ICML 読み会: Graph Neural PDE Solvers with Conservation and Similarity-Equivariance

(SBQI/FVSBM1%&4PMWFST XJUI$POTFSWBUJPOBOE4JNJMBSJUZ&RVJWBSJBODF ೥݄೔ *$.-ಡΈձ ࿦จஶऀ.BTBOPCV)PSJF /BPUP.JUTVNF ൃදऀງߐਖ਼৴ גࣜձࣾ3*$04ɹஜ೾େֶ ਖ਼ղσʔλ
༧ଌ݁Ռ

ݚڀͷ֓ཁϞνϕʔγϣϯ ࠞ߹ ೤ྲྀମ ࣗ༝ද໘ྲྀΕ • طଘͷ਺஋ղੳͷ൚༻ੑΛࣦΘͣʹߴ଎ɾߴਫ਼౓Խ͠༌ૹɾ೤ྲྀମݱ৅ͳͲʹ׆༻͍ͨ͠

ݚڀͷ֓ཁϞνϕʔγϣϯ ࠞ߹ ೤ྲྀମ ࣗ༝ද໘ྲྀΕ • طଘͷ਺஋ղੳͷ൚༻ੑΛࣦΘͣʹߴ଎ɾߴਫ਼౓Խ͠༌ૹɾ೤ྲྀମݱ৅ͳͲʹ׆༻͍ͨ͠ • ػցֶशΛ༻͍ͯɺ൚༻ੑ͕ߴ͘ߴ଎ɾߴਫ਼౓ͳภඍ෼ํఔࣜιϧόΛ࣮ݱ͍ͨ͠

ݚڀͷ֓ཁϞνϕʔγϣϯ ࠞ߹ ೤ྲྀମ ࣗ༝ද໘ྲྀΕ • ػցֶशΛ༻͍ͯɺ൚༻ੑ͕ߴ͘ߴ଎ɾߴਫ਼౓ͳภඍ෼ํఔࣜιϧόΛ࣮ݱ͍ͨ͠ • طଘͷ਺஋ղੳͷ൚༻ੑΛࣦΘͣʹߴ଎ɾߴਫ਼౓Խ͠༌ૹɾ೤ྲྀମݱ৅ͳͲʹ׆༻͍ͨ͠ •
෺ཧݱ৅ͷಛੑΛ൓өͤ͞Δ͜ͱͰ൚༻ੑɾਫ਼౓Λ޲্͍ͤͨ͞

ݚڀͷ֓ཁϞνϕʔγϣϯ ࠞ߹ ೤ྲྀମ ࣗ༝ද໘ྲྀΕ • طଘͷ਺஋ղੳͷ൚༻ੑΛࣦΘͣʹߴ଎ɾߴਫ਼౓Խ͠༌ૹɾ೤ྲྀମݱ৅ͳͲʹ׆༻͍ͨ͠ • ෺ཧݱ৅ͷಛੑΛ൓өͤ͞Δ͜ͱͰ൚༻ੑɾਫ਼౓Λ޲্͍ͤͨ͞ •
ہॴอଘੑɺ૬ࣅม׵ରশੑΛݫີʹຬͨ͢ػցֶशϞσϧ͕࣮ݱ[Horie & Mitsume. ICML. 2024] • ػցֶशΛ༻͍ͯɺ൚༻ੑ͕ߴ͘ߴ଎ɾߴਫ਼౓ͳภඍ෼ํఔࣜιϧόΛ࣮ݱ͍ͨ͠

• ہॴอଘੑɺ૬ࣅม׵ରশੑΛݫີʹຬͨ͢ػցֶशϞσϧ͕࣮ݱ[Horie & Mitsume. ICML. 2024] ݚڀͷ֓ཁ·ͱΊ ࿨෼ /VNFSJDBMGMVY
ʹରԠ͢Δ// ࣌ࠁU ࣌ࠁU ϶U • ہॴอଘੑ ༗ݶମੵ๏Λಋೖ

ݚڀͷ֓ཁ·ͱΊ ࿨෼ /VNFSJDBMGMVY ʹରԠ͢Δ// ࣌ࠁU ࣌ࠁU ϶U • ہॴอଘੑ
༗ݶମੵ๏Λಋೖ • ૬ࣅม׵ରশੑ ཭ࢄԽεέʔϧΛಋೖ <latexit sha1_base64="f69CgSP6DjgH/TDaBunZ3wxr8as=">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</latexit> ' • ہॴอଘੑɺ૬ࣅม׵ରশੑΛݫີʹຬͨ͢ػցֶशϞσϧ͕࣮ݱ[Horie & Mitsume. ICML. 2024]

ݚڀͷ֓ཁ·ͱΊ • ہॴอଘੑ ༗ݶମੵ๏Λಋೖ ࿨෼ /VNFSJDBMGMVY ʹରԠ͢Δ// ࣌ࠁU ࣌ࠁU
϶U • ૬ࣅม׵ରশੑ ཭ࢄԽεέʔϧΛಋೖ • ਺஋࣮ݧ ࠞ૬ྲྀݱ৅΁ͷద༻ <latexit 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</latexit> ' • ہॴอଘੑɺ૬ࣅม׵ରশੑΛݫີʹຬͨ͢ػցֶशϞσϧ͕࣮ݱ[Horie & Mitsume. ICML. 2024]

໨࣍ ࿨෼ /VNFSJDBMGMVY ʹରԠ͢Δ// ࣌ࠁU ࣌ࠁU ϶U • ہॴอଘੑ
༗ݶମੵ๏Λಋೖ • ૬ࣅม׵ରশੑ ཭ࢄԽεέʔϧΛಋೖ • ਺஋࣮ݧ ࠞ૬ྲྀݱ৅΁ͷద༻ <latexit 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</latexit> '

• '7.͸อଘܗࣜ Ҡྲྀ֦ࢄ΍/4ํఔࣜͷҰൠԽ Λ཭ࢄԽ͢Δ͜ͱʹΑΓಘΒΕΔ ہॴอଘੑ༗ݶମੵ๏ '7. ΛػցֶशϞσϧԽ <latexit sha1_base64="0GUpCmTdZxWL5yxai3Hk9Ia+V2Y=">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</latexit>
@ @t Z VP dV u = I @VP dSn · F (u) อଘܗࣜ

