数式表現学習の現在地と工学への道筋 / Current Status of Mathematical Representation Learning and its Application to Engineering

2025年度日本神経回路学会時限研究会「情報表現の統一を通じた知識表現構造の顕像」発表資料
URL: https://sites.google.com/view/knowledge-structure/%E3%83%9B%E3%83%BC%E3%83%A0

Transcript

1. ਺ࣜදݱֶशͷݱࡏ஍ͱ޻ֶ΁ͷಓے 加藤 祥太 shota@human.sys.i.kyoto-u.ac.jp 京都大学大学院 情報学研究科 ヒューマンシステム論分野 手法紹介における図表や事例は紹介論文からの引用です 2025-09-05 2025年度日本神経回路学会時限研究会「情報表現の統一を通じた知識表現構造の顕像」
2. r
   ژେӃɾ޻ֶݚڀՊɾԽֶ޻ֶઐ߈ म࢜ʢ޻ֶʣ r
   ژେӃɾ৘ใֶݚڀՊɾγεςϜՊֶઐ߈ ത࢜ʢ৘ใֶʣ r ژେӃɾ৘ใֶݚڀՊɾॿڭ 
   r +45
   "$59
   ݚڀһ "$59ʮ੡଄ϓϩηεͷઐ໳༻ޠͱ਺ࣜΛཧղ͢Δ෺ཧϞσϧࣗಈߏங"*ͷ։ൃʯ r
   ϚϯνΣελʔେֶ ٬һݚڀһ r ࢈૯ݚ টᡈݚڀһ • ઐ໳ɿԽֶ޻ֶ Y
   σʔλαΠΤϯε Y
   ࣗવݴޠॲཧ • ݚڀ಺༰ • ੡଄ϓϩηεͷϞσϦϯάɺ੍ޚ • ෺ཧϞσϧߏஙͷޮ཰Խ ࣗݾ঺հɿՃ౻
   ↅଠ 1 ෺ཧϞσϧ ࿦จ΍ॻ੶ ࣮ݧσʔλ ࣌ؒ ೱ౓
3. 2
   ҎԼͷࣜ͸ຒΊࠐΈۭؒͰͲ͏ฒͿ΂͖͔ʁ  𝑥! + 𝑦! + 2𝑥𝑦  𝑥 + 𝑦 !  𝑥! + 𝑦! + 𝑥𝑦  2(𝑥 + 𝑦)  2 ਺ࣜͷຒΊࠐΈͷΠϝʔδ 2 1 2 3 5 d/dx 1 2 3 5 4 ୅਺తͳૢ࡞Λදݱ (例えば[Valentino+, NAACL, 2024]) ߏ଄తͳྨࣅੑΛදݱ (例えば[Mansouri+, ICTIR, 2019]) "
   λεΫʹΑͬͯҟͳΔ͕ɺຊࢿྉͰ͸਺ࣜͷҙຯʹै͏΂͖ͱ͢Δɻ 4 d/dx
4. ਺ࣜͷҙຯ͸ɺͲ͜ʹ͋Δͷ͔ʁ 3 จ຺ [Krstovski&Blei, arXiv, 2018] ୅਺త౳Ձੑ [Gangwar&Kani, TMLR, 2023] ਺஋ڍಈ [Meidani+, ICLR, 2024]
5. ʮ਺ࣜͷҙຯʯΛϕΫτϧͰ࣋ͨͤΔํ๏Λɺ̏؍఺Ͱ੔ཧ͢Δɻ  पลςΩετ͔ΒͳΔจ຺ [Krstovski&Blei, arXiv, 2018]  ୅਺త౳Ձͳ਺ࣜϖΞ [Gangwar&Kani, TMLR, 2023]
   
