using non-differentiable activation functions such as the rectiﬁed linear unit (ReLU) (Nair & Hinton, 2010).

Original: For GCN and its variants, we simply replaced the IsoGCN layers with the corresponding ones.
Revised: For GCN and its variants, we simply replaced the IsoGCN layers with the corresponding ones.

Original: We stacked m (= 2 , 5) layers for GCN, GIN, GCNII
Revised: We stacked m (= 2 , 5) layers for GCN, GIN, GCNII

Original: Cluster-GCN.
Revised: Cluster-GCN.

Original: We used an m hop adjacency matrix for SGCN.
Revised: We used an m hop adjacency matrix for SGCN.

Original: For the TFN and SE(3)-Transformer, we set the hyperparameters to have almost the same number of parameters as in the IsoGCN model.
Revised: For the TFN and SE(3)-Transformer, we set the hyperparameters to have almost the same number of parameters as in the IsoGCN model.

Original: The settings of the hyperparameters are shown in Table 5.
Revised: The settings of the hyperparameters are shown in Table 5.

Original: E. R ESULT DETAILS
Revised: E. R ESULT DETAILS

Original: Table 6 represents the detailed comparison of training results.
Revised: Table 6 represents the detailed comparison of training results.

Original: The results show that an IsoGCN outperforms other GCN models for all settings.
Revised: The results show that an IsoGCN outperforms other GCN models for all settings.

Original: Compared to other equivariant models, IsoGCN has competitive performance compared to equivariant models with shorter inference time as shown in
Revised: Compared to other equivariant models, IsoGCN has competitive performance compared to equivariant models with shorter inference time as shown in

Original: Table 7.
Revised: Table 7.

Original: Therefore, it can be found out the proposed model has a strong expressive power to express differential regarding space with less computation resources compared to the TFN and SE(3)-
Revised: Therefore, it can be found out the proposed model has a strong expressive power to express differential regarding space with less computation resources compared to the TFN and SE(3)-

Original: Transformer.
Revised: Transformer

Original:  F E XPERIMENTS DETAILS : ANISOTROPIC NONLINEAR HEAT EQUATION DATASET 
Revised: . F E XPERIMENTS DETAILS : ANISOTROPIC NONLINEAR HEAT EQUATION DATASET F. D ATASET

Original: F D ATASET

Original: The purpose of the experiment was to solve the anisotropic nonlinear heat diffusion under an adiabatic boundary condition.
Revised: The purpose of the experiment was to solve the anisotropic nonlinear heat diffusion under an adiabatic boundary condition.

Original: The governing equation is deﬁned as follows:
Revised: The governing equation is deﬁned as follows:

Original: [Equation]
Revised: [Equation]

Original: where T is the temperature ﬁeld, T 0 .
Revised: where T is the temperature ﬁeld, T 0 .

Original: 0 is the initial temperature ﬁeld,
Revised: 0 is the initial temperature ﬁeld,

Original: C ∈ R d
Revised: C ∈ R d

Original: × d is an anisotropic diffusion tensor and n ( x b ) is the normal vector at x b ∈ ∂ Ω .
Revised: × d is an anisotropic diffusion tensor and n ( x b ) is the normal vector at x b ∈ ∂ Ω .

Original: Here, C depends on temperature thus the equation is nonlinear.
Revised: Here, C depends on temperature thus the equation is nonlinear.

Original: We randomly generate C ( T = − 1) for it to be a positive semidefinite symmetric tensor with eigenvalues varying from 0.0 to 0.02.
Revised: We randomly generate C ( T = − 1) for it to be a positive semidefinite symmetric tensor with eigenvalues varying from 0.0 to 0.02.

Original: Then, we deﬁned the linear temperature dependency the slope of which is − C
Revised: Then, we deﬁned the linear temperature dependency the slope of which is − C

Original: ( T = − 1) / 4 .
Revised: ( T = − 1) / 4 .

Original: The function of the anisotropic diffusion tensor is uniform for each sample.
Revised: The function of the anisotropic diffusion tensor is uniform for each sample.

Original: The task is deﬁned to predict the temperature ﬁeld at t = 0 .
Revised: The task is deﬁned to predict the temperature ﬁeld at t = 0 .

Original: ( T 0 .
Revised: ( T 0 .

Original: 2 , T 0 .
Revised: 2 , T 0 .

Original: 4 , T 0 .
Revised: 4 , T 0 .

Original: 6 , T 0 .
Revised: 6 , T 0 .

Original: 8 , T 1 . 0 )
Revised: 8 , T 1 . 0 )

Original: from the given initial temperature ﬁeld, material property, and mesh geometry.
Revised: from the given initial temperature ﬁeld, material property, and mesh geometry.

Original: However, the performance is evaluated only with T 1 . 0 to focus on the predictive performance.
Revised: However, the performance is evaluated only with T 1 . 0 to focus on the predictive performance.

Original: We inserted other output features to stabilize the trainings.
Revised: We inserted other output features to stabilize the trainings.