@ @t Z VP dV u = I @VP dSn · F (u) อଘܗࣜ /BWJFSr4UPLFTํఔࣜͷ৔߹

@ @t Z VP dV u = I @VP dSn · F (u) ηϧɾ໘த৺ͷ஋Ͱ୅දͤͯ͞཭ࢄԽ อଘܗࣜ ηϧiͱηϧjΛ ִͯΔ໘(i, j)ͷ໘ੵ

@ @t Z VP dV u = I @VP dSn · F (u) อଘܗࣜ ηϧiͱηϧjΛ ִͯΔ໘(i, j)ͷ໘ੵ ໘(i, j) = (j, i) ͷΈʹண໨ ηϧɾ໘த৺ͷ஋Ͱ୅දͤͯ͞཭ࢄԽ

• '7.͸อଘܗࣜ Ҡྲྀ֦ࢄ΍/4ํఔࣜͷҰൠԽ Λ཭ࢄԽ͢Δ͜ͱʹΑΓಘΒΕΔ ہॴอଘੑ༗ݶମੵ๏ '7. ΛػցֶशϞσϧԽ ηϧiͱηϧjΛ ִͯΔ໘(i,
j)ͷ໘ੵ ໘(i, j) = (j, i) ͷΈʹண໨

• '7.͸อଘܗࣜ Ҡྲྀ֦ࢄ΍/4ํఔࣜͷҰൠԽ Λ཭ࢄԽ͢Δ͜ͱʹΑΓಘΒΕΔ ہॴอଘੑ༗ݶମੵ๏ '7. ΛػցֶशϞσϧԽ ˠ໘(i, j)Λ௨ͨ͠෺ཧྔͷ΍ΓऔΓ͸อଘ͞ΕΔ
ہॴอଘੑ ηϧiͱηϧjΛ ִͯΔ໘(i, j)ͷ໘ੵ ໘(i, j) = (j, i) ͷΈʹண໨

• '7.͸อଘܗࣜ Ҡྲྀ֦ࢄ΍/4ํఔࣜͷҰൠԽ Λ཭ࢄԽ͢Δ͜ͱʹΑΓಘΒΕΔ ہॴอଘੑ༗ݶମੵ๏ '7. ΛػցֶशϞσϧԽ /VNFSJDBMGMVYFΛػցֶशϞσϧԽ ηϧiͱηϧjΛ
ִͯΔ໘(i, j)ͷ໘ੵ

• '7.͸อଘܗࣜ Ҡྲྀ֦ࢄ΍/4ํఔࣜͷҰൠԽ Λ཭ࢄԽ͢Δ͜ͱʹΑΓಘΒΕΔ ہॴอଘੑ༗ݶମੵ๏ '7. ΛػցֶशϞσϧԽ /VNFSJDBMGMVYFΛػցֶशϞσϧԽ ηϧiͱηϧjΛ
ִͯΔ໘(i, j)ͷ໘ੵ ֶशՄೳύϥϝʔλΛ࣋ͭؔ਺

• '7.͸อଘܗࣜ Ҡྲྀ֦ࢄ΍/4ํఔࣜͷҰൠԽ Λ཭ࢄԽ͢Δ͜ͱʹΑΓಘΒΕΔ ہॴอଘੑ༗ݶମੵ๏ '7. ΛػցֶशϞσϧԽ /VNFSJDBMGMVYFΛػցֶशϞσϧԽ ͕ඞཁ
• ہॴอଘੑɹɹɹɹɹɹɹɹɹΛຬͨ͢ʹ͸ ηϧiͱηϧjΛ ִͯΔ໘(i, j)ͷ໘ੵ ֶशՄೳύϥϝʔλΛ࣋ͭؔ਺

• '7.͸อଘܗࣜ Ҡྲྀ֦ࢄ΍/4ํఔࣜͷҰൠԽ Λ཭ࢄԽ͢Δ͜ͱʹΑΓಘΒΕΔ ہॴอଘੑ༗ݶମੵ๏ '7. ΛػցֶशϞσϧԽ /VNFSJDBMGMVYFΛػցֶशϞσϧԽ ͕ඞཁ
• ہॴอଘੑɹɹɹɹɹɹɹɹɹΛຬͨ͢ʹ͸ • ஔ׵ෆมͳػցֶशϞσϧͷඞཁे෼৚݅Ͱ͋Δ%FFQ4FU[Zaheer+ 2017]Λ࢖༻ ηϧiͱηϧjΛ ִͯΔ໘(i, j)ͷ໘ੵ ֶशՄೳύϥϝʔλΛ࣋ͭؔ਺

ہॴอଘੑάϥϑχϡʔϥϧωοτϫʔΫͱͷྨࣅੑ • ఏҊख๏͸೚ҙͷϝογϡʹద༻Ͱ͖Δػցֶशख๏Ͱ͋ΔάϥϑχϡʔϥϧωοτϫʔΫ (// ͷಛघͳྫͱΈͳͤΔ ࣌ࠁU ࣌ࠁU ϶U (//

ہॴอଘੑάϥϑχϡʔϥϧωοτϫʔΫͱͷྨࣅੑ • ఏҊख๏͸೚ҙͷϝογϡʹద༻Ͱ͖Δػցֶशख๏Ͱ͋ΔάϥϑχϡʔϥϧωοτϫʔΫ (// ͷಛघͳྫͱΈͳͤΔ ࣌ࠁU ࣌ࠁU ϶U &ODPEFS
(// ߴ࣍ݩԽ͞Εͨ෺ཧྔ

.FTTBHF ؔ਺ 6QEBUF ؔ਺ (// ߴ࣍ݩԽ͞Εͨ෺ཧྔ

.FTTBHF ؔ਺ 6QEBUF ؔ਺ (// %FDPEFS ߴ࣍ݩԽ͞Εͨ෺ཧྔ

.FTTBHF ؔ਺ 6QEBUF ؔ਺ (// %FDPEFS ఏҊ ख๏

.FTTBHF ؔ਺ 6QEBUF ؔ਺ (// %FDPEFS ˠఏҊख๏͸೚ҙͷϝογϡʹద༻Մೳ ఏҊ ख๏

.FTTBHF ؔ਺ 6QEBUF ؔ਺ (// %FDPEFS ˠఏҊख๏͸೚ҙͷϝογϡʹద༻Մೳ • ہॴอଘੑͷͨΊͷΤϯίʔμɾσίʔμɾVQEBUFؔ਺ʹ͍ͭͯͷ੍໿Λ਺ֶతʹಋग़ ఏҊ ख๏ ઢܗFODPEFS ઢܗ ߃౳ VQEBUF ؔ਺ &ODPEFSͷ ࠨٯࣸ૾

