物理シミュレーションのための同変グラフニューラルネットワーク
Deep learning and Physics
2021 年 4 月 22 日
堀江正信
株式会社科学計算総合研究所
筑波大学システム情報工学研究群
1背景：物理シミュレーションを軽量化したい

構造

熱

流体

• シミュレーションは製品の物理特性の評価において重要• 計算時間が長く、シミュレーションを用いた最適化が困難な場合がある➡ 軽量なモデル（サロゲートモデル）で重いシミュレーションを置き換えたい2

関連研究：物理シミュレーションの機械学習

Umetani+ 2018
• リアルタイムで車体の空力性能予測• メッシュ構造固定
3

関連研究：物理シミュレーションの機械学習

Umetani+ 2018
• リアルタイムで車体の空力性能予測

Sanchez-Gonzalez+ 2020
• 任意点群での学習• 高速化は未達成• メッシュ構造固定
4

関連研究：物理シミュレーションの機械学習

Umetani+ 2018
• リアルタイムで車体の空力性能予測

Sanchez-Gonzalez+ 2020

Chang+ 2020

• 任意点群での学習• 任意梁構造での学習• 高速化は未達成• 梁部材選定の最適化• メッシュ構造固定
5

関連研究：物理シミュレーションの機械学習
Isometric transformation!

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Isometric transformation!

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JNXRGtDC/QmTyZlkOoHUtP7vMQCVQaPUjzr/fDqmX57x/w9xSyzb6HKAvdfmrfHng0hSVgb90D0udjGBIghn+n/HjHeIdVvsekq2DplrfUrpPksMjnTFMWxgnXjXuZinIN2dx/fxQWM42Cj9QpyXqaJfni/mPd+eLnoIPMMNLfZxal70+9VF1dopuhgjBZ6PwzjbKIe26FTBPrvFHCX+4EOlkvAs1YGEYg4E1leF/QKHjbEzjSXenlyXQCGU7GLn30z+KUUozy/21j1HuecFfp0zsu/RY2XkRkA/h6u66tvNdTnpG2IV90ujVqxmfn/iRYDFNRAv+GXgBoGI4yzJL8flRDcffpubbP9pzzLF9evNjb/fuPl+Zu1X2/Xf+bz85W/Wvmble9XNlf+YeXXK4cr/ZWrlXDlX1d+t/JfK//95jdv/u3Nv7/5D9v0pz+p+/zlivfz5j//F6Y5aFc=</latexit>

Ground Truth (FEA)<latexit 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QV6nSfjMkj1A6nJfV5igSqDRyme9b579Uy/vWP+nmKa2bdQZYH7L83bY88GkCSsjXsg+lxsYgLEkM/0f48Y75Dqt9h0FWydstb6FdJ8GpmcaYriWMGqcS9zMcpBu7uP7+ICRnGw1noFOS/TRD+8n896372cd5B5Bppb7+LUven3qosrNFN0MEYLve8GcTZWj+3QKQL9dwq4y6eBDpY+4FkrgwhEnIksrwt6BQ9rYmuSS708uS4Aw4nYxs++GfxKimGe360te49zTgp9Ouflb1DjZWQGgL8Hq/rqaw31OWkb4lW3S6NWbGb+f6LFBQrqQr/hl4AaBEOMsyS/H5YQ3H38ZmW9/ac8ixdXr9bW/3nt5dmblT9s1n/m8+OlXyz9cunbpfWlf1n6w9L+0unS5VK4NF/6r6X/WfrfN7dv/u3Nv7/5D9v0hz+o+/zjkvfz5j//DKReZfk=</latexit>

IsoGCN (Ours)

Training samples<latexit 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CZPJmWQ6gdS0/u8xAJVBo9SPOt9/+qZfnvH/D3FLLNvocoC91+at8eeDSFJWBv3QPS52MYEiCGf6f8eMd4h1W+x6SrYOmWt9Suk+SwyOdMUxbGCdeNe5mKcg3Z3H9/FBYzjYKP1CnJepol+eL+Y975/uegg8ww0t9nFqXvT71UXV2im6GCMFnrfD+Nsoh7boVME+u8UcJfPAx0sfcCzVgYRiDgTWV4X9AoeNsTONJd6eXJdAIZTsYuffTP4lRSjPL/bWPUe55wV+nTOy79FjZeRGQD+Hq7rq2811OekbYhX3S6NWrGZ+f+JFgMU1EC/4ZeAGgYjjLMkvx+VENx9+m5ts/2nPMsX1682Nv9+4+XFm7Xfbdd/5vPTlb9a+euVX69srvxm5XcrhyvnK1cr4co/r/zHyn+t/Pebj2/+5c2/vvk32/THP6r7/HLF+3nz7/8LMapoSw==</latexit>

Umetani+ 2018
• リアルタイムで車体の空力性能予測

Sanchez-Gonzalez+ 2020

Chang+ 2020

Horie+ 2020 (IsoGCN)• 任意点群での学習• 任意梁構造での学習• 任意メッシュでの学習• 高速化は未達成• 梁部材選定の最適化• 高速化を達成• メッシュ構造固定• 物理現象の対称性を考慮
6

関連研究：物理シミュレーションの機械学習
Graph Neural Network の適用

同変性の導入Isometric transformation!

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y IsoGCN?
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y IsoGCN
Isometric transformation!

<latexit 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JNXRGtDC/QmTyZlkOoHUtP7vMQCVQaPUjzr/fDqmX57x/w9xSyzb6HKAvdfmrfHng0hSVgb90D0udjGBIghn+n/HjHeIdVvsekq2DplrfUrpPksMjnTFMWxgnXjXuZinIN2dx/fxQWM42Cj9QpyXqaJfni/mPd+eLnoIPMMNLfZxal70+9VF1dopuhgjBZ6PwzjbKIe26FTBPrvFHCX+4EOlkvAs1YGEYg4E1leF/QKHjbEzjSXenlyXQCGU7GLn30z+KUUozy/21j1HuecFfp0zsu/RY2XkRkA/h6u66tvNdTnpG2IV90ujVqxmfn/iRYDFNRAv+GXgBoGI4yzJL8flRDcffpubbP9pzzLF9evNjb/fuPl+Zu1X2/Xf+bz85W/Wvmble9XNlf+YeXXK4cr/ZWrlXDlX1d+t/JfK//95jdv/u3Nv7/5D9v0pz+p+/zlivfz5j//F6Y5aFc=</latexit>

Ground Truth (FEA)<latexit 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QV6nSfjMkj1A6nJfV5igSqDRyme9b579Uy/vWP+nmKa2bdQZYH7L83bY88GkCSsjXsg+lxsYgLEkM/0f48Y75Dqt9h0FWydstb6FdJ8GpmcaYriWMGqcS9zMcpBu7uP7+ICRnGw1noFOS/TRD+8n896372cd5B5Bppb7+LUven3qosrNFN0MEYLve8GcTZWj+3QKQL9dwq4y6eBDpY+4FkrgwhEnIksrwt6BQ9rYmuSS708uS4Aw4nYxs++GfxKimGe360te49zTgp9Ouflb1DjZWQGgL8Hq/rqaw31OWkb4lW3S6NWbGb+f6LFBQrqQr/hl4AaBEOMsyS/H5YQ3H38ZmW9/ac8ixdXr9bW/3nt5dmblT9s1n/m8+OlXyz9cunbpfWlf1n6w9L+0unS5VK4NF/6r6X/WfrfN7dv/u3Nv7/5D9v0hz+o+/zjkvfz5j//DKReZfk=</latexit>

IsoGCN (Ours)

Training samples<latexit sha1_base64="2kJMke9faGRQ78wZW0RD0+/K0VU=">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</latexit>

Umetani+ 2018
• リアルタイムで車体の空力性能予測• メッシュ構造固定

Sanchez-Gonzalez+ 2020

Chang+ 2020

Horie+ 2020 (IsoGCN)• 任意点群での学習• 任意梁構造での学習• 任意メッシュでの学習• 高速化は未達成• 梁部材選定の最適化• 高速化を達成

7• 物理現象の対称性を考慮目次• Graph neural network• 同変性• IsoGCN: 同変性＋Graph Neural Network
• まとめ

8

目次

• Graph neural network• 同変性• IsoGCN: 同変性＋Graph Neural Network
• まとめ

9

3D データをどう扱うか？

Ahmed+ 201810

3D データをどう扱うか？

複雑形状・3 次元の場の表現に不向きAhmed+ 201811

3D データをどう扱うか？

・形状の滑らかさを表現しにくい
・データ構造として非効率的

Ahmed+ 201812

3D データをどう扱うか？

物理シミュレーションでよく用いられるデータ構造

Ahmed+ 201813

3D データをどう扱うか？
物理シミュレーションでよく用いられるデータ構造

しかし、扱いが難しい

Ahmed+ 201814

グラフの難しさ• グラフ：頂点（vertex）と辺（edge）からなるデータ構造

辺
頂点

15

グラフの難しさ• グラフ：頂点（vertex）と辺（edge）からなるデータ構造- メッシュもグラフとみなせる

辺
頂点

16

グラフの難しさ• グラフ：頂点（vertex）と辺（edge）からなるデータ構造- メッシュもグラフとみなせる• グラフの難しいところ：- 頂点の数が不定- 頂点から伸びているエッジの数が不定

17グラフの難しさ• グラフ：頂点（vertex）と辺（edge）からなるデータ構造- メッシュもグラフとみなせる• グラフの難しいところ：- 頂点の数が不定- 頂点から伸びているエッジの数が不定

より扱いやすい例：
画像（格子グラフ）18全結合ネットワークと Graph Neural Network

全結合ニューラルネットワーク• 全結合のネットワークではグラフが変わると対応できない

19

全結合ネットワークと Graph Neural Network?

全結合ニューラルネットワーク• 全結合のネットワークではグラフが変わると対応できない

20

全結合ネットワークと Graph Neural Network?

全結合ニューラルネットワーク

GNN• 全結合のネットワークではグラフが変わると対応できない• Graph Neural Network (GNN) はグラフの構造が変わっても同じモデルが使える
21

全結合ネットワークと Graph Neural Network?

全結合ニューラルネットワーク

GNN• 全結合のネットワークではグラフが変わると対応できない• Graph Neural Network (GNN) はグラフの構造が変わっても同じモデルが使える
22

全結合ネットワークと Graph Neural Network?

全結合ニューラルネットワーク

GNN• 全結合のネットワークではグラフが変わると対応できない• Graph Neural Network (GNN) はグラフの構造が変わっても同じモデルが使える➡ 一般の GNN は計算が重く、大きいグラフ（~ 1M 節点）に使用できない
23

GCN: 軽量な GNN• GCN: Graph Convolutional Network (Kipf+ 2017)- ノード間の相互作用を線形化しモデルを軽量化<latexit 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vtX+Csoce3LnLUcLC3ATl4RrQ0t0Js8mZRBql9ITe/zEgtUGTxK8az3w6tn+vSO+XuKWWZPocoC11+a02PPhpAkrI17IfpcbGMCxJDP9H+PGO+Q6lNsugq2TllrfYQ0n0UmZ5qiOFawbtzLXIxz0O7u47u4gHEcbLSOIOdlmuiX9/O698PLeQeZZ6C5zS5O3Zt+r7q4QjNFB2O00PthGGcT9dgOnSLQf6eAq9wPdLAMAPdaGUQg4kxkeVPQK3jYEDvTXOrpyXUBGE7FLn72zeAXUozy/G5j1Xudc1bo3Tkv/xY1XkbmAfD3cF1ffauh3idtQ7zqdmnUis3M/0+0uERBXeoTfgmoYTDCOEvy+1EJwd2n79Y223/Ks3xx/Wpj8x82Xp6/WfvVdvNnPj9b+euVv1n5fmVz5R9XfrVyuNJfuVoJV36z8j8r/7vy2zf//uY/3vznm/+yTX/6k6bPX614P2/++/8Av/h3IA==</latexit>Â = D̃