Original: Accordingly, the diffusion number of this problem is 
Revised: Accordingly, the diffusion number of this problem is C

Original: [Equation]
Revised: [Equation]

Original: Figure 5 represents the process of generating the dataset.
Revised: Figure 5 represents the process of generating the dataset.

Original: We generated up to 9 FEA results for each
Revised: We generated up to 9 FEA results for each

Original: CAD shape.
Revised: CAD shape.

Original: To avoid data leakage in terms of the CAD shapes, we ﬁrst split them into training, validation, and test datasets, and then applied the following process.
Revised: To avoid data leakage in terms of the CAD shapes, we ﬁrst split them into training, validation, and test datasets, and then applied the following process.

Original: [Table] [Table]
Revised: [Table] [Table]

Original: SE(3)-Trans
Revised: SE(3)-Trans

Original: [Table]
Revised: [Table]

Original: Using one CAD shape
Revised: Using one CAD shape

Original: [Table]
Revised: [Table]

Original: we generated up to three meshes using clscale (a control parameter of the mesh characteristic lengths) = 0.20, 0.25, and 0.30.
Revised: we generated up to three meshes using clscale (a control parameter of the mesh characteristic lengths) = 0.20, 0. [Table] and 0.30.

Original: To facilitate the training process
Revised: To facilitate the training process

Original: [Table]
Revised: [Table]

Original: we scaled the meshes to ﬁt into a cube with an edge length equal to 1.
Revised: we scaled the meshes to ﬁt into a cube with an edge length equal to 1.

Original: Using one mesh
Revised: Using one mesh

Original: [Table]
Revised: [Table]

Original: we generated three initial conditions randomly using a Fourier series of the 2nd to 10th orders.
Revised: we generated three initial conditions randomly using a Fourier series of the 2nd to 10th orders.

Original: We then applied an FEA to each initial condition and material property determined randomly as described above.
Revised: We then applied an FEA to each initial condition and material property determined randomly as described above.

Original: We applied an implicit method to solve time evolutions and a direct method to solve the linear equations.
Revised: We applied an implicit method to solve time evolutions and a direct method to solve the linear equations.

Original: The FEA time step ∆ t was set to 0.01.
Revised: The FEA time step ∆ t was set to 0.01.

Original: During this process
Revised: During this process

Original: [Table]
Revised: [Table]

Original: some of the meshes or FEA results may not have been available due to excessive computation time or non-convergence.
Revised: some of the meshes or FEA results may not have been available due to excessive computation time or non-convergence.

Original: Therefore, the size of the dataset was not exactly equal to the number multiplied by 9.
Revised: Therefore, the size of the dataset was not exactly equal to the number multiplied by 9.

Original: Finally, we obtained 439 FEA results for the training dataset, 143 FEA results for the validation dataset, and 140 FEA results for the test dataset.
Revised: Finally, we obtained 439 FEA results for the training dataset, 143 FEA results for the validation dataset, and 140 FEA results for the test dataset.

Original: [Table]
Revised: [Table]

Original:  results
Revised: FEA results

Original: [Table]
Revised: [Table]

Original: Material property
Revised: /nbjDqddyTzunHk9bZ2+ruHqJH6DF6glx0is7QBzRAQ0TRJfqOfqCfzlNn4Iydz2Xr/ [Table] [Table] =</

Revised: [Table]

Revised: <latexit

Original: [Table]
Revised: [Table]

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Original: A smaller clscale parameter generates smaller meshes.
Revised: A smaller clscale parameter generates smaller meshes.

Original: F. I NPUT FEATURES
Revised: F. I NPUT FEATURES

Original: To express the geometry information, we extracted the effective volume of the i th vertex V eﬀective i and the mean volume of the
Revised: To express the geometry information, we extracted the effective volume of the i th vertex V eﬀective i and the mean volume of the

Original: i th vertex
Revised: i th vertex

Original: V mean i , which are deﬁned as follows:
Revised: V mean i , which are deﬁned as follows:

Original: [Equation]
Revised: [Equation]

Original: where N ei is the set of elements, including the i th vertex.
Revised: where N ei is the set of elements, including the i th vertex.

Original: For GCN or its variant models, we tested several combinations of input vertex features T 0 . 0 , C ,
Revised: For GCN or its variant models, we tested several combinations of input vertex features T 0 . 0 , C ,

Original: V eﬀective , V mean , and x (Table 9).
Revised: V eﬀective , V mean , and x (Table 9).

Original: For the IsoGCN model, inputs were T 0 . 0 , C , V eﬀective , and
Revised: For the IsoGCN model, inputs were T 0 . 0 , C , V eﬀective , and

Original: V mean .
Revised: V mean .

Original: F. M ODEL ARCHITECTURES
Revised: F. M ODEL ARCHITECTURES

Original: Figure 6 represents the IsoGCN model used for the anisotropic nonlinear heat equation dataset.
Revised: Figure 6 represents the IsoGCN model used for the anisotropic nonlinear heat equation dataset.

Original: We used the tanh activation function as a nonlinear activation function because we expect the target temperature ﬁeld to be smooth.
Revised: We used the tanh activation function as a nonlinear activation function because we expect the target temperature ﬁeld to be smooth.