૬ࣅม׵ରশੑϞνϕʔγϣϯ • ෺ཧݱ৅ʹ͸૬ࣅม׵ ฏߦҠಈɾճసɾڸөɾ֦େॖখ ͷରশੑ͕͋Δ DG3FZOPMET૬ࣅଇɺ୯Ґܥͷม׵ <latexit sha1_base64="f69CgSP6DjgH/TDaBunZ3wxr8as=">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</latexit> '
෺ཧతʹಉ͡

• ௨ৗͷػցֶशϞσϧͰ͸૬ࣅม׵ͷ΋ͱͰ෺ཧྔ͕֎ૠʹͳΓਫ਼౓͕௿Լ͢Δ ෺ཧతʹಉ͡

• ௨ৗͷػցֶशϞσϧͰ͸૬ࣅม׵ͷ΋ͱͰ෺ཧྔ͕֎ૠʹͳΓਫ਼౓͕௿Լ͢Δ ෺ཧతʹಉ͡ ˠ૬ࣅม׵ͷରশੑΛػցֶशϞσϧʹಋೖ͢Δඞཁ͕͋Δ

૬ࣅม׵ରশੑख๏ۭ࣌ؒ཭ࢄԽεέʔϧΛಋೖ • ఏҊख๏Ͱඇઢܗ//Λ༻͍Δͷ͸FML ͷΈͳͷͰͦΕ͚ͩΛߟ͑Ε͹े෼ ఏҊख๏

૬ࣅม׵ରশੑख๏ۭ࣌ؒ཭ࢄԽεέʔϧΛಋೖ • ఏҊख๏Ͱඇઢܗ//Λ༻͍Δͷ͸FML ͷΈͳͷͰͦΕ͚ͩΛߟ͑Ε͹े෼ • //ʹೖྗ͢Δ෺ཧྔ͸ϊϧϜΛͱΓۭ࣌ؒ཭ࢄԽεέʔϧͰہॴతʹن֨Խ ྲྀ଎ $PVSBOU਺ ఏҊख๏
DGHuang+ ICLR 2022

૬ࣅม׵ରশੑख๏ۭ࣌ؒ཭ࢄԽεέʔϧΛಋೖ • ఏҊख๏Ͱඇઢܗ//Λ༻͍Δͷ͸FML ͷΈͳͷͰͦΕ͚ͩΛߟ͑Ε͹े෼ • //ʹೖྗ͢Δ෺ཧྔ͸ϊϧϜΛͱΓۭ࣌ؒ཭ࢄԽεέʔϧͰہॴతʹن֨Խ ྲྀ଎ $PVSBOU਺ ఏҊख๏
૬ࣅม׵ʹରͯ͠ෆม ૬ࣅม׵ʹରͯ͠ڞม ςϯιϧͷ࠲ඪม׵ଇΛຬͨ͢ DGHuang+ ICLR 2022

૬ࣅม׵ରশੑख๏ۭ࣌ؒ཭ࢄԽεέʔϧΛಋೖ • ఏҊख๏Ͱඇઢܗ//Λ༻͍Δͷ͸FML ͷΈͳͷͰͦΕ͚ͩΛߟ͑Ε͹े෼ • //ʹೖྗ͢Δ෺ཧྔ͸ϊϧϜΛͱΓۭ࣌ؒ཭ࢄԽεέʔϧͰہॴతʹن֨Խ ྲྀ଎ $PVSBOU਺ ➡ྲྀ଎ͷ৔߹ɺ//΁ͷ௚઀ͷೖྗ͸$PVSBOU਺ͱͳΓ
FML ͸OVNFSJDBMGMVYΛܭࢉ͢Δࡍͷ҆ఆԽख๏Λֶश͍ͯ͠ΔͱղऍͰ͖Δ ఏҊख๏ ૬ࣅม׵ʹରͯ͠ෆม ૬ࣅม׵ʹରͯ͠ڞม ςϯιϧͷ࠲ඪม׵ଇΛຬͨ͢ DGHuang+ ICLR 2022