1
2(A + I)D̃

1
2<latexit 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= Activation

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GCN: 軽量な GNN|V| : グラフのノード数<latexit 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QOp6X1eYoEqg0cpng2+f/VMv71j/p5iltm3UGWB+y/N22PPhpAkrI17IPpcbGICxJDP9H+PGO+Q6rfYdBVsnbLW+hXSfBaZnGmK4ljBqnEvczHOQbu7j+/iAsZxsNZ5BTkv00Q/vJ/Xg+9fznvIPAPNrfdx6t70e9XHFZopehijhcH3wzibqMdu6BSB/jsF3OXTQAfLOeBZK4MIRJyJLG8KegUPa2Jrmku9PLkuAMOp2MbPvhn8RopRnt+tLXuPc04KfTrn5d+ixsvIDAB/D1f11bca6nPSNsSrfpdGrdjM/P9EiwsU1IV+wy8BNQxGGGdJfj8qIbj79N3KevdPeRYvrl6trf/92st3b1Z+t9n8mc/Pln699NdLv11aX/qHpd8t7S+dLl0uhUvzpf9c+u+l/3lz++Zf3/zbm3+3TX/8o6bPXy15P2/+4/8ABQdmoQ==</latexit>|V|⇥|V| : 隣接行列
• GCN: Graph Convolutional Network (Kipf+ 2017) A 2 R:の度数行列- ノード間の相互作用を線形化しモデルを軽量化 D̃ A + I(k)|V|⇥f (k) : 特徴量行列
1
1
h 2R
2
2
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= D̃(A + I)D̃<latexit sha1_base64="BxHdfQeRDDmOvNmizi6x5n/gNIw=">AAA1IniclVtLc9y6cta9ed0or3OTZTaoyK57TkpWLNupZJGkrt6SNXpYb9vjo+Jweji0+DKBoSTzTn5NNvklqcoulVWq8heyTbZpAAS7waF8ElXZYn8f0CSAr9E9HGhUJLFUL1/+x89+/lu//Tu/+3u/+P3lP/jDP/rjP/nul396JfNZGcJlmCd5eTMKJCRxBpcqVgncFCUE6SiB69HdluavKyhlnGcX6rGAT2kQZfEkDgOF0O13wfS2ruY/1t/f/TAXfy+GaaCmZVpvhCquTJOGGyYwUd+LoZylt/VsLobTQNUbc+yMxlRDut2LdfRy3XgblnE0VT/cfrfycu2l+RGLF+vNxcpS83N6+8vX/zwc5+EshUyFSSDlx/WXhfpUB6WKwwTmy8OZhCII74IIPuJlFqQgP9VmLubiOSJjMclL/JcpYVDeow5SKR/TEbbUY5VdToO93Cjt3FlN/vZTHWfFTEEW2htPZolQudATLcZxCaFKHvEiCMsYn12E06AMQoXL4XlW8d1Xz3et767yPJEIZ3Af5mkaZON6OI4nk/nHV5/q4QQd1WOxsj7X/7+az4V4npfjOAtKfcfJBEqcvjhIfAdFx8Ow0LMaJMYRGejP7xdnbUd9U/FXwtwVbxpnL7T2xP/h5jhXdHu6MTrjd2Y+Hfq0y1kWq/nHdfTXSvfjyvonMx2aE/exmgr5ZRaUIEY44jtQ0ncRKPtAWuBr+nGsbn9zWy9OwxgSnO8OeIkzN0rry27jK4tfIf5ceAxYBhZnuShzepq/w6dZxSlpnugfFvxcBaUdfDN0bZuew5oGMly8/47Xa6e/jz9JI2n76Ja/YbM0X15efq5/xPHOtTjauNgX2zu7B8cHFwcnx+fCUMt9MbaKv3WEylUdWPiER0F5JySGDG46UuQTEQaFvdbBXAIqoIyzSMfXOK5i6ZpN4mhWQmdREdRPOq9xslPx/QCvf+iOCduEqCkoXastY/W1M2N1zc7MttbTSuWFa3ORF30tRrlSeeoabRproV0z7sA1C55qMXItRk+1CF2L8KkWY9dirFvgMuzj6BI9QhEIbK/3L5hgMI4Fzk3q+8BrDRploKInehcxghC7ZlHsquHuCKsiye+hfBFiqlpbxjiamGmFycp6bRfwH4do1cZBX3d83FgFyZrYRTFIhRuDXnupVwx563HXedztejS0us/dPVdeNXeVwjUSOKLGeOV64LYxpi4rr1feLHRbbfu4q9fc1RsznHOr6m9OByrfPnwTAt589DhwE2J7n7ve5z29z1wvk5vu8zbK1tqJsXeXZmbaGHxiaroOpyVA1yXzt/J60SPNGvP9etF3kAnARdCde6YMvtgxuyZPD9rzMysKKIX2Y93sNG52+txsiDK4p3nvOHvx4kVQ5fFYzKTemuKJKHIpYyy+rOsiCTB0Gv9PP50uDAqMpJ4xasZ2b9r8vwfZONpqHW39pCMccxaB2YNtW2l9GLh9IpSKo52rFy+elAk+XZBEORZC07RnnMjZp2sbfXOgzNXCSDecq40eV07w7n44iNbXtzeDC6/Txk92WphUXc00I2fq06h9XH31rUWx/bvqPW37n/r93UjbG+BT6+snH7gRHMSJFmuiL2yi11eNv0mS56WhzZXlzWXTYN1UNuvdbKNKDIS5LTnCIKm3uw2qIInHvMFtW59Yar7gEqTq72CYeTsiKKTOcYWME/w4Y/PTGbrIU1EFZRxgtDp9gwpq7flBZXmZotdnQ4Sezd10lh06IGbkMyNiQp8JiRn7zJgY8BkgZuIzE2Iin4mImfrMlJjYZ2JiPvvMZ2LufOaOmMRnkrmRcZmKWGLE4sfS8aPe7OwKrorPM6nEOM9+pYT+DIdyfNQ7j7cwIm18Z77vjO6a+0xOTOEzBTFffOYLMaXPlMRIn5HEKJ9RxMx8ZkZM5TMVMfc+c0/Mg888EPPoM4/EfPWZr3Nb5rkAwMyct9t71QRJbUNpNGFh0z63mtoocS2MzXjGcXhEMIuNKiSYBUY1JphFRQUEs5CoJgSzeKgiglkwVFOCWSRUM4JZGFSfCWYxUN0RzAKgSghOGJwSnDKYTTSf4ZxgJuaqIJgpufpCMJNxVRLMNFxJgiVfVIJV/5xw6VYEM91W9wQz0VYPBDPFVo8EM7lWXwl2Wt1JQL8LMp/2yh7dghVd774MVnm9OzNY+fXuzWA12Ls7gxVi7/4MVo29OzRYSfbu0WB12btLI/fkPg1Wob07NViZ9u7VYLXa3a0dl/pcyrknd2Kw0u3di8Hqt3c3Bivi3v0YrJJ7d2Swcu7dk8FqundXBivs3n0ZrLp7d2awEu/dm8HqvHd3Biv23v0ZrOKf3qExFso4bCuUdIPiY4PCJt0keJPBWwRvMXib4G0G7xC8w+BdgncZvEfwHoP3Cd5n8AHBBwx+S/BbBh8SfMjgAcEDBh8RfMTgY4KPGXxC8AmDTwk+ZfA7gt8x+IzgMwafE3zO4AuCLxh8SfAlg68IvmLwNcHXDL4h+IbB7wl+z+APBH94env1RQdWdUyjG0y/RnqM2+Tcls9tcW7b57Y5t+NzO5zb9bldzu353B7n9n1un3MHPnfAubc+95Zzhz53yLmBzw04d+RzR5w79rljzp343AnnTn3ulHPvfO4d58587oxz5z53zrkLn7vg3KXPXXLuyueuOHftc9ecu/G5G86997n3nPvgc072V7yEqL6C+RyBn11ftn2rPIPafZ51WDqz0DClpNHWxBr36+GqgRkysgjVIaYKQYSqD1N7IEI1R9U8CVUaps5AhOoLU10gQlWFqSkQoVrCVBKIUAVh6gdEqG4wVQMiVC2YWgGRhM2DRagyMHUBIhmbP4tQFWBqAEQo95vMjwhlfJPvEaE8b7I8IpJNuEUop1fNsrBFqSxC+dtkb0Qoa5ucjQjlapOpEaEMbfIzIn3VqF+GVkFSTPV6m9+tAqtRIw6jCwfSRy16M9FQSZCOxrqHvSAiTyHSuPlNsJGklqMD0CEi+D9BMo5S3dX8JtgJtxFtO5C65s9fa7E6C8UakoVCHbNB1fZLLWuhQCdkoTgjslCYU7LwcdmzoiA/k4VivGNzU2sRtiOvtQCdhZPJZhHFl7MpqbXonIWi+0IWCq5kM1VrobUTVGuROQsnmk0zCqwiC8V1TxYK64EsFNUjWSior/PmOy/Msw8WNzkWdUa51WRWRCijmnyKCOVRk0URoexpcucoZTnTZExEKFOaPIkI5UeTHRGhrGhyIiKUC00mRIQyoMl/iFDeM1kPEcp2JtchQjnOZDhEKLOZvIYI5TOTzRChLGZyGCKUu0zmQoQylslXiFCeMlkKEcpOJjfZ72sdclW7b2odcm0Ryj8m+yBCWcfkHEQo15hMg8gHtoKUF0Y8LaSn02YjHuIVmz0X+poZNOHfDq6JYc2d2zg2KrqATOpDDtsQJkEJKKrpht6B8I622JOTWL8qhSzMx3EWobNglmhETtrrdF5L/Zb3HNRTDkZ5Mv4pN6OHOQZh901t1nx7bPJm48+8pW6Gpmx9mUmmfrXpMNK/2nIYRYDadhjFgNpxGEWB2nUYxYHacxhFgtp3GMWCOnAYRYN66zCKB3XoMIoINXAYxYQ6chhFhTp2GMWFOnEYRYY6dRjFhnrnMIoOdeYwig917jCKEHXhMIoRdekwihJ15TCKE3XtMIoUdeMwihX13mEULeqDw2xFhkLeK4NiatnIfc4NvY8b0SaDSRfRFoNJGtE2g0kd0Q6DSSDRLoNJI9Eeg0km0T6DSSnRAYNJLNFbBpNeokMGk2SiAYNJNdERg0k40TGDSTvRCYNJPtEpg0lB0TsGk4iiMwaTjqJzBpOUogsGk5qiSwaToKIrBpOmomsGk6yiGwaTsqL3DCZxRR8Y7Cr+c3P2R9OyfYsyYuKSm4SStuQWoSQtuU2oUdZzsW2+yZhJEIGQoATeOoGx2FkVIwgDjatpLMV9PkvGCKEFQprvPbCWnJVCn0rLE3Skz7vAQ4G1pfky132lvkt3JHXKPUJJnHKfUNKmPCCUpCnfEkrKlIeEkjDlgFDSpTwilGQpjwklVcoTQkmU8pRQ0qR8RyhJUp4RSoqU54SSIOUFoaRHeUkoyVFeEUpqlNeEkhjlDaGkRfmeUJKi/EBo+8olw7oPzEeIwL5saYpAoApg4Bf/ujzcIAulukkWSnSLLJTmNlm42e2QhSLaJQvFs0cWimafLBTLAVkokrdkoTgOyUJRDMhCMRyRhSI4JgsX/4QsXPRTsnCx35GFi3xGFi7uOVm4qBdk4WJekoWLeEUWLt41WbhoN2ThYr0nCxfpA7tfU2k1VZZeMuBLpmzFhVuKjl9z0BSD2KKr5sxhPlMCyx1xjymtgNIviIAqIq8aam6vWg2YhguFIJhyCTr1EpiCCToVE5iSCTo1E5iiCTpVE5iyCTp1E5jCCTqVE5jSCTq1E5jiCTrVE5jyCTr1E5gCCjoVFJgSCjo1FJgiCjpVFJgyCjp1FJhCCjqVFJhSCjq1FJhiCjrVFJhyCjr1FJiCCjoVFZiSCjo1FZiiCjpVFZiyCjp1FZjCCjqVFZjSCjq1FZjiCjrVFZjyCjr1FZgCC1iFhZ8UMOWocgZilo2hTB71oaVxoAIRQQYlZhttxxKVPprp1NM9PIxN53VxWw/LtDaGSXzaK6RFXMahOaxL/duzg6NHk+7MMRB9E8yPHd/uhIg+V17M/Vt4LU95y9N538Ok+RiSbw3ENGhHYq2F+zSNTr/VqFBxMoam5dAY7dO3PXCbUHk4DaQ+Ax7MVG4+QUHpPWHnAGth27TP2HRZfIAxeO2s2dOuRAI3HdfOmqiF0LxD8xsnQZEEIczbEzWDBpiL56K59qd3uXO+uM14O90HGUh2aGfQZc/mPLd3ds20YHPcIZNy7t67+UQJ0bx9k9alQkVj1FY8iaHsupb5RKXBA7V0QLcdJovcHGKyL9kWvRTJT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<latexit 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PsyZy0vFjZgK6+I1oYW6E2eTMog1Q+kpvd5iQWqDB6leDb49OqZfnvH/D3FLLNvocoC91+at8eeDSFJWBv3QPS52MQEiCGf6f8eMd4h1W+x6SrYOmWt9Suk+SwyOdMUxbGCVeNe5mKcg3Z3H9/FBYzjYK3zCnJepol+eD+vB59eznvIPAPNrfdx6t70e9XHFZopehijhcGnYZxN1GM3dIpA/50C7vJpoIPlAvCslUEEIs5EljcFvYKHNbE1zaVenlwXgOFUbONn3wx+JcUoz+/Wlr3HOSeFPp3z8q9R42VkBoC/h6v66nsN9TlpG+JVv0ujVmxm/n+ixSUK6lK/4ZeAGgYjjLMkvx+VENx9/mFlvfunPIsX16/W1v927eXZm5XfbDZ/5vOLpb9c+qulH5fWl/5u6TdL+0unS1dL4dI/L/3n0n8t/fcb9eZf3vzrm3+zTX/+s6bPXyx5P2/+438BXCZv+Q==</latexit>

(k)

2R

f (k

1)

⇥f (k):
学習するパラメータ行列

W

Activation(k) :<latexit 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/JkUgapfiA1vc9LLFBl8CjFs8EPr57pt3fM31PMMvsWqixw/6V5e+zZEJKEtXEPRJ+LTUyAGPKZ/u8R4x1S/RabroKtU9Zav0KazyKTM01RHCtYNe5lLsY5aHf38V1cwDgO1jqvIOdlmuiH9/N68MPLeQ+ZZ6C59T5O3Zt+r/q4QjNFD2O0MPhhGGcT9dgNnSLQf6eAu3wa6GC5ADxrZRCBiDOR5U1Br+BhTWxNc6mXJ9cFYDgV2/jZN4NfSTHK87u1Ze9xzkmhT+e8/BvUeBmZAeDv4aq++lZDfU7ahnjV79KoFZuZ/59ocYmCutRv+CWghsEI4yzJ70clBHefvltZ7/4pz+LF9au19b9be3n2ZuU3m82f+fx86a+W/nrp+6X1pb9f+s3S/tLp0tVSuPQvS/+59N9L//Nm+OZf3/zbm3+3TX/6k6bPXyx5P2/+4/8AyftnuA==</latexit>

成分ごとに作用する非線形関数

h(0)
v
h(0)
u
25

GCN: 軽量な GNN|V| : グラフのノード数<latexit 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QOp6X1eYoEqg0cpng2+f/VMv71j/p5iltm3UGWB+y/N22PPhpAkrI17IPpcbGICxJDP9H+PGO+Q6rfYdBVsnbLW+hXSfBaZnGmK4ljBqnEvczHOQbu7j+/iAsZxsNZ5BTkv00Q/vJ/Xg+9fznvIPAPNrfdx6t70e9XHFZopehijhcH3wzibqMdu6BSB/jsF3OXTQAfLOeBZK4MIRJyJLG8KegUPa2Jrmku9PLkuAMOp2MbPvhn8RopRnt+tLXuPc04KfTrn5d+ixsvIDAB/D1f11bca6nPSNsSrfpdGrdjM/P9EiwsU1IV+wy8BNQxGGGdJfj8qIbj79N3KevdPeRYvrl6trf/92st3b1Z+t9n8mc/Pln699NdLv11aX/qHpd8t7S+dLl0uhUvzpf9c+u+l/3lz++Zf3/zbm3+3TX/8o6bPXy15P2/+4/8ABQdmoQ==</latexit>|V|⇥|V| : 隣接行列
• GCN: Graph Convolutional Network (Kipf+ 2017) A 2 R:の度数行列- ノード間の相互作用を線形化しモデルを軽量化 D̃ A + I(k)|V|⇥f (k) : 特徴量行列
1
1
h 2R
2
2
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= D̃(A + I)D̃<latexit 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(k)

2R

f (k

1)

⇥f (k):
学習するパラメータ行列

W

Activation(k) :<latexit 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成分ごとに作用する非線形関数h(0)v<latexit sha1_base64="PH0JrDHV0Rbepy/bJaNyQlsEN/4=">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(1)
Âvu h(0)
u W

h(0)
u
26

GCN: 軽量な GNN|V| : グラフのノード数<latexit 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QOp6X1eYoEqg0cpng2+f/VMv71j/p5iltm3UGWB+y/N22PPhpAkrI17IPpcbGICxJDP9H+PGO+Q6rfYdBVsnbLW+hXSfBaZnGmK4ljBqnEvczHOQbu7j+/iAsZxsNZ5BTkv00Q/vJ/Xg+9fznvIPAPNrfdx6t70e9XHFZopehijhcH3wzibqMdu6BSB/jsF3OXTQAfLOeBZK4MIRJyJLG8KegUPa2Jrmku9PLkuAMOp2MbPvhn8RopRnt+tLXuPc04KfTrn5d+ixsvIDAB/D1f11bca6nPSNsSrfpdGrdjM/P9EiwsU1IV+wy8BNQxGGGdJfj8qIbj79N3KevdPeRYvrl6trf/92st3b1Z+t9n8mc/Pln699NdLv11aX/qHpd8t7S+dLl0uhUvzpf9c+u+l/3lz++Zf3/zbm3+3TX/8o6bPXy15P2/+4/8ABQdmoQ==</latexit>|V|⇥|V| : 隣接行列
• GCN: Graph Convolutional Network (Kipf+ 2017) A 2 R:の度数行列- ノード間の相互作用を線形化しモデルを軽量化 D̃ A + I(k)|V|⇥f (k) : 特徴量行列
1
1
h 2R
2
2
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vtX+Csoce3LnLUcLC3ATl4RrQ0t0Js8mZRBql9ITe/zEgtUGTxK8az3w6tn+vSO+XuKWWZPocoC11+a02PPhpAkrI17IfpcbGMCxJDP9H+PGO+Q6lNsugq2TllrfYQ0n0UmZ5qiOFawbtzLXIxz0O7u47u4gHEcbLSOIOdlmuiX9/O698PLeQeZZ6C5zS5O3Zt+r7q4QjNFB2O00PthGGcT9dgOnSLQf6eAq9wPdLAMAPdaGUQg4kxkeVPQK3jYEDvTXOrpyXUBGE7FLn72zeAXUozy/G5j1Xudc1bo3Tkv/xY1XkbmAfD3cF1ffauh3idtQ7zqdmnUis3M/0+0uERBXeoTfgmoYTDCOEvy+1EJwd2n79Y223/Ks3xx/Wpj8x82Xp6/WfvVdvNnPj9b+euVv1n5fmVz5R9XfrVyuNJfuVoJV36z8j8r/7vy2zf//uY/3vznm/+yTX/6k6bPX614P2/++/8Av/h3IA==</latexit>Â = D̃(A + I)D̃<latexit sha1_base64="BxHdfQeRDDmOvNmizi6x5n/gNIw=">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</latexit>

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⇥f (k):
学習するパラメータ行列

W

Activation(k) :<latexit sha1_base64="MWG5lUqnJdBVRQLRm3/SphDVBBo=">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</latexit>

成分ごとに作用する非線形関数

h(0)
v

(1)
Âvu h(0)u W!

<latexit 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h(1)
u
27

GCN: 軽量な GNN|V| : グラフのノード数<latexit 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QOp6X1eYoEqg0cpng2+f/VMv71j/p5iltm3UGWB+y/N22PPhpAkrI17IPpcbGICxJDP9H+PGO+Q6rfYdBVsnbLW+hXSfBaZnGmK4ljBqnEvczHOQbu7j+/iAsZxsNZ5BTkv00Q/vJ/Xg+9fznvIPAPNrfdx6t70e9XHFZopehijhcH3wzibqMdu6BSB/jsF3OXTQAfLOeBZK4MIRJyJLG8KegUPa2Jrmku9PLkuAMOp2MbPvhn8RopRnt+tLXuPc04KfTrn5d+ixsvIDAB/D1f11bca6nPSNsSrfpdGrdjM/P9EiwsU1IV+wy8BNQxGGGdJfj8qIbj79N3KevdPeRYvrl6trf/92st3b1Z+t9n8mc/Pln699NdLv11aX/qHpd8t7S+dLl0uhUvzpf9c+u+l/3lz++Zf3/zbm3+3TX/8o6bPXy15P2/+4/8ABQdmoQ==</latexit>|V|⇥|V| : 隣接行列
• GCN: Graph Convolutional Network (Kipf+ 2017) A 2 R:の度数行列- ノード間の相互作用を線形化しモデルを軽量化 D̃ A + I(k)|V|⇥f (k) : 特徴量行列
1
1
h 2R
2
2
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W

Activation(k) :<latexit sha1_base64="MWG5lUqnJdBVRQLRm3/SphDVBBo=">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</latexit>

成分ごとに作用する非線形関数

h(0)
v

h(1)
v

h(k)
v

<latexit sha1_base64="PH0JrDHV0Rbepy/bJaNyQlsEN/4=">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</latexit>(1)
Âvu h(0)
u W

h(0)
u

7!

⇥f (k)

7! · · · 7!