    ೖग़ྗͷ਺஋తৼΔ෣͍ [Meidani+, ICLR, 2024] ਺ࣜຒΊࠐΈʹඞཁͳͦΕҎ֎ͷݕ౼߲໨  ܇࿅ɾධՁσʔλɿจݙ͔Βऩूɺ౳Ձू߹ͷੜ੒  σʔλͷදݱํ๏ɿςΩετܗࣜɺτʔΫφΠζɺߏ଄Խ  Ϟσϧͱֶशํ๏ɿXPSEWFDɺTFRTFRɺରরֶशʢDPOTUSBTUJWF
   MFBSOJOHʣ  ධՁํ๏ɺԼྲྀλεΫɺԠ༻ઌ ࠓ೔ͷΰʔϧ 4
6. ʮ਺ࣜͷҙຯʯΛϕΫτϧͰ࣋ͨͤΔํ๏Λɺ̏؍఺Ͱ੔ཧ͢Δɻ  पลςΩετ͔ΒͳΔจ຺ [Krstovski&Blei, arXiv, 2018]  ୅਺త౳Ձͳ਺ࣜϖΞ [Gangwar&Kani, TMLR, 2023]
   
    ೖग़ྗͷ਺஋తৼΔ෣͍ [Meidani+, ICLR, 2024] ਺ࣜຒΊࠐΈʹඞཁͳͦΕҎ֎ͷݕ౼߲໨  ܇࿅ɾධՁσʔλɿจݙ͔Βऩूɺ౳Ձू߹ͷੜ੒  σʔλͷදݱํ๏ɿςΩετܗࣜɺτʔΫφΠζɺߏ଄Խ  Ϟσϧͱֶशํ๏ɿXPSEWFDɺTFRTFRɺରরֶशʢDPOTUSBTUJWF
   MFBSOJOHʣ  ධՁํ๏ɺԼྲྀλεΫɺԠ༻ઌ ࠓ೔ͷΰʔϧ 5
7. ؍఺̍ɿपลςΩετ͔ΒͳΔจ຺ [Krstovski&Blei, arXiv, 2018] 6 • &RVBUJPO
   FNCFEEJOH
   &R&NC ɿ਺ࣜΛ୯ޠͱΈͳͯ͠ɺ पล୯ޠͷ෼෍͔ΒຒΊࠐΈΛֶशɻ • &RVBUJPO
   VOJU
   FNCFEEJOH
   &R&NC6 ɿ਺ࣜΛߏ੒͢ΔϢχοτ ʢԋࢉࢠͱඃԋࢉࢠʣΛ୯ޠͱͨ͠ &R&NCɻ Ϣχοτʹ෼ׂͨ͠਺ࣜ
8. • BS9JW
   ͷ࿦จʢ/-1
   *3
   "*
   .-
   ͷ෼໺ʣ͔Βऩूͨ͠୯ޠͱ਺ࣜɻ • ୯ޠɿ/-5,
   <#JSE
   > Λ༻͍֤ͯ࿦จ͔Β໊ࢺ۟ͱܗ༰ࢺΛநग़͔ͯ͠Βɺ ετοϓϫʔυͷ࡟আ΍ώϡʔϦεςΟΫεॲཧʹΑͬͯબ୒ɻ • ਺ࣜɿσΟεϓϨΠ਺ࣜɻ୯Ұม਺͕ଟ͍ΠϯϥΠϯ਺ࣜ͸ෆ࢖༻ɻ &R&NC
   ͷߏஙʹ༻͍Δσʔλͱදݱํ๏ 7
9. ਺ࣜΛߏ੒͢ΔϢχοτͷ࡞੒ 8 • ਺ࣜͷߏ੒ཁૉ͸ԋࢉࢠͱඃԋࢉࢠɻ • ਺ࣜΛ 4ZNCPM
   MBZPVU USFF 4-5
   දݱ <;BOJCCJ 4*(*3
   > ʹม׵ɻ
10. • #FSOPVMMJ
    FNCFEEJOHT <3VEPMQI
    /*14
    > Λ࠾༻ɻ 𝑝 𝑤! 𝑤"! = 𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖 𝑏# = 𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖 𝜎 𝜌#! $ / %&' |"!| 𝛼#" 𝑝 𝑤! 𝑤"! : probability of a word 𝑤! given its context 𝑤"! 𝜎: logistic function; 𝜌#! : embedding of 𝑤! ; 𝛼#" : embedding of context word 𝑤$ • ୯ޠຒΊࠐΈɿ𝑏# = 𝜎 𝜌#! $ ∑ %&' |"!| 𝛼#" + ∑ )&' |"! #| 𝛼*$ 𝛼%#  any equations that appear in a possibly larger window ( 𝑐! & = 16 > 𝑐! = 4) • ਺ࣜຒΊࠐΈɿ𝑏* = 𝜎 𝜌*% $ ∑ +&' |"%| 𝛼#& • ໨తؔ਺ɿ 𝐿 𝜌,, 𝛼#, 𝜌*, 𝛼* = ∑!&' , log(𝑝 𝑤!|𝑤"! ) + ∑-&' . log(𝑝 𝑒-|𝑒"% ) &R&NC
    ͷֶशํ๏ 9
11. • &R&NC
    ಉ༷ɺ#FSOPVMMJ FNCFEEJOHT
    Λ࠾༻ɻ 𝑝 𝑤! 𝑤"! = 𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖 𝑏# = 𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖 𝜎 𝜌#! $ / %&' |"!| 𝛼#" 𝑝 𝑤! 𝑤"! : probability of a word 𝑤! given its context 𝑤"! 𝜎: logistic function; 𝜌#! : embedding of 𝑤! ; 𝛼#" : embedding of context word 𝑤$ • ୯ޠຒΊࠐΈɿ𝑏# = 𝜎 𝜌#! $ ∑ %&' |"!| 𝛼#" + ∑ )&' |"! #| ∑ +&' |*$'() | 𝛼/$& 𝛼%#  any equations that appear in a possibly larger window ( 𝑐! & = 16 > 𝑐! = 4) • ਺ࣜϢχοτຒΊࠐΈɿ𝑏/ = 𝜎 𝜌/! $ ∑ +&' |"& *| 𝛼/& ਺ࣜຒΊࠐΈɿ𝜌* = ' |*'()| ∑ %&' |*'()| 𝜌/" • ໨తؔ਺ɿ 𝐿 𝜌,, 𝛼#, 𝜌/, 𝛼/ = ∑!&' , log(𝑝 𝑤!|𝑤"! ) + ∑!&' 0 log(𝑝 𝑢!|𝑢"! ) ˞਺ࣜຒΊࠐΈʹ͍ͭͯɺ&R&NC
    ͸पล୯ޠɺ&R&NC6
    ͸਺ࣜͷΈΛ࢖༻ɻ &R&NC6
    ͷֶशํ๏ 10
12. • ݕূɾςετσʔλʹର͢Δର਺໬౓Λൺֱɻ • ఏҊϞσϧ͸͍ͣΕ΋ैདྷϞσϧΛ্ճΓɺ&R&NC6
    ͕࠷ߴੑೳɻ &R&NC ͱ &R&NC6
    ͷධՁ 11
13. • ਺ࣜ·ͨ͸୯ޠΛΫΤϦͱͨ͠ݕࡧ &R&NC ͱ &R&NC6 ͷԠ༻ 12
14. ʮ਺ࣜͷҙຯʯΛϕΫτϧͰ࣋ͨͤΔํ๏Λɺ̏؍఺Ͱ੔ཧ͢Δɻ  पลςΩετ͔ΒͳΔจ຺ [Krstovski&Blei, arXiv, 2018]  ୅਺త౳Ձͳ਺ࣜϖΞ [Gangwar&Kani, TMLR, 2023]
    