਺஋࣮ݧ໰୊ઃఆ <latexit sha1_base64="d+8Z5nE+DF/eVlRBORKBZKi6O9w=">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</latexit> prgh = p ⇢g · x
<latexit sha1_base64="8cnaa+CXoMLLSI46jr5hnN/MdCo=">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</latexit> @ @t (⇢u) = r · ⇥ ⇢u ⌦ u + p⇢gh µ r ⌦ u + (r ⌦ u)> ⇤ + (g · x)r⇢ <latexit sha1_base64="9SdWHa8nNU82DsIkWQcZoE/FO3Q=">AAA13niclVtZdxxLUtYM2yCWuQOPvORB9pk7IAnL9xp4AM5otWS19tV2+5rq6ujqsmpzZXZJck3zBm8cXvkH/Bpe4fBviMysrIisLvmCzrFV8X2ZUbl8kRFdXRoVSSzVixf/85Of/tZv/87v/t7Pfn/5D/7wj/7459/84k+uZT4rQ7gK8yQvb0eBhCTO4ErFKoHbooQgHSVwM7rb1vxNBaWM8+xSPRbwIQ2iLJ7EYaAQ+vjN5rCc5uLvxTBIimkgtPVxmAZqWqZ1kt/PxV+KbzfEWsP/ym8wjaPp/OM3Ky/WX5gfsXix0VysLDU/px9/8f0/D8d5OEshU2ESSPl+40WhPtRBqeIwgfnycCahCMK7IIL3eJkFKcgPtZnrXDxHZCwmeYn/MiUMynvUQSrlYzrClnqUsstpsJcbpd6d60ftX3ZGoyZ/+6GOs2KmIAvtYCazRKhc6MUV47iEUCWPeBGEZYzzEeE0KINQ4RZ43lV898W/nx6RyvPEv2VdBeV9PMYxe2gRlPIuLnzQrpAHPfRg+tZr4RjBDO7DPE2DbFwPx/FkMn//8kM9nOB467FY2Zjr/1/O50I8z8txnAWlnthkAiXuXBwkvoOi42FY6A0NEuOIDPTn94uztqO+qfgrYe6KN42zNS1r8X+4OW4J3Z5ujM74nZlPhz7tcpbFav5+A/0Jp/f3KxsfzHJoTtzHairk51lQghjhjO9ASd9FoOyAEpiodT2cYYkRo37zsV5chjEkuN4dUM6Kwozhh/rblY1fdfsgPWpoO7Qubdk2XnEnOk2ucGtGaX3Vxa8tfo34c+ExYBlY3MaizGm6f4fTXcU1b6b8Dwt+roPSrm4zNm2bnsOaVmq4eP9dr9dufx9/F0bS9tEtf8O2Yb68vPxc/4jj3RtxtHm5L3Z29w6ODy4PTo4vhKGW+86PVfxtTodVfWjgCI+C8k5IDH08U6XIJyIMCnutD6oSUGJlnEX6nBjHVSxds0kczUroqAZBPdJ5jYudim8HeL2w89gmRNFC6VptG6uvnZmra3aujb5WKi9cm8u86GsxypXKU9doy1gL7Zp5B65Z8FSLkWsxeqpF6FqET7UYuxZj3QK3YR9nl+gZikBge30OwwSjfSxwbVLfB15r0CgDFT0RJji0lz2zKXbX8JSHVYGJEMq1ENPs+jIG6sQsK0xWNmq7gf80RMtGV193HG6sgmRd7KEYpMKTR++91DuGvPW45zzudT0aWt3n7p4rL5u7SuEaCZxRY7x0PfBcGlOXle9Wvl/ottr2cVffcVffm+lcWFV/dTlQ+XbwTQh469HjwC2I7X3hel/09D53vUyOvc/bKFtvF8beXZqVaWPwiaXpOpyWAF2XzN/Kd4seadWY7+8WfQeZANwE3blnyeCznbNr8vSkPT+YD6AU2o91s9u42e1zsynK4J7WveNsbW0tqPJ4LGZSH03xRBS5lDEWjtZ1kQQYOo3/p0enC5wCI6lnjpqx3Zs2/+9JNo62W0fbP+oI55xFYM5g21ZaHwZuR4RScbRztbb2pExwdEES5VjQTdOeeSJnR9c2+upEmauFmW46V5s9rpzg3f1wEq2vrx8Gl16nzR/ttLCoulxqZs7Up1E7XH31tU2x/bvqPW37n/r93UzbG+Co9fWTA24EB3GixZroC5vo9VXjb5LkeWloc2V5c9k02DCVzUKNpEoMhLktOcIgqXe6Daogice8wce2PrHUfMElSNXfwTDzdkZQSJ3jChknedbkp3N0kacCPxjEAUar0zeooNaeH1SWlyl6fTZE6NncLWfZoQNiRj4zIib0mZCYsc+MiQGfAWImPjMhJvKZiJipz0yJiX0mJuaTz3wi5s5n7ohJfCaZGxmXqYglRix+pB4/6sPO7uCq+DSTSozz7JdK6M+nKMdHffJ4GyPSxnfm+87orrnP5MQUPlMQ89lnPhNT+kxJjPQZSYzyGUXMzGdmxFQ+UxFz7zP3xDz4zAMxjz7zSMwXn/kyt2WeCwDMzHl7vFdNkNQ2lEYTFjbtuNXURolrYWzGM47DI4JZbFQhwSwwqjHBLCoqIJiFRDUhmMVDFRHMgqGaEswioZoRzMKg+kQwi4HqjmAWAFVCcMLglOCUwWyh+QrnBDMxVwXBTMnVZ4KZjKuSYKbhShIs+aYSrPrXhEu3IpjptronmIm2eiCYKbZ6JJjJtfpCsNPqbgL6OZf5tFf26Bas6HrPZbDK6z2Zwcqv92wGq8He0xmsEHvPZ7Bq7D2hwUqy94wGq8veUxq5J89psArtPanByrT3rAar1e5p7bjU51LOPXkSg5Vu71kMVr+9pzFYEfeex2CV3Hsig5Vz75kMVtO9pzJYYfeey2DV3Xsyg5V479kMVue9pzNYsfeez2AV//QJjbFQxmFboaSbFB+bFDbpFsFbDN4meJvBOwTvMHiX4F0G7xG8x+DXBL9m8D7B+ww+IPiAwW8IfsPgQ4IPGTwgeMDgI4KPGHxM8DGDTwg+YfApwacMPiP4jMHnBJ8z+ILgCwZfEnzJ4CuCrxh8TfA1g28IvmHwLcG3DH5L8FsGvyP43dPHqy86sKpjGt1k+jXSY9wW57Z9bptzOz63w7ldn9vl3J7P7XHutc+95ty+z+1z7sDnDjj3xufecO7Q5w45N/C5AeeOfO6Ic8c+d8y5E5874dypz51y7sznzjh37nPnnLvwuQvOXfrcJeeufO6Kc9c+d825G5+74dytz91y7q3PveXcO59zsr/mJUT1BcznCPzs+qLtW+UZ1O7zrMPSmYWGKSWNtibWuF8PVw3MkJFFqA4xVQgiVH2Y2gMRqjmqZiRUaZg6AxGqL0x1gQhVFaamQIRqCVNJIEIVhKkfEKG6wVQNiFC1YGoFRBK2DhahysDUBYhkbP0sQlWAqQEQodxvMj8ilPFNvkeE8rzJ8ohItuAWoZxeNdvCNqWyCOVvk70RoaxtcjYilKtNpkaEMrTJz4j0VaN+GVqZ73extfndKrAaNeIwunAgfdSiJxMNlQTpaKx72Asi8hQijZvfBBtJajk6AB0igv8TJOMo1V3Nb4KdcBvRthOpaz7+WovVWSjWkCwU6phNqrZfalkLBTohC8UZkYXCnJKFw2VjRUF+IgvFeMfWptYibGdeawE6CxeTrSKKL2dLUmvROQtF95ksFFzJVqrWQmsXqNYicxYuNFtmFFhFForrniwU1gNZKKpHslBQX+bNd16YZx8sbnIs6oxyq8msiFBGNfkUEcqjJosiQtnT5M5RynKmyZiIUKY0eRIRyo8mOyJCWdHkREQoF5pMiAhlQJP/EKG8Z7IeIpTtTK5DhHKcyXCIUGYzeQ0RymcmmyFCWczkMEQod5nMhQhlLJOvEKE8ZbIUIpSdTG6y39c65Lp239Q65MYilH9M9kGEso7JOYhQrjGZBpF3bAcpL4x4WkhPp81BPMQrtnou9DUzaMK/nVwTw5q7sHFsVHQJmdQvcOxAmAQloKimm/oEwjvaYk9OYv2oFLIwH8dZhM6CWaIROWmv03kt9VPeC1BPORjlyfjH3Iwe5hiE3Se1WfPtscmbjT/zlLqZmrL1ZSaZ+tWWw0j/atthFAFqx2EUA2rXYRQFas9hFAfqtcMoEtS+wygW1IHDKBrUG4dRPKhDh1FEqIHDKCbUkcMoKtSxwygu1InDKDLUqcMoNtSZwyg61LnDKD7UhcMoQtSlwyhG1JXDKErUtcMoTtSNwyhS1K3DKFbUW4dRtKh3DrMVGQr5dRkUU8tG7nNu6H3ciLYYTLqIthlM0oh2GEzqiHYZTAKJ9hhMGoleM5hkEu0zmJQSHTCYxBK9YTDpJTpkMEkmGjCYVBMdMZiEEx0zmLQTnTCY5BOdMpgUFJ0xmEQUnTOYdBRdMJikFF0ymNQUXTGYBBVdM5g0Fd0wmGQV3TKYlBW9ZTCJK3rHYFfxX5iXizQt26coIyYuuUUoaUtuE0rSkjuEGmU9Fzvmm4yZBBEICUrgrRMYi91VMYIw0LiaxlLc57NkjBBaIKT53gNryVkp9Bt3eYKO9Psu8FBgbWm+zHVfqe/RHUmd8jWhJE65TyhpUx4QStKUbwglZcpDQkmYckAo6VIeEUqylMeEkirlCaEkSnlKKGlSnhFKkpTnhJIi5QWhJEh5SSjpUV4RSnKU14SSGuUNoSRGeUsoaVG+JZSkKN8R2j5yybDuA/MRIrAPW5oiEKgCGPjFvy4PN8lCqW6RhRLdJguluUMWHna7ZKGI9shC8bwmC0WzTxaK5YAsFMkbslAch2ShKAZkoRiOyEIRHJOFm39CFm76KVm42Wdk4Safk4Wbe0EWbuolWbiZV2ThJl6ThZt3QxZu2i1ZuFlvycJNesfu11RaTZWltwz4lilbceGRouPXvESLQWzRVfNSYz5TAssdcY8prYDSL4iAKiKvGmpur1oNmIYLhSCYcgk69RKYggk6FROYkgk6NROYogk6VROYsgk6dROYwgk6lROY0gk6tROY4gk61ROY8gk69ROYAgo6FRSYEgo6NRSYIgo6VRSYMgo6dRSYQgo6lRSYUgo6tRSYYgo61RSYcgo69RSYggo6FRWYkgo6NRWYogo6VRWYsgo6dRWYwgo6lRWY0go6tRWY4go61RWY8go69RWYAgtYhYWfFDDlqHIGYpaNoUwe9UtL40AFIoIMSsw22o4lKn0006mn+3YyNp3Xxcd6WKa1MUzi014hLeIyDs3bwNS/fXdw9GjSnXkNRN8E82PHt3tDZBoo/KTu38Jrecpbns77BpPmY0i+NhHToJ2JtRbu0zQ6/VqjQsXJGJqWQ2O0o2974DGh8nAaSP0uezBTufkEBaU3ws4LrIVt046x6bI4gDF47azZ065EAg8d186aqIXQPEPzGydBkQQhzNs3agYNMBfPRXPtL+9y5/3iNuPtdgcykOylnUGXPZ/z3N45NdOCrXGHTMq5e+7mEyVE8/ZJWpcKFc1RW/EkhrLrWuYTlQYP1NIB3XaYLHLzEpN9yLbopUhmevZf9JMAnz0czPkbTIeDhQ3U73Z7L3p3/Sv8FZS492XOWl4sbMB2XhGtDS3QmzyZlEGqH0hN7/MSC1QZPErxbPDDy2f67R3ztyKzzL6FKgvcf2neHns2hCRhbdwD0ediCxMghnym/3vEeIdUv8Wmq2DrlLXWr5Dms8jkTFMUxwpWjXuZi3EO2t19fBcXMI6D9c4ryHmZJvrh/bwe/PBi3kPmGWhuo49T96bfyz6u0EzRwxgtDH4YxtlEPXZDpwj0H0LgLp8GOlguAM9aGUQg4kxkeVPQK3hYF9vTXOrlyXUBGE7FDn72zeCXUozy/G592Xucc1Lo0zkv/wI1XkZmAPh7uKqvvtZQn5O2IV71uzRqxWbm/ydaXKKgLvUbfgmoYTDCOEvy+1EJwd3Hb1Y2un+mtHhx/XJ946/XX519v/LrreZPmH629GdLf7707dLG0t8s/Xppf+l06WopXPqPpf9c+q+l/371j6/+5dW/vvo32/SnP2n6/OmS9/Pq3/8X+pfL8A==</latexit> ⇢ = ↵⇢low + (1 ↵)⇢high <latexit sha1_base64="KqFZV0zbnWFJebFRVAf2JlrKhGo=">AAA14niclVtJc9xKcuaMtzG9zBvP0ZcKU4p54yBpUe9p7IPtGK4ixea+Smo9BRrIRkPEJlQ1SKqnffPFN4ev/gf+NT7a/jPOqkIhs9Cgns0Iicjvq0rU8mVlNhoclWki1YsX//WTn/7e7//BH/7Rz/54+U/+9M/+/Off/OIvrmUxrUK4Cou0qG5HgYQ0yeFKJSqF27KCIBulcDO629b8TQ2VTIr8Uj2W8CEL4jwZJ2GgEPr4ze6wjJLxeDafqbkYBmk5CcQ/iDUxzINRGohhGBVKfDscZbNpy6+JnZY3yK8/frPyYv2F+RGLFxvNxcpS83P68Rff//MwKsJpBrkK00DK9xsvSvVhFlQqCVOYLw+nEsogvAtieI+XeZCB/DAz852L54hEYlxU+C9XwqC8xyzIpHzMRtgyC9REdjkN9nKjzLvz7FH7l53RqPHffZgleTlVkId2MONpKlQh9AKLKKkgVOkjXgRhleB8RDgJqiBUuA2ed5XcffHvp0ekiiL1bzmrg+o+iXDMHloGlbxLSh+0K+RBDz2YvvVaGCGYw31YZFmQR7OhVsL8/csPs+EYxzuLxMrGXP//cj4X4nlRRUkeVHpi4zFUuHNJkPoOyo6HYak3NEiNIzLQn98vyduO+qbib4S5K940yde0tMX/4ea4JXR7ujE643dmPh36tMtpnqj5+w30J4Z6b6ps9n5l44NZDs2J+0RNhPw8DSoQI5zxHSjpuwiUHVAKY7WuhzOskniifvdxtrgMEaS43h1QTsvSjOGH2bcrG7/u9kF61NB2aF3ash/d+HEnOk2ucGswvq+6+LXFrxF/LjwGLAOL21hWBU3373G6q7jmzZT/ccHPdVDZ1W3Gpm3TczijlRou3n/X67Xb38ffhZG0fXTL37FtmC8vLz/XP+J490YcbV7ui53dvYPjg8uDk+MLYajlvvNjFX+b02FVHxo4wqOguhMSQx/PVSmKsQiD0l7rg6oClFiV5LE+J6KkTqRrNk7iaQUd1SCoRzqf4WJn4tsBXi/sPLYJUbRQuVbbxuprZ+bqmp1ro6+VKkrX5rIo+1qMCqWKzDXaMtZCu2begWsWPNVi5FqMnmoRuhbhUy0i1yLSLXAb9nF2qZ6hCAS21+cwjDHaI4Frk/k+8FqDRhmo6LEwwaG97JlNsbuGpzysirS4h2otxFS7voyBOjbLCuOVjZndwH8aomWjq687DjdRQbou9lAMUuHJo/de6h1D3nrccx73uh4Nre4Ld8+Vl81dpXCNBM6oMV66HnguRdRl5buV7xe6rbZ93NV33NX3ZjoXVtVfXQ5Uvh18EwLeevQ4cAtie1+43hc9vc9dL5Nj74s2ytbbhbF3l2Zl2hh8Ymm6DicVQNcl87fy3aJHWjXm+7tF30EuADdBd+5ZMvhs5+yaPD1pzw/mA6iE9mPd7DZudvvcbIoquKd17zhbW1sL6iKJxFTqoykZi7KQMsHi0bou0wBDp/H/9Oh0gVNiJPXMUTO2e9Pm/z3JxtF262j7Rx3hnPMYzBls20rrw8DtiFAqjnau1taelAmOLkjjAgu6SdYzT+Ts6NpGX50oc7Uw003narPHlRO8ux9OovX19cPg0uu0+aOdFhZVl0vNzJn6NGqHq6++tim2f1e9p23/U7+/m2l7Axy1vn5ywI3gIEm1WFN9YRO9vmr8jdOiqAxtrixvLpsGG6ayWaiRVIWBMLclRxiks51ugzpIk4g3+NjWJ5aaL7gEqfo7GGbezghKqXNcKZO0yJv8dI4uikzgB4MEP4eB0zeoYKY9P6i8qDL0+myI0LO5W86qQwfEjHxmREzoMyExkc9ExIDPADFjnxkTE/tMTMzEZybEJD6TEPPJZz4Rc+czd8SkPpPOjYyrTCQSIxY/VkeP+rCzO7gqPk2lElGR/0oJ/fkU5fioTx5vY0TW+M593zndtfCZgpjSZ0piPvvMZ2Iqn6mIkT4jiVE+o4iZ+syUmNpnamLufeaemAefeSDm0WceifniM1/mtsxzAYCZuWiP97oJkpkNpdGYhU07bjWxUeJaGJvxjOPwiGAWG3VIMAuMOiKYRUUNBLOQqMcEs3ioY4JZMNQTglkk1FOCWRjUnwhmMVDfEcwCoE4JThmcEZwxmC00X+GCYCbmuiSYKbn+TDCTcV0RzDRcS4Il31SCVf+acOnWBDPd1vcEM9HWDwQzxdaPBDO51l8IdlrdTUE/5zKf9qoe3YIVXe+5DFZ5vSczWPn1ns1gNdh7OoMVYu/5DFaNvSc0WEn2ntFgddl7SiP35DkNVqG9JzVYmfae1WC12j2tHZf5XMa5J09isNLtPYvB6rf3NAYr4t7zGKySe09ksHLuPZPBarr3VAYr7N5zGay6e09msBLvPZvB6rz3dAYr9t7zGazinz6hMRaqJGwrlGyT4mOTwibbIniLwdsEbzN4h+AdBu8SvMvgPYL3GPya4NcM3id4n8EHBB8w+A3Bbxh8SPAhgwcEDxh8RPARg48JPmbwCcEnDD4l+JTBZwSfMfic4HMGXxB8weBLgi8ZfEXwFYOvCb5m8A3BNwy+JfiWwW8JfsvgdwS/e/p49UUHVnVMo5tMv0Z6jNvi3LbPbXNux+d2OLfrc7uc2/O5Pc699rnXnNv3uX3OHfjcAefe+Nwbzh363CHnBj434NyRzx1x7tjnjjl34nMnnDv1uVPOnfncGefOfe6ccxc+d8G5S5+75NyVz11x7trnrjl343M3nLv1uVvOvfW5t5x753NO9te8hKi/gPkcgZ9dX7R