<latexit 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ycpXgx+evNCv71j/p5iltm3UGWB+y/N22MvhpAkrI17IPpSbGICxJDP9H9PGO+Q6rfYdBVsnbLW+hXSfBaZnGmK4ljBqnEvczHOQbt7iO/jAsZxsNZ5BTkv00Q/vJ/Xg59ez3vIPAPNrfdx6sH0e9PHFZopehijhcFPwzibqKdu6BSB/jsF3OWzQAfLBeBZK4MIRJyJLG8KegWPa2Jrmku9PLkuAMOp2MbPvhn8RopRnt+vLXuPc04LfTrn5d+jxsvIDAB/D1f11fca6nPSNsSrfpdGrdjM/P9Mi0sU1KV+wy8BNQxGGGdJ/jAqIbj//MPKevdPeRYvrt+srf/j2uv371Z+t9n8mc+vlv526e+WflxaX/qnpd8t7S+dLV0thUvp0n8s/efSf707fFe8e3j3ZJv+8hdNn79Z8n7e/fv/AoaoX/o=</latexit>

h(1)
u
28

<latexit 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ycpXgx+evNCv71j/p5iltm3UGWB+y/N22MvhpAkrI17IPpSbGICxJDP9H9PGO+Q6rfYdBVsnbLW+hXSfBaZnGmK4ljBqnEvczHOQbt7iO/jAsZxsNZ5BTkv00Q/vJ/Xg59ez3vIPAPNrfdx6sH0e9PHFZopehijhcFPwzibqKdu6BSB/jsF3OWzQAfLBeBZK4MIRJyJLG8KegWPa2Jrmku9PLkuAMOp2MbPvhn8RopRnt+vLXuPc04LfTrn5d+jxsvIDAB/D1f11fca6nPSNsSrfpdGrdjM/P9Mi0sU1KV+wy8BNQxGGGdJ/jAqIbj//MPKevdPeRYvrt+srf/j2uv371Z+t9n8mc+vlv526e+WflxaX/qnpd8t7S+dLV0thUvp0n8s/efSf707fFe8e3j3ZJv+8hdNn79Z8n7e/fv/AmSoYDQ=</latexit>

h(k)
u

目次

• Graph neural network• 同変性• IsoGCN: 同変性＋Graph Neural Network
• まとめ

29

目次

• Graph neural network• 同変性• IsoGCN: 同変性＋Graph Neural Network
• まとめ

30

不変性と同変性：現象の持つ対称性
G を群、X、Y を左 G-集合とする• 関数 f: X → Y が群 G の作用に対して不変（invariant）である：<latexit sha1_base64="+lD/DBh6ya7/pS2FVlmcpZU+JUA=">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</latexit>

8g 2 G, 8x 2 X, f (g · x) = f (x)

X

X

f

• 例：温度は合同変換（回転・鏡映・平行移動）に対して不変

Ti

Ti
Isometric transformation

!

31

g·

f

Y

不変性と同変性：現象の持つ対称性
G を群、X、Y を左 G-集合とする• 関数 f: X → Y が群 G の作用に対して同変（equivariant）である：<latexit sha1_base64="5ZYyyFqCOKeG/NzVXEbOAjqJ7pg=">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</latexit>

8g 2 G, 8x 2 X, f (g · x) = g · f (x)X• 例：変位は直交変換（回転・鏡映）に対して同変- 変位は平行移動に対して不変

g·

f

Y

f
g·

- 不変性は同変性の一種なので、変位は合同変換に対して同変

diOrthogonal transformation

!

32

X

g · diY不変性と同変性：現象の持つ対称性<latexit 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/NkVAapfiA1vs9LLFBl8CjFs963L5/pt3fMf7KYZPYtVFng/kvz9tizPiQJa+MeiD4XG5gAMeQz/esR4x1S/RabroKtU9Zav0KaTyKTM01RHCtYNu5lLoY5aHf38V1cwDAOVlqvIOdlmuiH97Np79vVWQeZZ6C5tS5O3Zt+L7u4QjNFB2O00Pu2H2cj9dgOnSLQ/08Bd/kk0MFyDnjWyiACEWciy+uCXsHDitgc51IvT64LwHAstvCzbwa/kGKQ53cri97jnONCn855+a+o8TIyA8C//WV99aWG+py0DfGq26VRKzYzv59ocYGCutBv+CWg+sEA4yzJ7wclBHcfvlpaa///nvmLq5cra79eWT1dXfpmY8H+/GjhXxZ+vvD1wtrCbxa+WdhbOFm4XAgXxgv/vfCnhf99ffv6P17/5+v/sk2//726z88WvJ/X//MXZ25pbg==</latexit>

合同変換の群（ユークリッド群）を E(n) とする• 物理量・物理法則は合同変換同変である- 例：フックの法則<latexit 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</latexit>

8e 2 E(n),=C" ) e ·=Orthogonal transformation!

33(e · C)(e · ")

不変性と同変性：現象の持つ対称性<latexit 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/NkVAapfiA1vs9LLFBl8CjFs963L5/pt3fMf7KYZPYtVFng/kvz9tizPiQJa+MeiD4XG5gAMeQz/esR4x1S/RabroKtU9Zav0KaTyKTM01RHCtYNu5lLoY5aHf38V1cwDAOVlqvIOdlmuiH97Np79vVWQeZZ6C5tS5O3Zt+L7u4QjNFB2O00Pu2H2cj9dgOnSLQ/08Bd/kk0MFyDnjWyiACEWciy+uCXsHDitgc51IvT64LwHAstvCzbwa/kGKQ53cri97jnONCn855+a+o8TIyA8C//WV99aWG+py0DfGq26VRKzYzv59ocYGCutBv+CWg+sEA4yzJ7wclBHcfvlpaa///nvmLq5cra79eWT1dXfpmY8H+/GjhXxZ+vvD1wtrCbxa+WdhbOFm4XAgXxgv/vfCnhf99ffv6P17/5+v/sk2//726z88WvJ/X//MXZ25pbg==</latexit>

合同変換の群（ユークリッド群）を E(n) とする• 物理量・物理法則は合同変換同変である- 例：フックの法則<latexit 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</latexit>

8e 2 E(n),eU e 2 O(n), Uiie 0 Ujj
0 i0 j 0 =<latexit sha1_base64="SnZWy75D1/rmIVbLqvbZSYxuR14=">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</latexit>=C" ) e ·=(e · C)(e · ")

e
e
e
e
e
0
0
0
0
Uiie 0 Ujj
U
U
C
U
U
0
kk0 ll0 i j k l kk00 ll00 "k00 l00=

eUiie 0 Ujj
0 k 0 k 00 l0 l00 Ci0 j 0 k 0 l0 "k 00 l00=

eUiie 0 Ujj
0 C i0 j 0 k 0 l 0 " k 0 l 0

Orthogonal transformation!

34

GCN と不変性・同変性|V| : グラフのノード数<latexit 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QOp6X1eYoEqg0cpng2+f/VMv71j/p5iltm3UGWB+y/N22PPhpAkrI17IPpcbGICxJDP9H+PGO+Q6rfYdBVsnbLW+hXSfBaZnGmK4ljBqnEvczHOQbu7j+/iAsZxsNZ5BTkv00Q/vJ/Xg+9fznvIPAPNrfdx6t70e9XHFZopehijhcH3wzibqMdu6BSB/jsF3OXTQAfLOeBZK4MIRJyJLG8KegUPa2Jrmku9PLkuAMOp2MbPvhn8RopRnt+tLXuPc04KfTrn5d+ixsvIDAB/D1f11bca6nPSNsSrfpdGrdjM/P9EiwsU1IV+wy8BNQxGGGdJfj8qIbj79N3KevdPeRYvrl6trf/92st3b1Z+t9n8mc/Pln699NdLv11aX/qHpd8t7S+dLl0uhUvzpf9c+u+l/3lz++Zf3/zbm3+3TX/8o6bPXy15P2/+4/8ABQdmoQ==</latexit>|V|⇥|V| : 隣接行列
• GCN: Graph Convolutional Network (Kipf+ 2017) A 2 R:の度数行列- ノード間の相互作用を線形化しモデルを軽量化 D̃ A + I(k)|V|⇥f (k) : 特徴量行列
1
1
h 2R
2
2
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W

Activation(k) :<latexit sha1_base64="MWG5lUqnJdBVRQLRm3/SphDVBBo=">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</latexit>

成分ごとに作用する非線形関数

h(0)
v

h(1)
v

h(k)
v

<latexit sha1_base64="PH0JrDHV0Rbepy/bJaNyQlsEN/4=">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</latexit>(1)
Âvu h(0)
u W

h(0)
u

7!

⇥f (k)

7! · · · 7!

<latexit 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ycpXgx+evNCv71j/p5iltm3UGWB+y/N22MvhpAkrI17IPpSbGICxJDP9H9PGO+Q6rfYdBVsnbLW+hXSfBaZnGmK4ljBqnEvczHOQbt7iO/jAsZxsNZ5BTkv00Q/vJ/Xg59ez3vIPAPNrfdx6sH0e9PHFZopehijhcFPwzibqKdu6BSB/jsF3OWzQAfLBeBZK4MIRJyJLG8KegWPa2Jrmku9PLkuAMOp2MbPvhn8RopRnt+vLXuPc04LfTrn5d+jxsvIDAB/D1f11fca6nPSNsSrfpdGrdjM/P9Mi0sU1KV+wy8BNQxGGGdJ/jAqIbj//MPKevdPeRYvrt+srf/j2uv371Z+t9n8mc+vlv526e+WflxaX/qnpd8t7S+dLV0thUvp0n8s/efSf707fFe8e3j3ZJv+8hdNn79Z8n7e/fv/AoaoX/o=</latexit>

h(1)
u
35

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と不変性・同変性|V| : グラフのノード数<latexit 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QOp6X1eYoEqg0cpng2+f/VMv71j/p5iltm3UGWB+y/N22PPhpAkrI17IPpcbGICxJDP9H+PGO+Q6rfYdBVsnbLW+hXSfBaZnGmK4ljBqnEvczHOQbu7j+/iAsZxsNZ5BTkv00Q/vJ/Xg+9fznvIPAPNrfdx6t70e9XHFZopehijhcH3wzibqMdu6BSB/jsF3OXTQAfLOeBZK4MIRJyJLG8KegUPa2Jrmku9PLkuAMOp2MbPvhn8RopRnt+tLXuPc04KfTrn5d+ixsvIDAB/D1f11bca6nPSNsSrfpdGrdjM/P9EiwsU1IV+wy8BNQxGGGdJfj8qIbj79N3KevdPeRYvrl6trf/92st3b1Z+t9n8mc/Pln699NdLv11aX/qHpd8t7S+dLl0uhUvzpf9c+u+l/3lz++Zf3/zbm3+3TX/8o6bPXy15P2/+4/8ABQdmoQ==</latexit>|V|⇥|V| : 隣接行列
• GCN: Graph Convolutional Network (Kipf+ 2017) A 2 R:の度数行列- ノード間の相互作用を線形化しモデルを軽量化 D̃ A + I(k)|V|⇥f (k) : 特徴量行列
1
1
h 2R
2
2
<latexit 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= D̃(A + I)D̃<latexit 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h(k)
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Activation
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• GCN は、入力が合同変換不変のときのみ不変- 不変性を保ったままノード位置を入力できない

36<latexit 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PsyZy0vFjZgK6+I1oYW6E2eTMog1Q+kpvd5iQWqDB6leDb49OqZfnvH/D3FLLNvocoC91+at8eeDSFJWBv3QPS52MQEiCGf6f8eMd4h1W+x6SrYOmWt9Suk+SwyOdMUxbGCVeNe5mKcg3Z3H9/FBYzjYK3zCnJepol+eD+vB59eznvIPAPNrfdx6t70e9XHFZopehijhcGnYZxN1GM3dIpA/50C7vJpoIPlAvCslUEEIs5EljcFvYKHNbE1zaVenlwXgOFUbONn3wx+JcUoz+/Wlr3HOSeFPp3z8q9R42VkBoC/h6v66nsN9TlpG+JVv0ujVmxm/n+ixSUK6lK/4ZeAGgYjjLMkvx+VENx9/mFlvfunPIsX16/W1v927eXZm5XfbDZ/5vOLpb9c+qulH5fWl/5u6TdL+0unS1dL4dI/L/3n0n8t/fcb9eZf3vzrm3+zTX/+s6bPXyx5P2/+438BXCZv+Q==</latexit>

(k)

2R

f (k

1)

⇥f (k):
学習するパラメータ行列

W

Activation(k) :<latexit 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/JkUgapfiA1vc9LLFBl8CjFs8EPr57pt3fM31PMMvsWqixw/6V5e+zZEJKEtXEPRJ+LTUyAGPKZ/u8R4x1S/RabroKtU9Zav0KazyKTM01RHCtYNe5lLsY5aHf38V1cwDgO1jqvIOdlmuiH9/N68MPLeQ+ZZ6C59T5O3Zt+r/q4QjNFD2O0MPhhGGcT9dgNnSLQf6eAu3wa6GC5ADxrZRCBiDOR5U1Br+BhTWxNc6mXJ9cFYDgV2/jZN4NfSTHK87u1Ze9xzkmhT+e8/BvUeBmZAeDv4aq++lZDfU7ahnjV79KoFZuZ/59ocYmCutRv+CWghsEI4yzJ70clBHefvltZ7/4pz+LF9au19b9be3n2ZuU3m82f+fx86a+W/nrp+6X1pb9f+s3S/tLp0tVSuPQvS/+59N9L//Nm+OZf3/zbm3+3TX/6k6bPXyx5P2/+4/8AyftnuA==</latexit>

成分ごとに作用する非線形関数

GCN と不変性・同変性|V| : グラフのノード数<latexit 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QOp6X1eYoEqg0cpng2+f/VMv71j/p5iltm3UGWB+y/N22PPhpAkrI17IPpcbGICxJDP9H+PGO+Q6rfYdBVsnbLW+hXSfBaZnGmK4ljBqnEvczHOQbu7j+/iAsZxsNZ5BTkv00Q/vJ/Xg+9fznvIPAPNrfdx6t70e9XHFZopehijhcH3wzibqMdu6BSB/jsF3OXTQAfLOeBZK4MIRJyJLG8KegUPa2Jrmku9PLkuAMOp2MbPvhn8RopRnt+tLXuPc04KfTrn5d+ixsvIDAB/D1f11bca6nPSNsSrfpdGrdjM/P9EiwsU1IV+wy8BNQxGGGdJfj8qIbj79N3KevdPeRYvrl6trf/92st3b1Z+t9n8mc/Pln699NdLv11aX/qHpd8t7S+dLl0uhUvzpf9c+u+l/3lz++Zf3/zbm3+3TX/8o6bPXy15P2/+4/8ABQdmoQ==</latexit>|V|⇥|V| : 隣接行列
• GCN: Graph Convolutional Network (Kipf+ 2017) A 2 R:の度数行列- ノード間の相互作用を線形化しモデルを軽量化 D̃ A + I(k)|V|⇥f (k) : 特徴量行列
1
1
h 2R
2
2
<latexit 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= D̃(A + I)D̃<latexit 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Activation
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• GCN は、入力が合同変換不変のときのみ不変- 不変性を保ったままノード位置を入力できない

37<latexit 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PsyZy0vFjZgK6+I1oYW6E2eTMog1Q+kpvd5iQWqDB6leDb49OqZfnvH/D3FLLNvocoC91+at8eeDSFJWBv3QPS52MQEiCGf6f8eMd4h1W+x6SrYOmWt9Suk+SwyOdMUxbGCVeNe5mKcg3Z3H9/FBYzjYK3zCnJepol+eD+vB59eznvIPAPNrfdx6t70e9XHFZopehijhcGnYZxN1GM3dIpA/50C7vJpoIPlAvCslUEEIs5EljcFvYKHNbE1zaVenlwXgOFUbONn3wx+JcUoz+/Wlr3HOSeFPp3z8q9R42VkBoC/h6v66nsN9TlpG+JVv0ujVmxm/n+ixSUK6lK/4ZeAGgYjjLMkvx+VENx9/mFlvfunPIsX16/W1v927eXZm5XfbDZ/5vOLpb9c+qulH5fWl/5u6TdL+0unS1dL4dI/L/3n0n8t/fcb9eZf3vzrm3+zTX/+s6bPXyx5P2/+438BXCZv+Q==</latexit>

(k)

2R

f (k

1)

⇥f (k):
学習するパラメータ行列

W

Activation(k) :<latexit 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/JkUgapfiA1vc9LLFBl8CjFs8EPr57pt3fM31PMMvsWqixw/6V5e+zZEJKEtXEPRJ+LTUyAGPKZ/u8R4x1S/RabroKtU9Zav0KazyKTM01RHCtYNe5lLsY5aHf38V1cwDgO1jqvIOdlmuiH9/N68MPLeQ+ZZ6C59T5O3Zt+r/q4QjNFD2O0MPhhGGcT9dgNnSLQf6eAu3wa6GC5ADxrZRCBiDOR5U1Br+BhTWxNc6mXJ9cFYDgV2/jZN4NfSTHK87u1Ze9xzkmhT+e8/BvUeBmZAeDv4aq++lZDfU7ahnjV79KoFZuZ/59ocYmCutRv+CWghsEI4yzJ70clBHefvltZ7/4pz+LF9au19b9be3n2ZuU3m82f+fx86a+W/nrp+6X1pb9f+s3S/tLp0tVSuPQvS/+59N9L//Nm+OZf3/zbm3+3TX/6k6bPXyx5P2/+4/8AyftnuA==</latexit>