     ೖग़ྗͷ਺஋తৼΔ෣͍ [Meidani+, ICLR, 2024] ਺ࣜຒΊࠐΈʹඞཁͳͦΕҎ֎ͷݕ౼߲໨  ܇࿅ɾධՁσʔλɿจݙ͔Βऩूɺ౳Ձू߹ͷੜ੒  σʔλͷදݱํ๏ɿςΩετܗࣜɺτʔΫφΠζɺߏ଄Խ  Ϟσϧͱֶशํ๏ɿXPSEWFDɺTFRTFRɺରরֶशʢDPOTUSBTUJWF
    MFBSOJOHʣ  ධՁํ๏ɺԼྲྀλεΫɺԠ༻ઌ ࠓ೔ͷΰʔϧ 13
15. • ݟͨ໨͕ҟͳΔ͕୅਺తʹ౳ՁͳࣜΛ͚ۙͮΔຒΊࠐΈΛ֫ಘɻ • 4&.&.#ɿ 4FR4FR
    Ͱ౳Ձࣜੜ੒Λֶश͠ɺ Τϯίʔμ࠷ऴ૚ͷ NBYQPPM
    ΛࣜϕΫτϧͱͯ͠ར༻ɻ ؍఺̎ɿ୅਺త౳Ձͳ਺ࣜϖΞ
    <(BOHXBS,BOJ
    5.-3
    > 14
16. • &RVJWBMFOU
    &YQSFTTJPOT
    %BUBTFUɿ໿ສϖΞͷݟͨ໨͕ҟͳΔ ౳ՁͳࣜΛ 4ZN1Z ͷؔ਺Λར༻ͯ͠ੜ੒ʢ̍ม਺ɺ࠷େԋࢉࢠʣɻ 4&.&.#
    ͷߏஙʹ༻͍Δσʔλ 15
17. • લஔʢ1PMJTIʣه๏ʢԋࢉࢠΛલʹஔ͘ʣͰԋࢉࢠ໦Λྻʹม׵ɻ ྫɿ-./(1) 34-(1) ˠ
    <EJW
    TJO
    Y
    DPT
    Y> • ೖྗ͸ POFIPU
    FODPEJOH
    ˠ
    ຒΊࠐΈ૚ ˠ ҐஔຒΊࠐΈɻ • ग़ྗੜ੒͸ϏʔϜ୳ࡧɻ 4&.&.#
    ͷߏஙʹ༻͍Δσʔλͱදݱํ๏ 16
18. • Ϟσϧ͸ 5SBOTGPSNFSɻ •  FODPEFS
    MBZFST
    