96yKHmfs867BsaqFhRkmjrYk17tfDdQMzZGQRqkNMFYIIVR+m9kCEao66GQlVGqbOQITqC1NdIEJVhakpEKFawlQSiFAFYeoHRKhuMFUDIlQtmFoBkZStg0WoMjB1ASI5Wz+LUBVgagBEKPebzI8IZXyT7xGhPG+yPCKSLbhFKKfXzbawTaktQvnbZG9EKGubnI0I5WqTqRGhDG3yMyJ91ahfhtbmy1xsbX63CqxHjTiMLhxIH7XoyURDpUE2inQPe0FEkUGscfObYCNJLUcHoENE8H+CZBJnuqv5TbATbiPadiKzGR//TIvVWSjWkCwUasQmNbNfalkLBTomC8UZk4XCnJCFw2VjRUF+IgvFeMfWZqZF2M58pgXoLFxMtooovoItyUyLzlkous9koeAqtlIzLbR2gWZaZM7ChWbLjAKryUJx3ZOFwnogC0X1SBYK6su8+c4L8+yDxU2ORZ1RbjWZFRHKqCafIkJ51GRRRCh7mtw5yljONBkTEcqUJk8iQvnRZEdEKCuanIgI5UKTCRGhDGjyHyKU90zWQ4Syncl1iFCOMxkOEcpsJq8hQvnMZDNEKIuZHIYI5S6TuRChjGXyFSKUp0yWQoSyk8lN9vtah1zP3De1DrmxCOUfk30Qoaxjcg4ilGtMpkHkHdtBygsjnhay00lzEA/xiq2eC33NDJrwbyfXxLDmLmwcGxVdQi71Cxw7EKZBBSiqyaY+gfCOttiT40Q/KoU8LKIkj9FZME01IsftdTafSf2U9wLUUw5GRRr9mJvRwxyDsPukNm++PTZ5s/FnnlI3U1O2vswlU7/achjpX207jCJA7TiMYkDtOoyiQO05jOJAvXYYRYLadxjFgjpwGEWDeuMwigd16DCKCDVwGMWEOnIYRYU6dhjFhTpxGEWGOnUYxYY6cxhFhzp3GMWHunAYRYi6dBjFiLpyGEWJunYYxYm6cRhFirp1GMWKeuswihb1zmG2IkMhv66CcmLZ2H3ODb2PG/EWg0kX8TaDSRrxDoNJHfEug0kg8R6DSSPxawaTTOJ9BpNS4gMGk1jiNwwmvcSHDCbJxAMGk2riIwaTcOJjBpN24hMGk3ziUwaTguIzBpOI4nMGk47iCwaTlOJLBpOa4isGk6DiawaTpuIbBpOs4lsGk7LitwwmccXvGOwq/gvzcpGmZfsUZcTEJbcIJW3JbUJJWnKHUKOs52LHfJMxlSACIUEJvHUKkdhdFSMIA42rSSLFfTFNI4TQAiHN9x5YS04rod+4K1J0pN93gYcSa0vzZa77Sn2P7kjqlK8JJXHKfUJJm/KAUJKmfEMoKVMeEkrClANCSZfyiFCSpTwmlFQpTwglUcpTQkmT8oxQkqQ8J5QUKS8IJUHKS0JJj/KKUJKjvCaU1ChvCCUxyltCSYvyLaEkRfmO0PaRS451H5iPEIF92NIUgUAVwMAv/nV5uEkWSnWLLJToNlkozR2y8LDbJQtFtEcWiuc1WSiafbJQLAdkoUjekIXiOCQLRTEgC8VwRBaK4Jgs3PwTsnDTT8nCzT4jCzf5nCzc3AuycFMvycLNvCILN/GaLNy8G7Jw027Jws16SxZu0jt2v6bSaqosvWXAt0zZiguPFB2/5iVaDGKLrpqXGoupEljuiHtMaSVUfkEEVBF51VBze9VqwDRcKATBlEvQqZfAFEzQqZjAlEzQqZnAFE3QqZrAlE3QqZvAFE7QqZzAlE7QqZ3AFE/QqZ7AlE/QqZ/AFFDQqaDAlFDQqaHAFFHQqaLAlFHQqaPAFFLQqaTAlFLQqaXAFFPQqabAlFPQqafAFFTQqajAlFTQqanAFFXQqarAlFXQqavAFFbQqazAlFbQqa3AFFfQqa7AlFfQqa/AFFjAKiz8pIApR1VTENM8gip91C8tRYEKRAw5VJhttJ1IVPpoqlNP9+1kbDqflR9nwyqbGcMkPu0VsjKpktC8DUz923cHR48m3ZnXQPRNMD92fLs3RCaBwk/q/i28lqe85em8bzBZEUH6tYmYBu1MrLVwn6bR6dcalSpJI2haDo3Rjr7tgceEKsJJIPW77MFUFeYTFFTeCDsvsJa2TTvGpsviACLw2lmzp12FBB46rp01UQuheYbmN06DMg1CmLdv1AwaYC6ei+baX97lzvvFbcbb7Q5kINlLO4Muez7nub1zamYlW+MOmVZz99zNJyqI5+2TtC4VKpqjtpJxAlXXtSzGKgseqKUDuu0wWRTmJSb7kG3RS5lO9ey/6CcBPns4mPM3mA4HCxuo3+32XvTu+lf4K6hw76uCtbxY2IDtoiZaG1qgN0U6roJMP5Ca3BcVFqgyeJTi2eCHl8/02zvmb0WmuX0LVZa4/9K8PfZsCGnK2rgHos/FFiZADPlc//eI8Q6ZfotNV8HWKWutXyEtprHJmaYoThSsGveyEFEB2t19cpeUECXBeucV5KLKUv3wfj4b/PBi3kMWOWhuo49T96bfyz6u1EzZwxgtDH4YJvlYPXZDpwz0H0LgLp8GOlguAM9aGcQgklzkRVPQK3hYF9uTQurlKXQBGE7EDn72zeFXUoyK4m592Xucc1Lq07mo/ho1XsVmAPh7uKqvvtZQn5O2IV71uzRqxWbm/ydaXKKgLvUbfimoYTDCOEuL+1EFwd3Hb1Y2un+mtHhx/XJ94zfrr86+X/ntVvMnTD9b+sulv1r6dmlj6W+Xfru0v3S6dLUULv3H0n8u/ffS/7yKXv3Lq3999W+26U9/0vT55ZL38+rf/xedM8yx</latexit> @ @t ↵ = r · (u↵ Dr↵) ਖ਼ղσʔλͷྫ • ࢧ഑ํఔࣜUXPGMVJENJYUVSFϞσϧɺݽཱܥ • ֶशσʔλৄࡉͳϝογϡɾ0QFO'0".Ͱ ͞·͟·ͳܗঢ়Ͱ࣍ݩղੳΛ͠ ૈ͍ϝογϡʹϚοϐϯάͨ͠΋ͷ • αϯϓϧ਺5SBJOɺWBMJEBUJPOɺUFTU