成分ごとに作用する非線形関数

GCN と不変性・同変性|V| : グラフのノード数<latexit 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QOp6X1eYoEqg0cpng2+f/VMv71j/p5iltm3UGWB+y/N22PPhpAkrI17IPpcbGICxJDP9H+PGO+Q6rfYdBVsnbLW+hXSfBaZnGmK4ljBqnEvczHOQbu7j+/iAsZxsNZ5BTkv00Q/vJ/Xg+9fznvIPAPNrfdx6t70e9XHFZopehijhcH3wzibqMdu6BSB/jsF3OXTQAfLOeBZK4MIRJyJLG8KegUPa2Jrmku9PLkuAMOp2MbPvhn8RopRnt+tLXuPc04KfTrn5d+ixsvIDAB/D1f11bca6nPSNsSrfpdGrdjM/P9EiwsU1IV+wy8BNQxGGGdJfj8qIbj79N3KevdPeRYvrl6trf/92st3b1Z+t9n8mc/Pln699NdLv11aX/qHpd8t7S+dLl0uhUvzpf9c+u+l/3lz++Zf3/zbm3+3TX/8o6bPXy15P2/+4/8ABQdmoQ==</latexit>|V|⇥|V| : 隣接行列
• GCN: Graph Convolutional Network (Kipf+ 2017) A 2 R:の度数行列- ノード間の相互作用を線形化しモデルを軽量化 D̃ A + I(k)|V|⇥f (k) : 特徴量行列
1
1
h 2R
2
2
<latexit 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vtX+Csoce3LnLUcLC3ATl4RrQ0t0Js8mZRBql9ITe/zEgtUGTxK8az3w6tn+vSO+XuKWWZPocoC11+a02PPhpAkrI17IfpcbGMCxJDP9H+PGO+Q6lNsugq2TllrfYQ0n0UmZ5qiOFawbtzLXIxz0O7u47u4gHEcbLSOIOdlmuiX9/O698PLeQeZZ6C5zS5O3Zt+r7q4QjNFB2O00PthGGcT9dgOnSLQf6eAq9wPdLAMAPdaGUQg4kxkeVPQK3jYEDvTXOrpyXUBGE7FLn72zeAXUozy/G5j1Xudc1bo3Tkv/xY1XkbmAfD3cF1ffauh3idtQ7zqdmnUis3M/0+0uERBXeoTfgmoYTDCOEvy+1EJwd2n79Y223/Ks3xx/Wpj8x82Xp6/WfvVdvNnPj9b+euVv1n5fmVz5R9XfrVyuNJfuVoJV36z8j8r/7vy2zf//uY/3vznm/+yTX/6k6bPX614P2/++/8Av/h3IA==</latexit>Â = D̃(A + I)D̃<latexit sha1_base64="BxHdfQeRDDmOvNmizi6x5n/gNIw=">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Activation
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• GCN は、入力が合同変換不変のときのみ不変<latexit 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PsyZy0vFjZgK6+I1oYW6E2eTMog1Q+kpvd5iQWqDB6leDb49OqZfnvH/D3FLLNvocoC91+at8eeDSFJWBv3QPS52MQEiCGf6f8eMd4h1W+x6SrYOmWt9Suk+SwyOdMUxbGCVeNe5mKcg3Z3H9/FBYzjYK3zCnJepol+eD+vB59eznvIPAPNrfdx6t70e9XHFZopehijhcGnYZxN1GM3dIpA/50C7vJpoIPlAvCslUEEIs5EljcFvYKHNbE1zaVenlwXgOFUbONn3wx+JcUoz+/Wlr3HOSeFPp3z8q9R42VkBoC/h6v66nsN9TlpG+JVv0ujVmxm/n+ixSUK6lK/4ZeAGgYjjLMkvx+VENx9/mFlvfunPIsX16/W1v927eXZm5XfbDZ/5vOLpb9c+qulH5fWl/5u6TdL+0unS1dL4dI/L/3n0n8t/fcb9eZf3vzrm3+zTX/+s6bPXyx5P2/+438BXCZv+Q==</latexit>(k)⇥f (k)

Activation(k) :<latexit sha1_base64="MWG5lUqnJdBVRQLRm3/SphDVBBo=">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</latexit>

成分ごとに作用する非線形関数• 一般に GCN は合同変換同変とならない

9e 2 E(d), e · h(k) = Activation(k) (Âe · h(k 1) W (k) )
38

1)

:
学習するパラメータ行列

W- 不変性を保ったままノード位置を入力できない<latexit sha1_base64="a4XalLJd//VZl1iR86fPH3G8F+Y=">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Tensor field network: 合同変換同変な機械学習モデル• TFN: Tensor field network (Thomas+ 2018)- 球面調和関数を用いた
合同変換不変・同変な機械学習モデル
XX
(l)
(l)
(k)
ll
lk
H̃out,i = w H̃in,i +W (xj xi ) H̃in,j<latexit 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J (kxk)JX

YJm (x/kxk)Qlk
Jm

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up3u70Xvdv+Ff4KStz7Mmct+wsbsJ1XRGtDC/QmT8ZlkOoHUpP7vMQCVQaPUjzr/fDqmX57x/ypxjSzb6HKAvdfmrfHng0gSVgb90D0udjCBIghn+n/HjHeIdVvsekq2DplrfUrpPk0MjnTFMWxglXjXuZilIN2dx/fxQWM4mCt9QpyXqaJfng/q3s/vJx1kHkGmlvv4tS96feqiys0U3QwRgu9HwZxNlaP7dApAv1nDLjLZ4EOlj7gWSuDCESciSxvCnoFD2tie5JLvTy5LgDDidjBz74Z/EqKYZ7frS17j3NOC3065+U/ocbLyAwAfw9W9dXXGupz0jbEq26XRq3YzPz/RItLFNSlfsMvATUIhhhnSX4/LCG4+/jNivvjIPH0xfWrtfXv1l6eb6z8fqv5C6KfL/390j8sfbu0vvTrpd8vHSydLV0thUvx0r8t/fvSf2z858Z/bfz3xv/Ypj/9SdPn75a8n40//i+/24Ox</latexit>

39

Tensor field network: 合同変換同変な機械学習モデル(l)• TFN: Tensor field network (Thomas+ 2018) H̃in,i 2 R2l+1 : type-l 入力（ l 階テンソルに対応）- 球面調和関数を用いた
xi : i 番目ノード位置
合同変換不変・同変な機械学習モデル
ll: 学習するパラメータ
w
XX
(l)
(l)
(k)
lk
H̃out,i = wll H̃in,i +W lk (xj xi ) H̃in,j
J : R 0 ! R : 学習する関数<latexit 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W lk (x) =k+lXk 0 j=ilk
J (kxk)JX

YJm (x/kxk)Qlk
Jm

m= J

J=|k l|

YJ : R3 ! R2J+1 :球面調和関数<latexit 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9FPAnz2sDfjbzAd9p5soH6323vRu+1f4a+gxL0vc9by/MkGbOYV0drQAr3Ok3EZpPqB1OQ+L7FAlcGjFC963795od/eMX+LMc3sW6iywP2X5u2xFwNIEtbGPRB9KTYwAWLIZ/q/R4x3SPVbbLoKtk5Za/0KaT6NTM40RXGsYNm4l7kY5aDd3cd3cQGjOFhpvYKcl2miH97P6t73r2cdZJ6B5la7OHVv+r3p4grNFB2M0ULv+0GcjdVjO3SKQP+dAu5yP9DBcg541sogAhFnIsubgl7Bw4rYnORSL0+uC8BwIrbws28Gv5BimOd3K4ve45yTQp/Oefk3qPEyMgPA34NlffW1hvqctA3xqtulUSs2M/8/0+ICBXWh3/BLQA2CIcZZkt8PSwjuPn6z5P76Rzx/cfVmZfXbldenb5d+s9H8idBPF/5y4a8WfrmwuvDrhd8s7C30Fy4XwoV/Wfi3hX9f+I+3//r2P9/+19vf2aY//lHT5y8WvJ+3//2/qQiHAw==</latexit>

J J =0.
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J =1.
<latexit 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</latexit>

xi<latexit 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P1ut/eid9u/wl9BiXtf5qxlf2EDtvOKaG1ogd7kybgMUv1AanKfl1igyuBRime9H14902/vmD/VmGb2LVRZ4P5L8/bYswEkCWvjHog+F1uYADHkM/3fI8Y7pPotNl0FW6estX6FNJ9GJmeaojhWsGrcy1yMctDu7uO7uIBRHKy1XkHOyzTRD+9nde+Hl7MOMs9Ac+tdnLo3/V51cYVmig7GaKH3wyDOxuqxHTpFoP+MAXf5LNDB0gc8a2UQgYgzkeVNQa/gYU1sT3KplyfXBWA4ETv42TeDX0kxzPO7tWXvcc5poU/nvPwn1HgZmQHg78GqvvpaQ31O2oZ41e3SqBWbmf+faHGJgrrUb/gloAbBEOMsye+HJQR3H79ZcX8cJJ6+uH61tv7d2svzjZXfbzV/QfTzpb9f+oelb5fWl3699Pulg6WzpaulcCle+relf1/6j43/3Pivjf/e+B/b9Kc/afr83ZL3s/HH/wWLJIOw</latexit>

J J =2.
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</latexit>

xj<latexit 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up3u70Xvdv+Ff4KStz7Mmct+wsbsJ1XRGtDC/QmT8ZlkOoHUpP7vMQCVQaPUjzr/fDqmX57x/ypxjSzb6HKAvdfmrfHng0gSVgb90D0udjCBIghn+n/HjHeIdVvsekq2DplrfUrpPk0MjnTFMWxglXjXuZilIN2dx/fxQWM4mCt9QpyXqaJfng/q3s/vJx1kHkGmlvv4tS96feqiys0U3QwRgu9HwZxNlaP7dApAv1nDLjLZ4EOlj7gWSuDCESciSxvCnoFD2tie5JLvTy5LgDDidjBz74Z/EqKYZ7frS17j3NOC3065+U/ocbLyAwAfw9W9dXXGupz0jbEq26XRq3YzPz/RItLFNSlfsMvATUIhhhnSX4/LCG4+/jNivvjIPH0xfWrtfXv1l6eb6z8fqv5C6KfL/390j8sfbu0vvTrpd8vHSydLV0thUvx0r8t/fvSf2z858Z/bfz3xv/Ypj/9SdPn75a8n40//i+/24Ox</latexit>

40Tensor field network: 合同変換同変な機械学習モデル<latexit 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</latexit>• TFN: Tensor field network (Thomas+ 2018)(l)
H̃in,i 2 R2l+1 : type-l 入力<latexit 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+asZX9hA7byimhtaIFe58m4DFL9QGpyl5dYoMrgQYonvW9fPNFv75i/xZhm9i1UWeD+S/P22JMBJAlr4x6IPhWbmAAx5DP93wPGO6T6LTZdBVunrLV+hTSfRiZnmqI4VrBq3MtcjHLQ7u7i27iAURystV5Bzss00Q/v57Pet8/nHWSegebWuzh1Z/q96OIKzRQdjNFC79tBnI3VQzt0ikD/nQLu8mmgg6UPeNbKIAIRZyLL64Jewf2a2JrkUi9PrgvAcCK28bNvBj+XYpjnt2vL3uOck0Kfznn5b6jxMjIDwN+DVX31pYb6nLQN8arbpVErNjP/P9LiAgV1od/wS0ANgiHGWZLfDUsIbj98teL++kc8fnH1Ym3967XnZ69Wfr9Z/4nQj5Z+tvSvS79YWl/6zdLvl/aXTpcul8IlWPqPpf9c+q9Xf3z136/+59X/2qbf/17d56dL3s+r//sLmHV2oA==</latexit>（ l 階テンソルに対応）
xi : i 番目ノード位置
wll : 学習するパラメータlk
J : R 0 ! R : 学習する関数<latexit 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+asZX9hA7byimhtaIFe58m4DFL9QGpyl5dYoMrgQYonvW9fPNFv75i/xZhm9i1UWeD+S/P22JMBJAlr4x6IPhWbmAAx5DP93wPGO6T6LTZdBVunrLV+hTSfRiZnmqI4VrBq3MtcjHLQ7u7i27iAURystV5Bzss00Q/v57Pet8/nHWSegebWuzh1Z/q96OIKzRQdjNFC79tBnI3VQzt0ikD/nQLu8mmgg6UPeNbKIAIRZyLL64Jewf2a2JrkUi9PrgvAcCK28bNvBj+XYpjnt2vL3uOck0Kfznn5b6jxMjIDwN+DVX31pYb6nLQN8arbpVErNjP/P9LiAgV1od/wS0ANgiHGWZLfDUsIbj98teL++kc8fnH1Ym3967XnZ69Wfr9Z/4nQj5Z+tvSvS79YWl/6zdLvl/aXTpcul8IlWPqPpf9c+q9Xf3z136/+59X/2qbf/17d56dL3s+r//sLmHV2oA==</latexit>- 球面調和関数を用いた
合同変換不変・同変な機械学習モデル
XX
(l)
(l)
(k)
ll
lk
H̃out,i = w H̃in,i +W (xj xi ) H̃in,j<latexit sha1_base64="OkSxLwGt30+efbwsqNIziX8PRjQ=">AAA113iclVtbcxxLUpaXBRZxOwuPvBTIZs+CJCT7mD1EALG6S9ZIGt0le2xFz0xOT1t9c1dNS3LvwBvBK7+HX8K/IauqazKrp+UDirDV+X1V2XX5sjKnp9XP40iqtbX/efGz3/v57//BH/7ijxb/+E/+9M/+/Ltf/sWVzCbFAC4HWZwVN/1AQhylcKkiFcNNXkCQ9GO47t9vaf66hEJGWXqhnnL4mARhGo2iQaAQuvvuv4XoqSgeQtVT+9O7qpcEalwkVTZR02URTXtykudVPBV/+y/i4VMV41WvvUeUeh0Whfh7gUZyV92LXghfxNW lk (x) =k+lX

J=|k l|k 0 j=ilk
J (kxk)JX

m= J<latexit 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YJ : R3 ! R2J+1 :球面調和関数
YJm (x/kxk)Qlk
Jm Qlk 2 R(2l+1)⇥(2k+1):Jm<latexit 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</latexit><latexit 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Clebsch-Gordan 係数

41

Tensor field network: 合同変換同変な機械学習モデル(l)• TFN: Tensor field network (Thomas+ 2018) H̃in,i 2 R2l+1 : type-l 入力（ l 階テンソルに対応）- 球面調和関数を用いた
xi : i 番目ノード位置
合同変換不変・同変な機械学習モデル
ll: 学習するパラメータ
w
XX
(l)
(l)
(k)
lk
H̃out,i = wll H̃in,i +W lk (xj xi ) H̃in,j
J : R 0 ! R : 学習する関数<latexit 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J (kxk)J
X<latexit 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</latexit>

m= J

J=|k l|

YJ : R3 ! R2J+1 :球面調和関数
YJm (x/kxk)Qlk
Jm Qlk 2 R(2l+1)⇥(2k+1):Jm<latexit sha1_base64="eqeBoJC9eBvi2uibBjnqrKmdSRk=">AAA1KHiclVvZctxKcuWMlxnT253xo18qTClm7KFoUdT1XDscjuG+NcnmTkqtq0Cjs9EQsQlVDZLCtP/Bv+Ef8Kv9B35zzKsj/B/OqkJ1ZoGgrs0IichzqhK1nKzMRoPDIomlev36dz/68e/9/h/84U9++keLf/wnf/pnf/7Nz35+JfNpGcJlmCd5eTMMJCRxBpcqVgncFCUE6TCB6+HdpuavKyhlnGcX6rGAD2kQZfE4DgOF0MdvfnX7sT6Y/YMYpIGaDIf12ez7NTFQO

Clebsch-Gordan 係数

type- J と type- k の入力から type- l の出力を返すときの係数<latexit 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725DdTvdnsverf9K/wVlLj3Zc5aXsxtwFZeEa0NLdCbPBmVQaofSI0f8hILVBk8SfGi98PrF/rtHfOnGpPMvoUqC9x/ad4ee9GHJGFt3APRl2ITEyCGfKZ/PGG8Q6rfYtNVsHXKWutXSPNJZHKmKYpjBcvGvczFMAft7iG+jwsYxsFK6xXkvEwT/fB+Nu39sDrrIPMMNLfWxakH0+91F1dopuhgjBZ6P/TjbKSe2qFTBPrPGHCXTwMdLBeAZ60MIhBxJrK8LugVPK6IrXEu9fLkugAMx2IbP/tm8GspBnl+v7LoPc45KfTpnJd/hxovIzMA/N1f1ldfa6jPSdsQr7pdGrViM/PzmRaXKKhL/YZfAqofDDDOkvxhUEJw//GbJffHQeL5i+vXK2vfrayerS/9brP+C6JfLPzNwt8ufLuwtvDbhd8t7C+cLlwthAvpwr8u/NvCv6//x/p/rv/X+n/bpj/9Sd3nrxe8f+v/+3/TyYNB</latexit>

J(J, k, l) .(0, 0, 0)<latexit 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対応物

掛け算

(0, 1, 1)(1, 0, 1)<latexit sha1_base64="5UmXG5F5pqvn7SwhwGZJSDsy55k=">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</latexit>