    EFDPEFS
    MBZFST
    
    BUUFOUJPO
    IFBET • 3F-6 BDUJWBUJPO
    
    ESPQPVU • .PEFM
    EJNFOTJPO
    
    PS
     • ֶश໨ඪɿೖྗࣜͱ਺ֶతʹ౳ՁͳࣜΛσίʔυͰ͖ΔΑ͏ʹ܇࿅ ʢ౳Ձͳ਺ࣜ͸̍ͭʹݶΒͳ͍ͨΊɺݕূ࣌͸ 4ZNQZ Ͱ൑ఆʣɻ • 4&.&.#
    ͷൺֱख๏ͱͯ͠ɺBVUPFODPEFS
    ͷΑ͏ʹ ೖྗΛ࠶ߏ੒͢ΔϞσϧ 4536$5&.#
    ΋ֶशɻ 4&.&.# ͷֶशํ๏ 17
19. • 1$"
    Ͱ࣍ݩʹϓϩοτɻ4&.&.#
    ͸ΧςΰϦ෼཭͕໌ྎɻ 4&.&.# ͷධՁ̍ɿ࣍ݩϓϩοτͷఆੑతධՁ 18
20. • ྨࣅ౓ΛίαΠϯڑ཭ͷٯ਺ʢ1/(1 − (𝑐𝑜𝑠𝑖𝑛𝑒 𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦))ʣͰఆٛɻ • 4536$56&.#
    ΑΓ΋ 4&.&.#
    ͸ߏ଄͕ྨࣅͨ͠਺ࣜΛग़ྗɻ 4&.&.# ͷධՁ̎ɿ਺ࣜݕࡧ 19
21. 4&.&.# ͷධՁ̏ɿΞφϩδʔ 20 • l𝑥' JT
    UP
    𝑦' BT
    𝑥1 JT
    UP
    𝑦1 zʹ ౰ͯ͸·Δ 𝑥1 Λ࣍ࣜͰ༧ଌɻ 𝑥' = 𝑒𝑚𝑏 𝑥( − 𝑒𝑚𝑏 𝑦( + 𝑒𝑚𝑏(𝑦') • 4536$5&.#
    ΑΓ΋ 4&.&.#
    ͷํ͕ߴਖ਼ղ཰ɻ • Ξφϩδʔͷਖ਼ޡ൑ఆ͸೉ͦ͠͏ɻ ྫɿ𝑥! = 𝑥"
    𝑦! = 𝑥
    𝑦" = log 𝑥 ͸ ޡΓͰ͸ͳ͍ͷͰ͸ʁ
22. ʮ਺ࣜͷҙຯʯΛϕΫτϧͰ࣋ͨͤΔํ๏Λɺ̏؍఺Ͱ੔ཧ͢Δɻ  पลςΩετ͔ΒͳΔจ຺ [Krstovski&Blei, arXiv, 2018]  ୅਺త౳Ձͳ਺ࣜϖΞ [Gangwar&Kani, TMLR, 2023]
    