਺஋࣮ݧ݁ՌఆੑධՁ ਖ਼ղσʔλ .11%&[Brandstetter+ 2022] 1&//[Horie & Mitsume 2022] ఏҊख๏[Horie
& Mitsume 2024] 5FTU ྲྀ଎

ਖ਼ղσʔλ .11%&[Brandstetter+ 2022] 1&//[Horie & Mitsume 2022] ఏҊख๏[Horie &
Mitsume 2024] 5FTU 5FTUΛ ճస ਺஋࣮ݧ݁ՌఆੑධՁ ྲྀ଎

Mitsume 2024] 5FTU 5FTUΛ ճస 5FTUΛ εέʔϦϯά ਺஋࣮ݧ݁ՌఆੑධՁ ྲྀ଎ 4DBMJOHͷ ྲྀ଎

Mitsume 2024] 5FTU 5FTUΛ ճస ྖҬͷ֎ૠ 5FTUΛ εέʔϦϯά ਺஋࣮ݧ݁ՌఆੑධՁ ྲྀ଎ 4DBMJOHͷ ྲྀ଎

਺஋࣮ݧ݁ՌఆྔධՁ • ఏҊख๏͸'7. ػցֶशͳ͠ͷఏҊख๏ ͱಉ౳ͷอଘੑΛ࣮ݱ • ఏҊख๏͸ྖҬͷ֎ૠʹ͍ͭͯਫ਼౓ͷྼԽ͕ͳ͍ˠߴ͍൚༻ੑ ख๏ σʔληοτ
ޡࠩ (×10-1) อଘੑͷޡࠩ (×10-5) FVM (ػցֶशͳ͠ͷఏҊख๏) test 20.034 0.01 MP-PDE[Brandstetter+ 2022] test 1.165 1001.95 PENN[Horie & Mitsume 2022] test 0.525 1356.62 ఏҊख๏[Horie & Mitsume 2024] test 0.898 0.06 FVM (ػցֶशͳ͠ͷఏҊख๏) ྖҬͷ֎ૠ ൃࢄ ൃࢄ MP-PDE[Brandstetter+ 2022] ྖҬͷ֎ૠ 4.557 16770.80 PENN[Horie & Mitsume 2022] ྖҬͷ֎ૠ 1.240 4799.89 ఏҊख๏[Horie & Mitsume 2024] ྖҬͷ֎ૠ 0.829 0.02

਺஋࣮ݧ݁ՌఆྔධՁ 100 101 102 Computation time [s] 10°1 100
101 102 103 104 Loss FluxGNN (Ours): test PENN: test MP-PDE: test FVM: test FluxGNN (Ours): taller PENN: taller MP-PDE: taller FVM: taller • ͞·͟·ͳઃఆͰੑೳΛଌఆ ޡࠩ $16ίΞͰͷܭࢉ࣌ؒ<T>

100 101 102 Computation time [s] 10°1 100 101 102
103 104 Loss FluxGNN (Ours): test PENN: test MP-PDE: test FVM: test FluxGNN (Ours): taller PENN: taller MP-PDE: taller FVM: taller ਺஋࣮ݧ݁ՌఆྔධՁ ఏҊख๏ 1&// .11%& • ͞·͟·ͳઃఆͰੑೳΛଌఆ '7. ػցֶशͳ͠ ͷఏҊख๏ ޡࠩ $16ίΞͰͷܭࢉ࣌ؒ<T>

਺஋࣮ݧ݁ՌఆྔධՁ 100 101 102 Computation time [s] 10°1 100
101 102 103 104 Loss FluxGNN (Ours): test PENN: test MP-PDE: test FVM: test FluxGNN (Ours): taller PENN: taller MP-PDE: taller FVM: taller ఏҊख๏ 1&// .11%& • ͞·͟·ͳઃఆͰੑೳΛଌఆ '7. ػցֶशͳ͠ ͷఏҊख๏ ޡࠩ $16ίΞͰͷܭࢉ࣌ؒ<T>

10°1 100 101 102 Computation time [s] 10°1 100 Loss
਺஋࣮ݧ݁ՌఆྔධՁ 100 101 102 Computation time [s] 10°1 100 101 102 103 104 Loss FluxGNN (Ours): test PENN: test MP-PDE: test FVM: test FluxGNN (Ours): taller PENN: taller MP-PDE: taller FVM: taller ఏҊख๏ 1&// .11%& ޡࠩ $16ίΞͰͷܭࢉ࣌ؒ<T> ബ͍৭UFTU ೱ͍৭ྖҬͷ֎ૠ '7. ػցֶशͳ͠ ͷఏҊख๏ • ͞·͟·ͳઃఆͰੑೳΛଌఆ

10°1 100 101 102 Computation time [s] 10°1 100 Loss
਺஋࣮ݧ݁ՌఆྔධՁ 100 101 102 Computation time [s] 10°1 100 101 102 103 104 Loss FluxGNN (Ours): test PENN: test MP-PDE: test FVM: test FluxGNN (Ours): taller PENN: taller MP-PDE: taller FVM: taller ఏҊख๏ 1&// .11%& ޡࠩ $16ίΞͰͷܭࢉ࣌ؒ<T> • ఏҊख๏͸ྖҬͷ֎ૠʹ͓͍ͯ طଘख๏ͷܭࢉ଎౓ɾਫ਼౓ͷ τϨʔυΦϑΛେ෯ʹվળ ബ͍৭UFTU ೱ͍৭ྖҬͷ֎ૠ '7. ػցֶशͳ͠ ͷఏҊख๏ • ͞·͟·ͳઃఆͰੑೳΛଌఆ

϶U • ૬ࣅม׵ରশੑ ཭ࢄԽεέʔϧΛಋೖ • ਺஋࣮ݧ ࠞ૬ྲྀݱ৅΁ͷద༻ <latexit 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</latexit> ' • ہॴอଘੑɺ૬ࣅม׵ରশੑΛݫີʹຬͨ͢ػցֶशϞσϧ͕࣮ݱ[Horie & Mitsume. ICML. 2024]

϶U • ૬ࣅม׵ରশੑ ཭ࢄԽεέʔϧΛಋೖ • ਺஋࣮ݧ ࠞ૬ྲྀݱ৅΁ͷద༻ <latexit 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</latexit> ' • ہॴอଘੑɺ૬ࣅม׵ରশੑΛݫີʹຬͨ͢ػցֶशϞσϧ͕࣮ݱ[Horie & Mitsume. ICML. 2024] • কདྷͷల๬ - ೤ྲྀମɾଟ૬ྲྀͷ࣮໰୊΁ͷద༻ͱੑೳͷධՁ - ࣮ݧɾ࣮ଌσʔλͷֶशʹΑΔطଘͷγϛϡϨʔγϣϯΛ্ճΔਫ਼౓ͷ࣮ݱ

ICML 読み会: Graph Neural PDE Solvers with Conserv...

ICML 読み会: Graph Neural PDE Solvers with Conservation and Similarity-Equivariance

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