ベクトルの
スカラー倍
42

<latexit 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725DdTvdnsverf9K/wVlLj3Zc5aXsxtwFZeEa0NLdCbPBmVQaofSI0f8hILVBk8SfGi98PrF/rtHfOnGpPMvoUqC9x/ad4ee9GHJGFt3APRl2ITEyCGfKZ/PGG8Q6rfYtNVsHXKWutXSPNJZHKmKYpjBcvGvczFMAft7iG+jwsYxsFK6xXkvEwT/fB+Nu39sDrrIPMMNLfWxakH0+91F1dopuhgjBZ6P/TjbKSe2qFTBPrPGHCXTwMdLBeAZ60MIhBxJrK8LugVPK6IrXEu9fLkugAMx2IbP/tm8GspBnl+v7LoPc45KfTpnJd/hxovIzMA/N1f1ldfa6jPSdsQr7pdGrViM/PzmRaXKKhL/YZfAqofDDDOkvxhUEJw//GbJffHQeL5i+vXK2vfrayerS/9brP+C6JfLPzNwt8ufLuwtvDbhd8t7C+cLlwthAvpwr8u/NvCv6//x/p/rv/X+n/bpj/9Sd3nrxe8f+v/+3/Tz4NB</latexit><latexit 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fTmNlC/2+296N32r/BXUOLelzlreT63AZt5RbQ2tECv82RUBql+IDW+z0ssUGXwKMWz3nevnum3d8yfakwy+xaqLHD/pXl77FkfkoS1cQ9En4sNTIAY8pn+8YjxDql+i01XwdYpa61fIc0nkcmZpiiOFSwb9zIXwxy0u/v4Li5gGAcrrVeQ8zJN9MP72bT33ctZB5lnoLnVLk7dm36vurhCM0UHY7TQ+64fZyP12A6dItB/xoC7fBLoYDkHPGtlEIGIM5HldUGv4GFFbI5zqZcn1wVgOBZb+Nk3g19LMcjzu5VF73HOcaFP57z8J9R4GZkB4O/+sr76UkN9TtqGeNXt0qgVm5mfT7S4QEFd6Df8ElD9YIBxluT3gxKCuw9fLbk/DhJPX1y9Wln9ZuXl6drSHzbqvyD66cI/LPzjwtcLqwu/X/jDwt7CycLlQriQLvzbwr8v/Mfaf67919p/r/2PbfrDH9R9frXg/Vv7v/8HCJaDQg==</latexit>(1, 1, 0) (1, 1, 1)
ドット積

クロス積

Tensor field network: 合同変換同変な機械学習モデル<latexit 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KeaDmjXw2CRm0NM9iXbopUimerRf9VvAnz2sDfjJ5gOewsLqM92ewe92/YVfgQlrn2Zs5rnCwuwlVdE64IW6HWejMsg1S+kJnd5iQmqDB6keNL7+OKJPr1j/lVjmtlTqLLA9Zfm9NiTASQJq+NeiD4VmxgA0eUz/ecB/R1SfYpNZ8HWKKutj5Dm08jETJMUxwpWjXmZi1EO2txdfBsXMIqDtdYR5LxME/3yflb3Pj6fdZB5Bppb7+LUnWn3oosrNFN0MEYLvY+DOBurh7brFIH+NwZc5X6gneUccK+VQQQizkSWNwm9gvs1sTXJpZ6eXCeA4URs47NvBr+VYpjnt2vL3uuck0Lvznn5j6jxMjIdwM/Bqr76VkW9T9qKeNVt0qgVq5m/j9S4QEFd6BN+CahBMEQ/S/K7YQnB7afvVtw/B4nHL65erK2/Xnt++mrlj5vNfxD9Yuk3S3+/9P3S+tLvl/64tL/UX7pcCpf+fek/lv5z6b9e/c/rpdc/f/1LW/WnP2na/N2S9/P61/8L5iWW1g==</latexit>• TFN: Tensor field network (Thomas+ 2018)(l)
H̃in,i 2 R2l+1 : type-l 入力<latexit 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+asZX9hA7byimhtaIFe58m4DFL9QGpyl5dYoMrgQYonvW9fPNFv75i/xZhm9i1UWeD+S/P22JMBJAlr4x6IPhWbmAAx5DP93wPGO6T6LTZdBVunrLV+hTSfRiZnmqI4VrBq3MtcjHLQ7u7i27iAURystV5Bzss00Q/v57Pet8/nHWSegebWuzh1Z/q96OIKzRQdjNFC79tBnI3VQzt0ikD/nQLu8mmgg6UPeNbKIAIRZyLL64Jewf2a2JrkUi9PrgvAcCK28bNvBj+XYpjnt2vL3uOck0Kfznn5b6jxMjIDwN+DVX31pYb6nLQN8arbpVErNjP/P9LiAgV1od/wS0ANgiHGWZLfDUsIbj98teL++kc8fnH1Ym3967XnZ69Wfr9Z/4nQj5Z+tvSvS79YWl/6zdLvl/aXTpcul8IlWPqPpf9c+q9Xf3z136/+59X/2qbf/17d56dL3s+r//sLmHV2oA==</latexit>（ l 階テンソルに対応）
xi : i 番目ノード位置
wll : 学習するパラメータlk
J : R 0 ! R : 学習する関数<latexit 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+asZX9hA7byimhtaIFe58m4DFL9QGpyl5dYoMrgQYonvW9fPNFv75i/xZhm9i1UWeD+S/P22JMBJAlr4x6IPhWbmAAx5DP93wPGO6T6LTZdBVunrLV+hTSfRiZnmqI4VrBq3MtcjHLQ7u7i27iAURystV5Bzss00Q/v57Pet8/nHWSegebWuzh1Z/q96OIKzRQdjNFC79tBnI3VQzt0ikD/nQLu8mmgg6UPeNbKIAIRZyLL64Jewf2a2JrkUi9PrgvAcCK28bNvBj+XYpjnt2vL3uOck0Kfznn5b6jxMjIDwN+DVX31pYb6nLQN8arbpVErNjP/P9LiAgV1od/wS0ANgiHGWZLfDUsIbj98teL++kc8fnH1Ym3967XnZ69Wfr9Z/4nQj5Z+tvSvS79YWl/6zdLvl/aXTpcul8IlWPqPpf9c+q9Xf3z136/+59X/2qbf/17d56dL3s+r//sLmHV2oA==</latexit>- 球面調和関数を用いた
合同変換不変・同変な機械学習モデル
XX
(l)
(l)
(k)
ll
lk
H̃out,i = w H̃in,i +W (xj xi ) H̃in,j<latexit sha1_base64="OkSxLwGt30+efbwsqNIziX8PRjQ=">AAA113iclVtbcxxLUpaXBRZxOwuPvBTIZs+CJCT7mD1EALG6S9ZIGt0le2xFz0xOT1t9c1dNS3LvwBvBK7+HX8K/IauqazKrp+UDirDV+X1V2XX5sjKnp9XP40iqtbX/efGz3/v57//BH/7ijxb/+E/+9M/+/Ltf/sWVzCbFAC4HWZwVN/1AQhylcKkiFcNNXkCQ9GO47t9vaf66hEJGWXqhnnL4mARhGo2iQaAQuvvuv4XoqSgeQtVT+9O7qpcEalwkVTZR02URTXtykudVPBV/+y/i4VMV41WvvUeUeh0Whfh7gUZyV92LXghfxNW lk (x) =k+lX

J=|k l|k 0 j=ilk
J (kxk)JX

m= J<latexit 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YJ : R3 ! R2J+1 :球面調和関数
YJm (x/kxk)Qlk
Jm Qlk 2 R(2l+1)⇥(2k+1):Jm<latexit 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Clebsch-Gordan 係数• Universality （同変な連続関数を任意の精度で一様に近似できる性質）が示されている（Dym+ 2020)• 計算に時間がかかるためシミュレーションの代替には不向き
43

目次• Graph neural network• 同変性• IsoGCN: 同変性＋Graph Neural Network
• まとめ

44

目次

• Graph neural network• 同変性• IsoGCN: 同変性＋Graph Neural Network
• まとめ

45

概要：対称性があり計算効率のよい GNN を開発• IsoGCN: 合同変換不変・同変で計算効率のよい GNN (Horie+ 2020)• IsoGCN は同等の精度の有限要素解析より高速• IsoGCN は微分演算子の近似とみなせる

スカラー場

勾配

勾配

（入力）

（正解）

（推論）

46

IsoGCN：合同変換不変・同変な GCN
頂点数 |V| のグラフが d 次元ユークリッド空間に埋め込まれているとする• IsoAM (Isometric Adjacency Matrix) G：<latexit 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qzlxcIGbOUV0drQAr3Jk0kZpPqB1PQ+L7FAlcGjFM8G3796pt/eMX9PMcvsW6iywP2X5u2xZ0NIEtbGPRB9LjYxAWLIZ/q/R4x3SPVbbLoKtk5Za/0KaT6LTM40RXGsYNW4l7kY56Dd3cd3cQHjOFjrvIKcl2miH97P68H3L+c9ZJ6B5tb7OHVv+r3q4wrNFD2M0cLg+2GcTdRjN3SKQP+dAu7yaaCD5QLwrJVBBCLORJY3Bb2ChzWxNc2lXp5cF4DhVGzjZ98MfiXFKM/v1pa9xzknhT6d8/JvUONlZAaAv4er+upbDfU5aRviVb9Lo1ZsZv5/osUlCupSv+GXgBoGI4yzJL8flRDcffpuZb37pzyLF9ev1tb/bu3l2ZuV3242f+bz86W/XPqrpV8vrS/9/dJvl/aXTpeulsKlf1n6r6X/XvqfN//85l/f/Pub/7BNf/qTps9fLHk/b/7z/wANt3cl</latexit>|V|⇥|V|⇥d

G2R X
Gijm =[Tijkl (xkxll<latexit sha1_base64="2OMewJLGHHWnr2uKuJJjSITozJg=">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</latexit>xl )]mk,l;k6=lk xk47

IsoGCN：合同変換不変・同変な GCN
頂点数 |V| のグラフが d 次元ユークリッド空間に埋め込まれているとする

xl• IsoAM (Isometric Adjacency Matrix) G：

G2R X
Gijm =[Tijkl (xk

l

Gij:

|V|⇥|V|⇥d<latexit 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)]mijk,l;k6=l• Tijkl : グラフごとに定まる 2 階テンソル<latexit 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JLixeDHNy/02zvm7ylmmX0LVRa4/9K8PfZiCEnC2rgHoi/FJiZADPlM//eE8Q6pfotNV8HWKWutXyHNZ5HJmaYojhWsGvcyF+MctLuH+D4uYBwHa51XkPMyTfTD+3k9+PH1vIfMM9Dceh+nHky/N31coZmihzFaGPw4jLOJeuqGThHov1PAXT4NdLBcAJ61MohAxJnI8qagV/C4JramudTLk+sCMJyKbfzsm8FvpBjl+f3asvc456TQp3Ne/gNqvIzMAPD3cFVffa+hPidtQ7zqd2nUis3M/8+0uERBXeo3/BJQw2CEcZbkD6MSgvvPP6ysd/+UZ/Hi+s3a+j+tvT57t/K7zebPfH659LdLf7f026X1pX9e+t3S/tLp0tVSuJQt/efSfy3997vDd1/fPb77Zpv+/GdNn79Z8n7e/cf/AU2eYQ4=</latexit>- Tijkl = il jk I とおけば<latexit sha1_base64="jNNPdscmZf0Ba95WP/eUEX88WQc=">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</latexit>

Gij: = xj<latexit 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</latexit>

xi
48

k xk

Key Idea: GCN with Vector-Valued Adjacency Matrix• GCNʼs message passing<latexit 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Hout = ÂHin W

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+qZfnvH/D3FLLNvocoC91+at8eeDSFJWBv3QPS52MYEiCGf6f8eMd4h1W+x6SrYOmWt9Suk+SwyOdMUxbGCdeNe5mKcg3Z3H9/FBYzjYKP1CnJepol+eL+Y975/uegg8ww0t9nFqXvT71UXV2im6GCMFnrfD+Nsoh7boVME+u8UcJf7gQ6WAeBZK4MIRJyJLK8LegUPG2Jnmku9PLkuAMOp2MXPvhn8VopRnt9trHqPc84KfTrn5b+ixsvIDAB/D9f11bca6nPSNsSrbpdGrdjM/P9Ei0sU1KV+wy8BNQxGGGdJfj8qIbj79N3aZvtPeZYvrl9tbP77xsvzN2t/2q7/zOdnK79e+c3K71Y2V/5j5U8rhyv9lauVcAVW/rrynyv/9Wb7zfRN/uaLbfrjH9V9/nnF+3nz9X8BAW5c/w==</latexit>Âij : Renormalized adjacency matrix<latexit 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49jKey Idea: GCN with Vector-Valued Adjacency Matrix• GCNʼs message passing<latexit 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Hout = ÂHin W

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: Renormalized adjacency matrix<latexit 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1fP9Ns75u8pZpl9C1UWuP/SvD32bAhJwtq4B6LPxTYmQAz5TP/3iPEOqX6LTVfB1ilrrV8hzWeRyZmmKI4VrBv3MhfjHLS7+/guLmAcBxutV5DzMk30w/vFvPf9y0UHmWeguc0uTt2bfq+6uEIzRQdjtND7fhhnE/XYDp0i0H+ngLvcD3SwDADPWhlEIOJMZHld0Ct42BA701zq5cl1ARhOxS5+9s3gt1KM8vxuY9V7nHNW6NM5L/8VNV5GZgD4e7iur77VUJ+TtiFedbs0asVm5v8nWlyioC71G34JqGEwwjhL8vtRCcHdp+/WNtt/yrN8cf1qY/PfN16ev1n703b9Zz4/W/n1ym9WfreyufIfK39aOVzpr1ythCuw8teV/1z5rzfbb6Zv8jdfbNMf/6ju888r3s+br/8LNcpdAA==</latexit>• IsoGCNʼs message passing:

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1

<latexit sha1_base64="xhWmIeX/O90J+AEuSZINeXKTprY=">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</latexit>- Scalar to vector0
1 0
1
Hout,xG::,x@ Hout,y A = @ G::,y A Hin W
Hout,zG::,z<latexit 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Gij,x@ Gij,y A = xj xi
Gij,zIsoAM :
<latexit 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1fP9Ns75u8pZpl9C1UWuP/SvD32bAhJwtq4B6LPxTYmQAz5TP/3iPEOqX6LTVfB1ilrrV8hzWeRyZmmKI4VrBv3MhfjHLS7+/guLmAcBxutV5DzMk30w/vFvPf9y0UHmWeguc0uTt2bfq+6uEIzRQdjtND7fhhnE/XYDp0i0H+ngLvcD3SwDADPWhlEIOJMZHld0Ct42BA701zq5cl1ARhOxS5+9s3gt1KM8vxuY9V7nHNW6NM5L/8VNV5GZgD4e7iur77VUJ+TtiFedbs0asVm5v8nWlyioC71G34JqGEwwjhL8vtRCcHdp+/WNtt/yrN8cf1qY/PfN16ev1n703b9Zz4/W/n1ym9WfreyufIfK39aOVzpr1ythCuw8teV/1z5rzfbb6Zv8jdfbNMf/6ju888r3s+br/8LNcpdAA==</latexit>

50

jrelative vertex positions

Key Idea: GCN with Vector-Valued Adjacency Matrix• GCNʼs message passing<latexit 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Hout = ÂHin W