Original: Therefore, we avoid using non-differentiable activation functions such as the rectiﬁed linear unit (ReLU) (Nair & Hinton, 2010).
Revised: Therefore, we avoid using non-differentiable activation functions such as the rectiﬁed linear unit (ReLU) (Nair & Hinton, 2010).

Original: Although the model looks complicated, one propagation block corresponds to the ﬁrst-order Taylor expansion T ( t +∆ t ) (cid:39) ∇ C (cid:12)∇ T ( t )
Revised: Although the model looks complicated, one propagation block corresponds to the ﬁrst-order Taylor expansion T ( t +∆ t ) (cid:39) ∇ C (cid:12)∇ T ( t )

Original: + T ( t ) because the propagation block is expressed as ˜ D (cid:12) C (cid:12) MLP( T )
Revised: + T ( t ) because the propagation block is expressed as ˜ D (cid:12) C (cid:12) MLP( T )

Original: ˜ D ∗ T + T , where T denotes the
Revised: ˜ D ∗ T + T , where T denotes the

Original: [Table]
Revised: [Table]

Original: +LcUoz+82Vr3XOWeFPp3z8p9R42VkBoC/h+v66VsV9TlpK+JTt0mjVqxm/n+ixiUK6lLf8EtADYMR+lmS349KCO4+fbe22f5TnuWH61cbm7/deHn+Zu2P282f+fx85R9W/nHlNyubK79b+ePK4Up/5WolXHlc+a+V/17505vrNz+8+fc3/2Gr/vQnTZtfrXg/ b/7zfwHCGmR0</latexit
Revised: +LcUoz+82Vr3XOWeFPp3z8p9R42VkBoC/h+v66VsV9TlpK+JTt0mjVqxm/n+ixiUK6lLf8EtADYMR+lmS349KCO4+fbe22f5TnuWH61cbm7/deHn+Zu2P282f+fx85R9W/nHlNyubK79b+ePK4Up/5WolXHlc+a+V/17505vrNz+8+fc3/2Gr/vQnTZtfrXg/b/7zfwHCGmR

Original: [Figure]
Revised: [Figure]

Original: [Table] [Table]
Revised: [Figure] [Table]

Original: rank-0 tensor input to the propagation block.
Revised: rank-0 tensor input to the propagation block.

Original: By stacking this propagation block p times, we can approximate the p th order Taylor expansion of the anisotropic nonlinear heat equation.
Revised: By stacking this propagation block p times, we can approximate the p th order Taylor expansion of the anisotropic nonlinear heat equation.

Original: For GCN and its variants, we simply replaced the IsoGCN layers with the corresponding ones.
Revised: For GCN and its variants, we simply replaced the IsoGCN layers with the corresponding ones.

Original: We stacked m (= 2 , 5) layers for GCN, GIN, GCNII, and Cluster-GCN.
Revised: We stacked m (= 2 , 5) layers for GCN, GIN, GCNII, and Cluster-GCN.

Original: We used an m hop adjacency matrix for SGCN.
Revised: We used an m hop adjacency matrix for SGCN.

Original: For the TFN and SE(3)-Transformer, we set the hyperparameters to as many parameters as possible that would ﬁt on the GPU because the TFN and SE(3)-Transformer with almost the same number of parameters as in IsoGCN did not ﬁt on the GPU we used (NVIDIA Tesla V100 with 32 GiB memory).
Revised: For the TFN and SE(3)-Transformer, we set the hyperparameters to as many parameters as possible that would ﬁt on the GPU because the TFN and SE(3)-Transformer with almost the same number of parameters as in IsoGCN did not ﬁt on the GPU we used (NVIDIA Tesla V100 with 32 GiB memory).

Original: The settings of the hyperparameters are shown in Table 8.
Revised: The settings of the hyperparameters are shown in Table 8.

Original: F. R ESULT DETAILS
Revised: F. R ESULT DETAILS

Original: Table 9 shows a detailed comparison of the training results.
Revised: Table 9 shows a detailed comparison of the training results.

Original: The inclusion of x in the input features of the baseline models did not improve the performance.
Revised: The inclusion of x in the input features of the baseline models did not improve the performance.

Original: In addition, if x is included in the input features, a loss of the generalization capacity for larger shapes compared to the training dataset may result as it extrapolates.
Revised: In addition, if x is included in the input features, a loss of the generalization capacity for larger shapes compared to the training dataset may result as it extrapolates.

Original: The proposed model achieved the best performance compared to the baseline models considered.
Revised: The proposed model achieved the best performance compared to the baseline models considered.

Original: Therefore, we concluded that the essential features regarding the mesh shapes are included in ˜ D .
Revised: Therefore, we concluded that the essential features regarding the mesh shapes are included in ˜ D .

Original: Besides, IsoGCN can scale up to meshes with 1M vertices as shown in
Revised: Besides, IsoGCN can scale up to meshes with 1M vertices as shown in

Original: Figure 7.
Revised: Figure 7.

Original: [Table] [Table]
Revised: [Table] [Table]

Revised: Training samples

Original:  [Figure]
Revised: [Table] [Figure]