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: Renormalized adjacency matrix<latexit 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1fP9Ns75u8pZpl9C1UWuP/SvD32bAhJwtq4B6LPxTYmQAz5TP/3iPEOqX6LTVfB1ilrrV8hzWeRyZmmKI4VrBv3MhfjHLS7+/guLmAcBxutV5DzMk30w/vFvPf9y0UHmWeguc0uTt2bfq+6uEIzRQdjtND7fhhnE/XYDp0i0H+ngLvcD3SwDADPWhlEIOJMZHld0Ct42BA701zq5cl1ARhOxS5+9s3gt1KM8vxuY9V7nHNW6NM5L/8VNV5GZgD4e7iur77VUJ+TtiFedbs0asVm5v8nWlyioC71G34JqGEwwjhL8vtRCcHdp+/WNtt/yrN8cf1qY/PfN16ev1n703b9Zz4/W/n1ym9WfreyufIfK39aOVzpr1ythCuw8teV/1z5rzfbb6Zv8jdfbNMf/6ju888r3s+br/8LNcpdAA==</latexit>• IsoGCNʼs message passing:

j

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1

<latexit sha1_base64="xhWmIeX/O90J+AEuSZINeXKTprY=">AAA1mXiclVvbchtJcuWub2uuL7P2475UmFLMrIOkRc3I9oN3Y3m/gSRI8CYKGkajkWi02Dd1FZqksNhv82/4B/xqf4KzqrqQWY2mZs0IiZ3nVGXX5WRlotEcFEks1Zs3//Wzn//Zn//FX/7VL/56+Zd/87d/9/ff/OofrmU+KUO4CvMkL28HgYQkzuBKxSqB26KEIB0kcDN42Nb8TQWljPPsUj0X8DENoiwexWGgELr/5q6fwEh91x9AFGfToCyD59k0nC3v30/jT6tPs+V+v75+Ztdf8BqyYd2+X8bRWP1m+bfL/erp/pNYE/p3fP/Nypv1N+ZHLF5s1BcrS/VP9/5XP3zbH+bhJIVMhUkg5YeNN4X6iHdRcZgA3nMioQjChyCCD3iZBSnIj1OzCDPxGpGhGOUl/suUMCjvMQ1SKZ/TAbZMAzWWTU6DrdwgbdxZjf794zTOiomCLLQ3Hk0SoXKhV1gM4xJClTzjRRCWMY5dhOOgDEKF++B5VvHDF8/3VN9d5XkifdhO0YOeWjDtby0cIpjBY5inaYB71B/Go9Hsw9uP0/4IBzEdipWNmf7/7WwmxOu8HMZZUOrRjkZQ4tLHQeI7KBoe+oXekSAxjshAf36/OJt31DcV/yLMXfGmcbamBSv+hJvjOtPt6cbojN+Z+XToyy4nWaxmHzbQn+jrBS/T6YeVjY9mOTQnHmM1FvLzJChBDHDGD6Ck7yJQdkA6eNb1cGwM/OF+urgMQ0hwvRugnBSFGcOP0+9WNn7T7CMnA8PeuwHiUjeaXOHaD9LpVRO/tvg14q+Fx4BlYHGfijKn+fwHzmcVF7We0+8W/FwHpV2+emzaNj37U1qK/uL9d71eu+19/GUeSNtHt/wDW+fZ8vLya/0jTndvxMnm5YHY2d07PD28PDw77QlDLbdF+Cr+1ueDXNVhjSM8CcoHITFg8TiUIh+JMCjstT5KSkANlXEW6egexlUsXbNRHE1KaMgCQT3S2RQXOxXfdfB6YWuxTYiqhNK12jZWWzszV9fswhyyLa1UXrg2l3nR1mKQK5WnrtGWsRba1fMOXLPgpRYD12LwUovQtQhfajF0LYa6BW7DAc4u0TMUgcD2+vSEEYbzUODapL4PvNagUQYqeiRMcGgve2ZT7K7h2QyrIskfoVwLMUOuL2MkjsyywmhlY2o38I99tGx0tXXH4cYqSNbFHopBKjxa9N5LvWPIW497zuNe06Oh1WPu7rnytr6rFK6RwBnVxlvXAw+eIXVZ+X7lh4Vuq/M+7up77uoHM52eVfVXlwOVbwdfh4C3Hi0O3ILY3j3Xu9fS+8L1MpnxMZ9H2fp8YezdpVmZeQy+sDRNh+MSoOmS+Vv5ftEjrRrz/f2i7yATgJugO7csGXy2c3ZNXp605wcPfCiF9mPd7NZudtvcbIoyeKR1bzhbW1sLqjweionUR1M8EkUuZYw1n3VdJAGGTu3/5dHpsqTASGqZo2Zs97rN/3uStaPtuaPtn3SEc84iMGewbSutDwPPR4RScbRztbb2okxwdEES5ViGjdOWeSJnRzdv9NWJMlcLM910rjZbXDnBu/vhJOa+vn4YXHqdNn+y08Ki6nqonjlTn0btcPXV1zbF9m+qtzvv3/X7u5nOb4Cj1tcvDrgWHMSJFmuiL2yi11e1v1GS56WhzZXlzWXdYMNUNgs1kioxEGa25AiDZLrTbFAFSTzkDe7n9YmlZgsuQar2DoaZzWcEhdQ5rpBxkmd1frpAF3kqqqCMA4xWp29QwVR7flJZXqbo9VUfoVczt5xlgw6IGfjMgJjQZ0Jihj4zJAZ8BogZ+cyImMhnImLGPjMmJvaZmJhPPvOJmAefeSAm8ZlkZmRcpiKWGLH4aXj4rA87u4Or4tNEKjHMs2+V0J8gUY7P+uTxNkakte/M953RXXOfyYkpfKYg5rPPfCam9JmSGOkzkhjlM4qYic9MiKl8piLm0WceiXnymSdinn3mmZgvPvNlZss8FwCYmfP58V7VQTK1oTQYsbCZj1uNbZS4FsZmPOM4PCCYxUYVEswCoxoSzKKiAoJZSFQjglk8VBHBLBiqMcEsEqoJwSwMqk8EsxioHghmAVAlBCcMTglOGcwWmq9wTjATc1UQzJRcfSaYybgqCWYariTBkm8qwap9Tbh0K4KZbqtHgploqyeCmWKrZ4KZXKsvBDut7iagn0SZT3tli27Biq71XAarvNaTGaz8Ws9msBpsPZ3BCrH1fAarxtYTGqwkW89osLpsPaWRe/GcBqvQ1pMarExbz2qwWm2e1o5LfS7l3IsnMVjptp7FYPXbehqDFXHreQxWya0nMlg5t57JYDXdeiqDFXbruQxW3a0nM1iJt57NYHXeejqDFXvr+QxW8S+f0BgLZRzOK5R0k+Jjk8Im3SJ4i8HbBG8zeIfgHQbvErzL4D2C9xi8T/A+gw8IPmDwIcGHDD4i+IjBxwQfM7hDcIfBJwSfMPiU4FMGnxF8xuAuwV0GnxN8zuALgi8Y3CO4x+BLgi8ZfEXwFYOvCb5m8A3BNwy+JfiWwe8Jfs/gO4LvXj5efdGBVR3T6CbTr5Ee47Y4t+1z25zb8bkdzu363C7n9nxuj3P7PrfPuQOfO+Dcoc8dcu7I5444d+xzx5zr+FyHcyc+d8K5U5875dyZz51xrutzXc6d+9w55y587oJzPZ/rce7S5y45d+VzV5y79rlrzt343A3nbn3ulnPvfe495+58zsn+mpcQ1RcwnyPws+ubed8qz2DqPs86LJ1YqJ9S0pjXxBr36+GqhhkysAjVIaYKQYSqD1N7IEI1R1WPhCoNU2cgQvWFqS4QoarC1BSIUC1hKglEqIIw9QMiVDeYqgERqhZMrYBIwtbBIlQZmLoAkYytn0WoCjA1ACKU+03mR4Qyvsn3iFCeN1keEckW3CKU06t6W9imVBah/G2yNyKUtU3ORoRytcnUiFCGNvkZkbZq1C9DqyApxnq/ze+5AqtBLQ6jCwfSRy16MlFTSZAOhrqHvSAiTyHSuPlNsJGklqMD0CEi+D9BMo5S3dX8JtgJtxbtfCLTKR//VIvVWSjWkCwU6pBNamq/1LIWCnREFoozIguFOSYLh8vGioL8RBaK8YGtzVSLcD7zqRags3Ax2Sqi+HK2JFMtOmeh6D6ThYIr2UpNtdDmCzTVInMWLjRbZhRYRRaK65EsFNYTWSiqZ7JQUF9m9XdemGefLG5yLOqMcqvJrIhQRjX5FBHKoyaLIkLZ0+TOQcpypsmYiFCmNHkSEcqPJjsiQlnR5EREKBeaTIgIZUCT/xChvGeyHiKU7UyuQ4RynMlwiFBmM3kNEcpnJpshQlnM5DBEKHeZzIUIZSyTrxChPGWyFCKUnUxust/XOuR66r6pdciNRSj/mOyDCGUdk3MQoVxjMg0id2wHKS8MeFpIu+P6IO7jFVs9F/qa6dThP59cHcOa69k4Niq6hEzqVyx2IEyCElBU4019AuEdbbEnR7F+VApZmA/jLEJnwSTRiBzNr9PZVOqnvD1QLzkY5Mnwp9wMnmYYhM0ntVn97bHJm7U/85S6npqy9WUmmfrVlsNI/2rbYRQBasdhFANq12EUBWrPYRQHat9hFAnqwGEUC+rQYRQN6shhFA/q2GEUEarjMIoJdeIwigp16jCKC3XmMIoM1XUYxYY6dxhFh7pwGMWH6jmMIkRdOoxiRF05jKJEXTuM4kTdOIwiRd06jGJFvXcYRYu6c5ityFDI+2VQjC0buc+5ofdxI9piMOki2mYwSSPaYTCpI9plMAkk2mMwaSTaZzDJJDpgMCklOmQwiSU6YjDpJTpmMEkm6jCYVBOdMJiEE50ymLQTnTGY5BN1GUwKis4ZTCKKLhhMOop6DCYpRZcMJjVFVwwmQUXXDCZNRTcMJllFtwwmZUXvGUziiu4Y7Cr+nnl7SNNy/hRlwMQltwglbcltQklacodQo6zXYsd8kzGRIAIhQQm8dQJDsbsqBhAGGlfjWIrHfJIMEUILhDTfe2AtOSmFficuT9CRft8FngqsLc2Xue4r9T26I6lT7hNK4pQHhJI25SGhJE15RCgpUx4TSsKUHUJJl/KEUJKlPCWUVCnPCCVRyi6hpEl5TihJUl4QSoqUPUJJkPKSUNKjvCKU5CivCSU1yhtCSYzyllDSonxPKElR3hE6f+SSYd0H5iNEYB+21EUgUAXQ8Yt/XR5ukoVS3SILJbpNFkpzhyw87HbJQhHtkYXi2ScLRXNAForlkCwUyRFZKI5jslAUHbJQDCdkoQhOycLNPyMLN71LFm72OVm4yRdk4eb2yMJNvSQLN/OKLNzEa7Jw827Iwk27JQs36z1ZuEl37H51pVVXWXrLgG+ZshUXHik6fs1rrhjEFl01by3mEyWw3BGPmNIKKP2CCKgi8qqh+vZqrgHTcKEQBFMuQaNeAlMwQaNiAlMyQaNmAlM0QaNqAlM2QaNuAlM4QaNyAlM6QaN2AlM8QaN6AlM+QaN+AlNAQaOCAlNCQaOGAlNEQaOKAlNGQaOOAlNIQaOSAlNKQaOWAlNMQaOaAlNOQaOeAlNQQaOiAlNSQaOmAlNUQaOqAlNWQaOuAlNYQaOyAlNaQaO2AlNcQaO6AlNeQaO+AlNgAauw8JMCphxVTkBMsiGUybN+aWkYqEBEkEGJ2UbbsUSlDyY69TRfP8ams2lxP+2X6dQYJvFpr5AWcRmH5nVf6j9/d3DwbNKdeQ1E3wTzY8O3e0NkHCj8pO7fwmvZ5S27s7bBpPkQkq9NxDSYz8RaC/epG3W/1qhQcTKEumXfGPPRz3vgMaHycBxI/QZ6MFG5+QQFpTfCxgushW0zH2PdZXEAQ/DaWbOlXYkEHjqunTVRC6F5huY3ToIiCUKYzd+o6dTATLwW9bW/vMuN94vnGW+3OZCOZC/tdJrsxYzn9sapmRZsjRtkUs7cczefKCGazZ+kNalQ0Ry1FY9iKJuuZT5SafBELR3QbIfJIjcvMdmHbIteimSiZ/9FPwnw2ePOjL/BdNxZ2ED9brf3onfTv8JfQYl7X+asZW9hA7bzimhtaIHe5MmoDFL9QGr8mJdYoMrgWYpXnR/fvtJv75i/5phk9i1UWeD+S/P22Ks+JAlr4x6IvhZbmAAx5DP93zPGO6T6LTZdBVunrLV+hTSfRCZnmqI4VrBq3MtcDHPQ7h7jh7iAYRysN15Bzss00Q/vZ9POj29mLWSegeY22jj1aPq9beMKzRQtjNFC58d+nI3UczN0ikD/pQPucjfQwdIDPGtlEIGIM5HldUGv4GldbI9zqZcn1wVgOBY7+Nk3g2+lGOT5w/qy9zjnrNCnc17+M2q8jMwA8Hd/VV99raE+J21DvGp3adSKzcz/L7S4REFd6jf8ElD9YIBxluSPgxKCh/tvVjaaf0i0eHH9dn3jX9c3zn9Y+f1W/UdGv1j69dI/LX23tLH0b0u/XzpY6i5dLYVL/7n030v/s/S/7379bvPdwbsj2/TnP6v7/OOS9/Ou938IHa5T</latexit>- Scalar to vector0
1 0
1
Hout,xG::,x@ Hout,y A = @ G::,y A Hin W
Hout,zG::,z<latexit 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Gij,x@ Gij,y A = xj xi
Gij,zIsoAM :
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1fP9Ns75u8pZpl9C1UWuP/SvD32bAhJwtq4B6LPxTYmQAz5TP/3iPEOqX6LTVfB1ilrrV8hzWeRyZmmKI4VrBv3MhfjHLS7+/guLmAcBxutV5DzMk30w/vFvPf9y0UHmWeguc0uTt2bfq+6uEIzRQdjtND7fhhnE/XYDp0i0H+ngLvcD3SwDADPWhlEIOJMZHld0Ct42BA701zq5cl1ARhOxS5+9s3gt1KM8vxuY9V7nHNW6NM5L/8VNV5GZgD4e7iur77VUJ+TtiFedbs0asVm5v8nWlyioC71G34JqGEwwjhL8vtRCcHdp+/WNtt/yrN8cf1qY/PfN16ev1n703b9Zz4/W/n1ym9WfreyufIfK39aOVzpr1ythCuw8teV/1z5rzfbb6Zv8jdfbNMf/6ju888r3s+br/8LNcpdAA==</latexit>

51

jrelative vertex positions

Key Idea: GCN with Vector-Valued Adjacency Matrix
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Hout = ÂHin W

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Gij,x@ Gij,y A = xj xi
Gij,zIsoAM :
<latexit sha1_base64="vfYDJtaEzJs8VlqgAivyd/lZ39E=">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</latexit>jrelative vertex positions• IsoGCNʼs message passing:- Scalar to vector0
1 0
1
Hout,xG::,x@ Hout,y A = @ G::,y A Hin W
Hout,zG::,z• IsoGCN incorporates
graphs' geometry information
by simply replacing
the adjacency matrices with IsoAMs<latexit 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</latexit>

52

Key Idea: GCN with Vector-Valued Adjacency Matrix
0<latexit 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</latexit>• GCNʼs message passing

1

Gij,x
@ Gij,y A = xj xi
iGij,zIsoAM :

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1fP9Ns75u8pZpl9C1UWuP/SvD32bAhJwtq4B6LPxTYmQAz5TP/3iPEOqX6LTVfB1ilrrV8hzWeRyZmmKI4VrBv3MhfjHLS7+/guLmAcBxutV5DzMk30w/vFvPf9y0UHmWeguc0uTt2bfq+6uEIzRQdjtND7fhhnE/XYDp0i0H+ngLvcD3SwDADPWhlEIOJMZHld0Ct42BA701zq5cl1ARhOxS5+9s3gt1KM8vxuY9V7nHNW6NM5L/8VNV5GZgD4e7iur77VUJ+TtiFedbs0asVm5v8nWlyioC71G34JqGEwwjhL8vtRCcHdp+/WNtt/yrN8cf1qY/PfN16ev1n703b9Zz4/W/n1ym9WfreyufIfK39aOVzpr1ythCuw8teV/1z5rzfbb6Zv8jdfbNMf/6ju888r3s+br/8LNcpdAA==</latexit>• IsoGCNʼs message passing:- Scalar to vector0
1 0
1
Hout,xG::,x@ Hout,y A = @ G::,y A Hin W
Hout,zG::,zjrelative vertex positions• IsoGCN incorporates
graphs' geometry information
by simply replacing
the adjacency matrices with IsoAMs<latexit sha1_base64="fn/QRNScX2RE0RkKT9FRqv/4bDE=">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</latexit>• IsoAM is equivariant
53

Key Idea: GCN with Vector-Valued Adjacency Matrix• IsoGCNʼs message passing:0<latexit sha1_base64="xhWmIeX/O90J+AEuSZINeXKTprY=">AAA1mXiclVvbchtJcuWub2uuL7P2475UmFLMrIOkRc3I9oN3Y3m/gSRI8CYKGkajkWi02Dd1FZqksNhv82/4B/xqf4KzqrqQWY2mZs0IiZ3nVGXX5WRlotEcFEks1Zs3//Wzn//Zn//FX/7VL/56+Zd/87d/9/ff/OofrmU+KUO4CvMkL28HgYQkzuBKxSqB26KEIB0kcDN42Nb8TQWljPPsUj0X8DENoiwexWGgELr/5q6fwEh91x9AFGfToCyD59k0nC3v30/jT6tPs+V+v75+Ztdf8BqyYd2+X8bRWP1m+bfL/erp/pNYE/p3fP/Nypv1N+ZHLF5s1BcrS/VP9/5XP3zbH+bhJIVMhUkg5YeNN4X6iHdRcZgA3nMioQjChyCCD3iZBSnIj1OzCDPxGpGhGOUl/suUMCjvMQ1SKZ/TAbZMAzWWTU6DrdwgbdxZjf794zTOiomCLLQ3Hk0SoXKhV1gM4xJClTzjRRCWMY5dhOOgDEKF++B5VvHDF8/3VN9d5XkifdhO0YOeWjDtby0cIpjBY5inaYB71B/Go9Hsw9uP0/4IBzEdipWNmf7/7WwmxOu8HMZZUOrRjkZQ4tLHQeI7KBoe+oXekSAxjshAf36/OJt31DcV/yLMXfGmcbamBSv+hJvjOtPt6cbojN+Z+XToyy4nWaxmHzbQn+jrBS/T6YeVjY9mOTQnHmM1FvLzJChBDHDGD6Ck7yJQdkA6eNb1cGwM/OF+urgMQ0hwvRugnBSFGcOP0+9WNn7T7CMnA8PeuwHiUjeaXOHaD9LpVRO/tvg14q+Fx4BlYHGfijKn+fwHzmcVF7We0+8W/FwHpV2+emzaNj37U1qK/uL9d71eu+19/GUeSNtHt/wDW+fZ8vLya/0jTndvxMnm5YHY2d07PD28PDw77QlDLbdF+Cr+1ueDXNVhjSM8CcoHITFg8TiUIh+JMCjstT5KSkANlXEW6egexlUsXbNRHE1KaMgCQT3S2RQXOxXfdfB6YWuxTYiqhNK12jZWWzszV9fswhyyLa1UXrg2l3nR1mKQK5WnrtGWsRba1fMOXLPgpRYD12LwUovQtQhfajF0LYa6BW7DAc4u0TMUgcD2+vSEEYbzUODapL4PvNagUQYqeiRMcGgve2ZT7K7h2QyrIskfoVwLMUOuL2MkjsyywmhlY2o38I99tGx0tXXH4cYqSNbFHopBKjxa9N5LvWPIW497zuNe06Oh1WPu7rnytr6rFK6RwBnVxlvXAw+eIXVZ+X7lh4Vuq/M+7up77uoHM52eVfVXlwOVbwdfh4C3Hi0O3ILY3j3Xu9fS+8L1MpnxMZ9H2fp8YezdpVmZeQy+sDRNh+MSoOmS+Vv5ftEjrRrz/f2i7yATgJugO7csGXy2c3ZNXp605wcPfCiF9mPd7NZudtvcbIoyeKR1bzhbW1sLqjweionUR1M8EkUuZYw1n3VdJAGGTu3/5dHpsqTASGqZo2Zs97rN/3uStaPtuaPtn3SEc84iMGewbSutDwPPR4RScbRztbb2okxwdEES5ViGjdOWeSJnRzdv9NWJMlcLM910rjZbXDnBu/vhJOa+vn4YXHqdNn+y08Ki6nqonjlTn0btcPXV1zbF9m+qtzvv3/X7u5nOb4Cj1tcvDrgWHMSJFmuiL2yi11e1v1GS56WhzZXlzWXdYMNUNgs1kioxEGa25AiDZLrTbFAFSTzkDe7n9YmlZgsuQar2DoaZzWcEhdQ5rpBxkmd1frpAF3kqqqCMA4xWp29QwVR7flJZXqbo9VUfoVczt5xlgw6IGfjMgJjQZ0Jihj4zJAZ8BogZ+cyImMhnImLGPjMmJvaZmJhPPvOJmAefeSAm8ZlkZmRcpiKWGLH4aXj4rA87u4Or4tNEKjHMs2+V0J8gUY7P+uTxNkakte/M953RXXOfyYkpfKYg5rPPfCam9JmSGOkzkhjlM4qYic9MiKl8piLm0WceiXnymSdinn3mmZgvPvNlZss8FwCYmfP58V7VQTK1oTQYsbCZj1uNbZS4FsZmPOM4PCCYxUYVEswCoxoSzKKiAoJZSFQjglk8VBHBLBiqMcEsEqoJwSwMqk8EsxioHghmAVAlBCcMTglOGcwWmq9wTjATc1UQzJRcfSaYybgqCWYariTBkm8qwap9Tbh0K4KZbqtHgploqyeCmWKrZ4KZXKsvBDut7iagn0SZT3tli27Biq71XAarvNaTGaz8Ws9msBpsPZ3BCrH1fAarxtYTGqwkW89osLpsPaWRe/GcBqvQ1pMarExbz2qwWm2e1o5LfS7l3IsnMVjptp7FYPXbehqDFXHreQxWya0nMlg5t57JYDXdeiqDFXbruQxW3a0nM1iJt57NYHXeejqDFXvr+QxW8S+f0BgLZRzOK5R0k+Jjk8Im3SJ4i8HbBG8zeIfgHQbvErzL4D2C9xi8T/A+gw8IPmDwIcGHDD4i+IjBxwQfM7hDcIfBJwSfMPiU4FMGnxF8xuAuwV0GnxN8zuALgi8Y3CO4x+BLgi8ZfEXwFYOvCb5m8A3BNwy+JfiWwe8Jfs/gO4LvXj5efdGBVR3T6CbTr5Ee47Y4t+1z25zb8bkdzu363C7n9nxuj3P7PrfPuQOfO+Dcoc8dcu7I5444d+xzx5zr+FyHcyc+d8K5U5875dyZz51xrutzXc6d+9w55y587oJzPZ/rce7S5y45d+VzV5y79rlrzt343A3nbn3ulnPvfe495+58zsn+mpcQ1RcwnyPws+ubed8qz2DqPs86LJ1YqJ9S0pjXxBr36+GqhhkysAjVIaYKQYSqD1N7IEI1R1WPhCoNU2cgQvWFqS4QoarC1BSIUC1hKglEqIIw9QMiVDeYqgERqhZMrYBIwtbBIlQZmLoAkYytn0WoCjA1ACKU+03mR4Qyvsn3iFCeN1keEckW3CKU06t6W9imVBah/G2yNyKUtU3ORoRytcnUiFCGNvkZkbZq1C9DqyApxnq/ze+5AqtBLQ6jCwfSRy16MlFTSZAOhrqHvSAiTyHSuPlNsJGklqMD0CEi+D9BMo5S3dX8JtgJtxbtfCLTKR//VIvVWSjWkCwU6pBNamq/1LIWCnREFoozIguFOSYLh8vGioL8RBaK8YGtzVSLcD7zqRags3Ax2Sqi+HK2JFMtOmeh6D6ThYIr2UpNtdDmCzTVInMWLjRbZhRYRRaK65EsFNYTWSiqZ7JQUF9m9XdemGefLG5yLOqMcqvJrIhQRjX5FBHKoyaLIkLZ0+TOQcpypsmYiFCmNHkSEcqPJjsiQlnR5EREKBeaTIgIZUCT/xChvGeyHiKU7UyuQ4RynMlwiFBmM3kNEcpnJpshQlnM5DBEKHeZzIUIZSyTrxChPGWyFCKUnUxust/XOuR66r6pdciNRSj/mOyDCGUdk3MQoVxjMg0id2wHKS8MeFpIu+P6IO7jFVs9F/qa6dThP59cHcOa69k4Niq6hEzqVyx2IEyCElBU4019AuEdbbEnR7F+VApZmA/jLEJnwSTRiBzNr9PZVOqnvD1QLzkY5Mnwp9wMnmYYhM0ntVn97bHJm7U/85S6npqy9WUmmfrVlsNI/2rbYRQBasdhFANq12EUBWrPYRQHat9hFAnqwGEUC+rQYRQN6shhFA/q2GEUEarjMIoJdeIwigp16jCKC3XmMIoM1XUYxYY6dxhFh7pwGMWH6jmMIkRdOoxiRF05jKJEXTuM4kTdOIwiRd06jGJFvXcYRYu6c5ityFDI+2VQjC0buc+5ofdxI9piMOki2mYwSSPaYTCpI9plMAkk2mMwaSTaZzDJJDpgMCklOmQwiSU6YjDpJTpmMEkm6jCYVBOdMJiEE50ymLQTnTGY5BN1GUwKis4ZTCKKLhhMOop6DCYpRZcMJjVFVwwmQUXXDCZNRTcMJllFtwwmZUXvGUziiu4Y7Cr+nnl7SNNy/hRlwMQltwglbcltQklacodQo6zXYsd8kzGRIAIhQQm8dQJDsbsqBhAGGlfjWIrHfJIMEUILhDTfe2AtOSmFficuT9CRft8FngqsLc2Xue4r9T26I6lT7hNK4pQHhJI25SGhJE15RCgpUx4TSsKUHUJJl/KEUJKlPCWUVCnPCCVRyi6hpEl5TihJUl4QSoqUPUJJkPKSUNKjvCKU5CivCSU1yhtCSYzyllDSonxPKElR3hE6f+SSYd0H5iNEYB+21EUgUAXQ8Yt/XR5ukoVS3SILJbpNFkpzhyw87HbJQhHtkYXi2ScLRXNAForlkCwUyRFZKI5jslAUHbJQDCdkoQhOycLNPyMLN71LFm72OVm4yRdk4eb2yMJNvSQLN/OKLNzEa7Jw827Iwk27JQs36z1ZuEl37H51pVVXWXrLgG+ZshUXHik6fs1rrhjEFl01by3mEyWw3BGPmNIKKP2CCKgi8qqh+vZqrgHTcKEQBFMuQaNeAlMwQaNiAlMyQaNmAlM0QaNqAlM2QaNuAlM4QaNyAlM6QaN2AlM8QaN6AlM+QaN+AlNAQaOCAlNCQaOGAlNEQaOKAlNGQaOOAlNIQaOSAlNKQaOWAlNMQaOaAlNOQaOeAlNQQaOiAlNSQaOmAlNUQaOqAlNWQaOuAlNYQaOyAlNaQaO2AlNcQaO6AlNeQaO+AlNgAauw8JMCphxVTkBMsiGUybN+aWkYqEBEkEGJ2UbbsUSlDyY69TRfP8ams2lxP+2X6dQYJvFpr5AWcRmH5nVf6j9/d3DwbNKdeQ1E3wTzY8O3e0NkHCj8pO7fwmvZ5S27s7bBpPkQkq9NxDSYz8RaC/epG3W/1qhQcTKEumXfGPPRz3vgMaHycBxI/QZ6MFG5+QQFpTfCxgushW0zH2PdZXEAQ/DaWbOlXYkEHjqunTVRC6F5huY3ToIiCUKYzd+o6dTATLwW9bW/vMuN94vnGW+3OZCOZC/tdJrsxYzn9sapmRZsjRtkUs7cczefKCGazZ+kNalQ0Ry1FY9iKJuuZT5SafBELR3QbIfJIjcvMdmHbIteimSiZ/9FPwnw2ePOjL/BdNxZ2ED9brf3onfTv8JfQYl7X+asZW9hA7bzimhtaIHe5MmoDFL9QGr8mJdYoMrgWYpXnR/fvtJv75i/5phk9i1UWeD+S/P22Ks+JAlr4x6IvhZbmAAx5DP93zPGO6T6LTZdBVunrLV+hTSfRCZnmqI4VrBq3MtcDHPQ7h7jh7iAYRysN15Bzss00Q/vZ9POj29mLWSegeY22jj1aPq9beMKzRQtjNFC58d+nI3UczN0ikD/pQPucjfQwdIDPGtlEIGIM5HldUGv4GldbI9zqZcn1wVgOBY7+Nk3g2+lGOT5w/qy9zjnrNCnc17+M2q8jMwA8Hd/VV99raE+J21DvGp3adSKzcz/L7S4REFd6jf8ElD9YIBxluSPgxKCh/tvVjaaf0i0eHH9dn3jX9c3zn9Y+f1W/UdGv1j69dI/LX23tLH0b0u/XzpY6i5dLYVL/7n030v/s/S/7379bvPdwbsj2/TnP6v7/OOS9/Ou938IHa5T</latexit>

1

Gij,x
@ Gij,y A = xj xi
iGij,zIsoAM :

- Scalar to vector0
1 0
1
Hout,xG::,x@ Hout,y A = @ G::,y A Hin W
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sha1_base64="vfYDJtaEzJs8VlqgAivyd/lZ39E=">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</latexit>- Vector to scalar
0
1 0
1
G::,x
Hin,x
Hout = @ G::,y A · @ Hin,y A W
G::,z
Hin,z

relative vertex positions

j<latexit sha1_base64="UijHg3lLPd+eIp9LabP4aA/Tcic=">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</latexit>- Scalar to rank-2 tensor
0<latexit 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</latexit>

Hout,xx@ Hout,yx
Hout,zx

Hout,xy
Hout,yy
Hout,zy1• IsoGCN can convert any rank tensors
to any rank tensors with keeping
equivariance0

Hout,xz
G::,x G::,x
Hout,yz A = @ G::,y G::,x
Hout,zz
G::,z G::,x
54

G::,x G::,y
G::,y G::,y
G::,z G::,y1

G::,x G::,z
G::,y G::,z A Hin W
G::,z G::,zIsoGCN：合同変換不変・同変な GCN• 有限要素解析に適用するため
IsoAM の具体例 D̃ijk を以下のように定義する
XD̃ijk = Dijk
Dilk
ij
l

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</latexit>

1 xjk
Dijk = M
kxj

X xl
Mi =
kxl
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</latexit>lxik Vje↵ective
Aije↵ective2
xi k Vixi
xl⌦
xi k kxl

xi Vje↵ectiveAile↵ective
xi k V i55

IsoGCN：合同変換不変・同変な GCN• 有限要素解析に適用するため
IsoAM の具体例 D̃ijk を以下のように定義する
XD̃ijk = Dijk
Dilk
ij
l

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</latexit>

1 xjk
Dijk = M
kxj

X xl
Mi =
kxl
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Vje↵ective
Aije↵ective2
xi k Vixi
xl⌦
xi k kxl

xi Vje↵ectiveAile↵ective
xi k V i

A ：隣接行列
Vie↵ective：頂点 i の体積<latexit 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8ZlkOoHUpP7vMQCVQaPUjzrffvqmX57x/w9xTSzb6HKAvdfmrfHng0gSVgb90D0udjEBIghn+n/HjHeIdVvsekq2DplrfUrpPk0MjnTFMWxglXjXuZilIN2dx/fxQWM4mCt9QpyXqaJfng/n/W+fTnvIPMMNLfexal70+9VF1dopuhgjBZ63w7ibKwe26FTBPrvFHCX+4EOlnPAs1YGEYg4E1leF/QKHtbE1iSXenlyXQCGE7GNn30z+IUUwzy/W1v2HuecFPp0zst/RY2XkRkA/h6s6quvNdTnpG2IV90ujVqxmfn/iRYXKKgL/YZfAmoQDDHOkvx+WEJw9/GblfX2n/IsXly9Wlv/97WXp29WfrdZ/5nPj5d+tvTzpV8urS/9aul3S/tL/aXLpXDpu6X/XvrT0v+8uX3zhzf/8eY/bdMf/qDu809L3s+b//o/HH9mow==</latexit>

D̃ij:

j
i
Vie↵ective<latexit sha1_base64="RbeWVNamq4lVmgipNykdZ5cK3D4=">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</latexit>

56

IsoGCN：合同変換不変・同変な GCN• 有限要素解析に適用するため
IsoAM の具体例 D̃ijk を以下のように定義する
XD̃ijk = Dijk
Dilk
ij
l

<latexit sha1_base64="oHb+BDeHNuzhTyK0+cEYBl1owNM=">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</latexit>

1 xjk
Dijk = M
kxj

X xl
Mi =
kxl
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Vje↵ective
Aije↵ective2
xi k Vixi
xl⌦
xi k kxl

xi Vje↵ectiveAile↵ective
xi k V i• D̃ がナブラ演算子の離散化（Tamai+ 2014）に対応している⌧e↵ective
X j
X
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x
x
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ik j
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Aij =
D̃ijk je↵ective@xk i
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IsoGCN：合同変換不変・同変な GCN<latexit 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</latexit>• 有限要素解析に適用するため
IsoAM の具体例 D̃ijk を以下のように定義する
XD̃ijk = Dijk
Dilk
ij

j

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Dijk = M

1

X xl
Mi =
kxl
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D̃ がナブラ演算子の離散化（Tamai+ 2014）に対応している⌧e↵ective
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IsoGCN：合同変換不変・同変な GCN<latexit 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</latexit><latexit 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</latexit>• 有限要素解析に適用するため
IsoAM の具体例 D̃ijk を以下のように定義する
XD̃ijk = Dijk
Dilk
ij

j

i

kxj

xi k
<latexit 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Dijk = M

1

X xl
Mi =
kxl
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xi Vje↵ectiveAile↵ective
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P1ut/eid9u/wl9BiXtf5qxlf2EDtvOKaG1ogd7kybgMUv1AanKfl1igyuBRime9H14902/vmD/VmGb2LVRZ4P5L8/bYswEkCWvjHog+F1uYADHkM/3fI8Y7pPotNl0FW6estX6FNJ9GJmeaojhWsGrcy1yMctDu7uO7uIBRHKy1XkHOyzTRD+9nde+Hl7MOMs9Ac+tdnLo3/V51cYVmig7GaKH3wyDOxuqxHTpFoP+MAXf5LNDB0gc8a2UQgYgzkeVNQa/gYU1sT3KplyfXBWA4ETv42TeDX0kxzPO7tWXvcc5poU/nvPwn1HgZmQHg78GqvvpaQ31O2oZ41e3SqBWbmf+faHGJgrrUb/gloAbBEOMsye+HJQR3H79ZcX8cJJ6+uH61tv7d2svzjZXfbzV/QfTzpb9f+oelb5fWl3699Pulg6WzpaulcCle+relf1/6j43/3Pivjf/e+B/b9Kc/afr83ZL3s/HH/wWLJIOw</latexit>• D̃ がナブラ演算子の離散化（Tamai+ 2014）に対応している⌧e↵ective
X j
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D̃ijk je↵ective@xk i
kxj xi k kxj xi k Vijj<latexit 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</latexit>

59

xi
xi k

IsoGCN：合同変換不変・同変な GCN• 有限要素解析に適用するため
IsoAM の具体例 D̃ijk を以下のように定義する
XD̃ijk = Dijk
Dilk
ij
l

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</latexit>

1 xjk
Dijk = M
kxj

X xl
Mi =
kxl
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Vje↵ective
Aije↵ective2
xi k Vixi
xl⌦
xi k kxl

xi Vje↵ectiveAile↵ective
xi k V i• D̃ がナブラ演算子の離散化（Tamai+ 2014）に対応している⌧e↵ective
X j
X
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x
x
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ik j
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Aij =
D̃ijk je↵ective@xk i
kxj xi k kxj xi k Vijj<latexit 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</latexit>60

実験：微分演算子の学習• X、Y 方向に 10 〜 100 要素ある
グリッドメッシュをランダムに生成• ランダムなフーリエ級数でスカラー場を生成
スカラー場
（入力）

勾配場
（正解）

• 微分を解析的に計算• training、validation、test dataset
はそれぞれ 100 サンプル61

実験：微分演算子の学習

スカラー場
（入力）

勾配場
（正解）

Cluster-GCN（推論）

SE(3)-Trans.
（推論）

IsoGCN（推論）• 既存の GCN 系の手法では空間の情報を扱えないが IsoGCN では可能
62

実験：微分演算子の学習

スカラー場
（入力）

勾配場
（正解）

Cluster-GCN（推論）

100 サンプルの推論時間：

SE(3)-Trans.
（推論）

IsoGCN
（推論）

4.0 s

0.4 s

• 既存の GCN 系の手法では空間の情報を扱えないが IsoGCN では可能
63

実験：微分演算子の学習
スカラー
↓
勾配

スカラー
↓
ヘッシアン

勾配
↓
ラプラシアン勾配↓
ヘッシアン• 既存の GCN 系の手法では空間の情報を扱えないが IsoGCN では可能
64

実験：微分演算子の学習
スカラー
↓
勾配

スカラー
↓
ヘッシアン

• IsoGCN は TFN、SE(3)-Transformer （TFN の改良）より高速

65

実験：非定常・異方的・非線型熱拡散問題<latexit 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</latexit>

初期条件<latexit 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</latexit>⌦ ⇢ R3⌦, ⇢ R3 ,@T (x, @Tt) (x, t)= r · C(T= r(x,· C(Tt))rT(x, (x,t))rTt) (x, t)@t@t
T (x, t T=(x,0) t==T0)(x)Tinit (x)init=rT (x,rTt)|x=x(x, bt)|· n(x= 0 b) = 0
x=xbb )· n(x有限要素解析

66

in ⌦, iin ⌦, ion @⌦ o

実験：非定常・異方的・非線型熱拡散問題<latexit 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初期条件<latexit 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</latexit>⌦ ⇢ R3⌦, ⇢ R3 ,@T (x, @Tt) (x, t)= r · C(T= r(x,· C(Tt))rT(x, (x,t))rTt) (x, t)@t@t
T (x, t T=(x,0) t==T0)(x)Tinit (x)init=rT (x,rTt)|x=x(x, bt)|· n(x= 0 b) = 0
x=xbb )· n(xin ⌦, iin ⌦, ion @⌦ o• ABC dataset (Koch+ 2018) から CAD データを抽出
有限要素解析

• CAD 形状ひとつにつきメッシュサイズ 3 通り
初期条件 3 通りの最大 9 サンプルを生成- Training dataset: 50 形状、439 サンプル
- Validation dataset: 16 形状、143 サンプル
- Test dataset: 16 shapes、140 サンプル67

実験：非定常・異方的・非線型熱拡散問題

T0.0

MLP
[3, 128, 512]
[tanh, tanh]<latexit sha1_base64="NUFrh3hqzQtVdWvHtZoBXM+NYUs=">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</latexit>Ve↵ective

IsoGCND̃ ⇤[512, 512]
[Identity]

IsoGCND̃ ⇤[512, 512]
[Identity]

MLP[512, 512, 128, 1][tanh, tanh, Identity]<latexit sha1_base64="NUFrh3hqzQtVdWvHtZoBXM+NYUs=">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</latexit><latexit sha1_base64="0gsloBDE5IYdEAYRX+YQSr4H/dc=">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</latexit>

Vmean

⌦

⌦

<latexit 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sha1_base64="h3S2Pl8cWfTrdptIA3FHqByE2Bs=">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</latexit>

MLP[512, 512]
[tanh]

IsoGCND̃
[512, 512]
[Identity]

MLP[512, 512]
[tanh]

IsoGCND̃
[512, 512]
[Identity]

Linear
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p+XWKDK4FGK5/0fXj/Xb++Yv6eYZfYtVFng/kvz9tjzISQJa+MeiL4Q25gAMeQz/d8jxjuk+i02XQVbp6y1foU0n0UmZ5qiOFawbtzLXIxz0O7u47u4gHEcbLReQc7LNNEP7xfz/g+vFh1knoHmNrs4dW/6ve7iCs0UHYzRQv+HYZxN1GM7dIpA/50C7vJZoINlAHjWyiACEWciy+uCXsHDhtiZ5lIvT64LwHAqdvGzbwa/lmKU53cbq97jnNNCn855+feo8TIyA8Dfw3V99b2G+py0DfGq26VRKzYz/z/R4hIFdanf8EtADYMRxlmS349KCO4+P1vbbP8pz/LF9euNzX/ceHX+du132/Wf+fx85Zcrv1r5zcrmyj+t/G7lYOVs5WolXKlW/nPlv1b+++3g7ePbf3v777bpT39S9/nbFe/n7X/8H9V6Y/o=</latexit>

Propagation 1

Propagation 2<latexit 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68

Decoder<latexit sha1_base64="miXsY/pjlE/gW+s1wuAK0wD685g=">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</latexit>

T0.2
T0.4
T0.6
T0.8
T1.0

実験：非定常・異方的・非線型熱拡散問題

T0.0

MLP
[3, 128, 512]
[tanh, tanh]<latexit sha1_base64="NUFrh3hqzQtVdWvHtZoBXM+NYUs=">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</latexit>Ve↵ective

IsoGCND̃ ⇤[512, 512]
[Identity]

IsoGCND̃ ⇤[512, 512]
[Identity]

MLP[512, 512, 128, 1][tanh, tanh, Identity]<latexit sha1_base64="NUFrh3hqzQtVdWvHtZoBXM+NYUs=">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</latexit><latexit sha1_base64="0gsloBDE5IYdEAYRX+YQSr4H/dc=">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</latexit>

Vmean

⌦

⌦

<latexit 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sha1_base64="h3S2Pl8cWfTrdptIA3FHqByE2Bs=">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</latexit>

MLP[512, 512]
[tanh]

IsoGCND̃
[512, 512]
[Identity]

MLP[512, 512]
[tanh]

IsoGCND̃
[512, 512]
[Identity]

Linear
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p+XWKDK4FGK5/0fXj/Xb++Yv6eYZfYtVFng/kvz9tjzISQJa+MeiL4Q25gAMeQz/d8jxjuk+i02XQVbp6y1foU0n0UmZ5qiOFawbtzLXIxz0O7u47u4gHEcbLReQc7LNNEP7xfz/g+vFh1knoHmNrs4dW/6ve7iCs0UHYzRQv+HYZxN1GM7dIpA/50C7vJZoINlAHjWyiACEWciy+uCXsHDhtiZ5lIvT64LwHAqdvGzbwa/lmKU53cbq97jnNNCn855+feo8TIyA8Dfw3V99b2G+py0DfGq26VRKzYz/z/R4hIFdanf8EtADYMRxlmS349KCO4+P1vbbP8pz/LF9euNzX/ceHX+du132/Wf+fx85Zcrv1r5zcrmyj+t/G7lYOVs5WolXKlW/nPlv1b+++3g7ePbf3v777bpT39S9/nbFe/n7X/8H9V6Y/o=</latexit>

T (x, t +Propagation 1

Propagation 2<latexit 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MmkDFL9QGp6n5dYoMrgUYpnve9fPdNv75i/p5hl9i1UWeD+S/P22LMhJAlr4x6IPhfbmAAx5DP93yPGO6T6LTZdBVunrLV+hTSfRSZnmqI4VrBu3MtcjHPQ7u7ju7iAcRxstF5Bzss00Q/vF/Pe9y8XHWSegeY2uzh1b/q96uIKzRQdjNFC7/thnE3UYzt0ikD/nQLucj/QwTIAPGtlEIGIM5HldUGv4GFD7ExzqZcn1wVgOBW7+Nk3g19KMcrzu41V73HOWaFP57z8B9R4GZkB4O/hur76VkN9TtqGeNXt0qgVm5n/n2hxiYK61G/4JaCGwQjjLMnvRyUEd5++W9ts/ynP8sX1q43NX2+8PH+z9rvt+s98frrytyt/t/Krlc2V36z8buVwpb9ytRKu/GHlP1b+a+W/33x48y9v/vXNv9mmP/5R3ecXK97Pm3//XzzPZts=</latexit><latexit 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' T (x, t) + r · C(T (x, t))rT (x, t) t
69

Decoder<latexit sha1_base64="miXsY/pjlE/gW+s1wuAK0wD685g=">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</latexit>

T0.2
T0.4
T0.6
T0.8
T1.0

実験：非定常・異方的・非線型熱拡散問題<latexit 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初期条件<latexit 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</latexit>⌦ ⇢ R3⌦, ⇢ R3 ,@T (x, @Tt) (x, t)= r · C(T= r(x,· C(Tt))rT(x, (x,t))rTt) (x, t)@t@t
T (x, t T=(x,0) t==T0)(x)Tinit (x)init=rT (x,rTt)|x=x(x, bt)|· n(x= 0 b) = 0
x=xbb )· n(xin ⌦, iin ⌦, ion @⌦ o• ABC dataset (Koch+ 2018) から CAD データを抽出
有限要素解析

• CAD 形状ひとつにつきメッシュサイズ 3 通り
初期条件 3 通りの最大 9 サンプルを生成- Training dataset: 50 形状、439 サンプル
- Validation dataset: 16 形状、143 サンプル
- Test dataset: 16 shapes、140 サンプル70

実験：非定常・異方的・非線型熱拡散問題
初期条件

有限要素解析

SGCN

SE(3)-Trans.

71

IsoGCN

実験：非定常・異方的・非線型熱拡散問題
初期条件

SGCN
（誤差）

SE(3)-Trans.
（誤差）

IsoGCN（誤差）

有限要素解析

SGCN

SE(3)-Trans.

IsoGCN72

実験：非定常・異方的・非線型熱拡散問題
SE(3)-Trans.
（誤差）

SGCN
（誤差）

IsoGCN（誤差）➡ IsoGCN は既存手法と比較して高精度➡ ただし、TFN と SE(3)-Transformer はメモリに載せるためにパラメータ数を少なくしている73

実験：非定常・異方的・非線型熱拡散問題➡ IsoGCN は同等の精度となるセッティングの FrontISTR と比較して高速➡ TFN・SE(3)-Transformer は大規模メッシュでは Out-of-Memory (500 GB)74

実験：非定常・異方的・非線型熱拡散問題• 学習データサンプルより大きなメッシュに対しても外挿可能
IsoGCN

学習データ
サンプル

有限要素解析
75

目次• Graph neural network• 同変性• IsoGCN: 同変性＋Graph Neural Network
• まとめ

76

目次

• Graph neural network• 同変性• IsoGCN: 同変性＋Graph Neural Network
• まとめ

77

関連研究：物理シミュレーションの機械学習
Graph Neural Network の適用

同変性の導入Isometric transformation!

?
?
y IsoGCN?
?
y IsoGCN
Isometric transformation!

<latexit 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JNXRGtDC/QmTyZlkOoHUtP7vMQCVQaPUjzr/fDqmX57x/w9xSyzb6HKAvdfmrfHng0hSVgb90D0udjGBIghn+n/HjHeIdVvsekq2DplrfUrpPksMjnTFMWxgnXjXuZinIN2dx/fxQWM42Cj9QpyXqaJfni/mPd+eLnoIPMMNLfZxal70+9VF1dopuhgjBZ6PwzjbKIe26FTBPrvFHCX+4EOlkvAs1YGEYg4E1leF/QKHjbEzjSXenlyXQCGU7GLn30z+KUUozy/21j1HuecFfp0zsu/RY2XkRkA/h6u66tvNdTnpG2IV90ujVqxmfn/iRYDFNRAv+GXgBoGI4yzJL8flRDcffpubbP9pzzLF9evNjb/fuPl+Zu1X2/Xf+bz85W/Wvmble9XNlf+YeXXK4cr/ZWrlXDlX1d+t/JfK//95jdv/u3Nv7/5D9v0pz+p+/zlivfz5j//F6Y5aFc=</latexit>

Ground Truth (FEA)<latexit 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QV6nSfjMkj1A6nJfV5igSqDRyme9b579Uy/vWP+nmKa2bdQZYH7L83bY88GkCSsjXsg+lxsYgLEkM/0f48Y75Dqt9h0FWydstb6FdJ8GpmcaYriWMGqcS9zMcpBu7uP7+ICRnGw1noFOS/TRD+8n896372cd5B5Bppb7+LUven3qosrNFN0MEYLve8GcTZWj+3QKQL9dwq4y6eBDpY+4FkrgwhEnIksrwt6BQ9rYmuSS708uS4Aw4nYxs++GfxKimGe360te49zTgp9Ouflb1DjZWQGgL8Hq/rqaw31OWkb4lW3S6NWbGb+f6LFBQrqQr/hl4AaBEOMsyS/H5YQ3H38ZmW9/ac8ixdXr9bW/3nt5dmblT9s1n/m8+OlXyz9cunbpfWlf1n6w9L+0unS5VK4NF/6r6X/WfrfN7dv/u3Nv7/5D9v0hz+o+/zjkvfz5j//DKReZfk=</latexit>

IsoGCN (Ours)

Training samples<latexit 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CZPJmWQ6gdS0/u8xAJVBo9SPOt9/+qZfnvH/D3FLLNvocoC91+at8eeDSFJWBv3QPS52MYEiCGf6f8eMd4h1W+x6SrYOmWt9Suk+SwyOdMUxbGCdeNe5mKcg3Z3H9/FBYzjYKP1CnJepol+eL+Y975/uegg8ww0t9nFqXvT71UXV2im6GCMFnrfD+Nsoh7boVME+u8UcJfPAx0sfcCzVgYRiDgTWV4X9AoeNsTONJd6eXJdAIZTsYuffTP4lRSjPL/bWPUe55wV+nTOy79FjZeRGQD+Hq7rq2811OekbYhX3S6NWrGZ+f+JFgMU1EC/4ZeAGgYjjLMkvx+VENx9+m5ts/2nPMsX1682Nv9+4+XFm7Xfbdd/5vPTlb9a+euVX69srvxm5XcrhyvnK1cr4co/r/zHyn+t/Pebj2/+5c2/vvk32/THP6r7/HLF+3nz7/8LMapoSw==</latexit>

Umetani+ 2018
• リアルタイムで車体の空力性能予測• メッシュ構造固定

Sanchez-Gonzalez+ 2020

Chang+ 2020

Horie+ 2020

• 任意点群での学習• 任意梁構造での学習• 任意メッシュでの学習• 高速化は未達成• 梁部材選定の最適化• 高速化を達成

78• 物理現象の対称性を考慮

概要：対称性があり計算効率のよい GNN を開発• IsoGCN: 合同変換不変・同変で計算効率のよい GNN (Horie+ 2020)• IsoGCN は同等の精度の有限要素解析より高速• IsoGCN は微分演算子の近似とみなせる

スカラー場

勾配

勾配

（入力）

（正解）

（推論）

79

概要：対称性があり計算効率のよい GNN を開発• IsoGCN: 合同変換不変・同変で計算効率のよい GNN (Horie+ 2020)• IsoGCN は同等の精度の有限要素解析より高速• IsoGCN は微分演算子の近似とみなせる

今後の展望：• IsoGCN の近似性能を評価• IsoGCN を多様な物理現象へ適用

80References• N. Umetani, and B. Bickel. Learning three-dimensional flow for interactive aerodynamic design. ACM Transactions on Graphics(TOG) 37.4: 1-10, 2018
• A. Sanchez-Gonzalez, J. Godwin, T. Pfaff, R. Ying, J. Leskovec, and P. W. Battaglia. Learning to simulate complex physics withgraph networks. In ICML, 2020.
• K. Chang and C. Cheng. Learning to simulate and design for structural engineering. In ICML, 2020
• M. Horie, N. Morita, T. Hishinuma, Y. Ihara, and N. Mitsume. Isometric Transformation Invariant and Equivariant Graph
Convolutional Networks. arXiv preprint arXiv:2005.06316, 2020
• E. Ahmed, A. Saint, A. E. R. Shabayek, K. Cherenkova, R. Das, G. Gusev, D. Aouada and B. Ottersten. Deep Learning
Advances on Different 3D Data Representations: A Survey. arXiv preprint arXiv:1808.01462, 2018
• T. N. Kipf, and M. Welling. Semi-supervised classification with graph convolutional networks. In ICLR, 2017.
• N. Thomas, T. Smidt, S. Kearnes, L. Yang, L. Li, K. Kohlhoff, and P. Riley. Tensor field networks: Rotation-and translationequivariant neural networks for 3d point clouds. arXiv preprint arXiv:1802.08219, 2018.
• N. Dym, and H. Maron. On the Universality of Rotation Equivariant Point Cloud Networks. arXiv preprint
arXiv:2010.02449, 2020
• T. Tamai and S. Koshizuka. Least squares moving particle semi-implicit method. Computational Particle Mechanics, 1(3):277–
305, 2014• S. Koch, A. Matveev, Z. Jiang, F. Williams, A. Artemov, E. Burnaev, M. Alexa, D. Zorin, and D. Panozzo. Abc: A big cad model
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