     ೖग़ྗͷ਺஋తৼΔ෣͍ [Meidani+, ICLR, 2024] ਺ࣜຒΊࠐΈʹඞཁͳͦΕҎ֎ͷݕ౼߲໨  ܇࿅ɾධՁσʔλɿจݙ͔Βऩूɺ౳Ձू߹ͷੜ੒  σʔλͷදݱํ๏ɿςΩετܗࣜɺτʔΫφΠζɺߏ଄Խ  Ϟσϧͱֶशํ๏ɿXPSEWFDɺTFRTFRɺରরֶशʢDPOTUSBTUJWF
    MFBSOJOHʣ  ධՁํ๏ɺԼྲྀλεΫɺԠ༻ઌ ࠓ೔ͷΰʔϧ 21
23. • 4/*1
    4ZNCPMJD/VNFSJD
    *OUFHSBUFE
    1SFUSBJOJOH
    NPEFM 
    ਺ࣜʢTZNCPMJDʣͱ਺஋σʔλʢOVNFSJDʣͷ֤ϞμϦςΟΛຒΊࠐΈɺ ରরֶशͰରԠ෇͚Δ͜ͱͰɺৼΔ෣͍ʹجͮ͘ҙຯΛଊ͑Δɻ ؍఺̏ɿೖग़ྗͷ਺஋తৼΔ෣͍
    <.FJEBOJ
    *$-3
    > 22
24. • ࣄલֶश༻ɿҎԼͷखॱͰ໿ ສϖΞੜ੒ɻ  ԋࢉࢠ໦͔Βࣜ
    𝑓 Λ߹੒ʢ࢛ଇɾࡾ֯ɾࢦ਺ɾର਺ͳͲʣɻ  ࢦఆͨ͠ఆٛҬ͔Βαϯϓϧͨ͠ 𝑥
    ʹରͯ͠ 𝑦 = 𝑓(𝑥) Λܭࢉ͠ɺ਺஋ྻΛੜ੒ɻ  σʔλͷඪ४Խɺ
    ఆٛҬ֎΍ۃ୺ͳ஋Λ࣋ͭ 𝑦 ͷআ֎ɾ࠶ੜ੒ɻ  ಉ͡ 𝑓 ͷʢ਺ࣜɼ਺஋ྻʣͷ૊Λਖ਼ྫɺଞͷ 𝑓 ͱͷ૊Λෛྫͱͯ͠αϯϓϧ࡞੒ɻ • ϑΝΠϯνϡʔχϯά༻ɿԼྲྀλεΫ͝ͱʹ࡞੒ɻ " $SPTTNPEBM
    1SPQFSUZ
    1SFEJDUJPOʢֶश
     ɺධՁ
     
    αϯϓϧʣ • ೖྗɿ਺ࣜͱ਺஋σʔλɺग़ྗɿ਺஋తੑ࣭ʢྫɿ/PO$POWFYJUZ
    3BUJP
    /$3 ʣ /$3
    ͷग़ྗ͸ತੑͷఔ౓ʢ׬શʹತ
    ׬શʹԜʣ # 4ZNCPMJD
    3FHSFTTJPO
    43#FODI <-B
    $BWB
    /FVS*14
    > Λ࠾༻ • ೖྗɿ਺஋σʔλɺग़ྗɿ਺ࣜ 4/*1
    ͷߏஙʹ༻͍Δσʔλ 23
25. • ਺ࣜɿલஔʢ1PMJTIʣه๏ͰτʔΫϯྻԽɻ
    
    • ਺஋ྻɿ(𝑥, 𝑦) Λ਺஋σʔλͱͯ͠ೖྗʢ௕͞ɾॱংΛࣄલʹ౷Ұʣɻ • ఆ਺ɿ base-10 floating-point notation [Charton, arXiv, 2022; Kamienny+, NeurIPS, 2022] • ਺஋จࣈྻΛ༗ݶޠኮͰѻ͍ɺεέʔϧʹؤ݈ʹɻ • نଇɿ
    TJHO
    NBOUJTTB
    FYQPOFOU
    ͷτʔΫϯ΁෼ղɻ
    
    • ྫɿ
    ˠ
    <
    
    &>ɼ
    ˠ
    <
    
    &> • ਺ࣜɾ਺஋ྻதͷఆ਺Λ͍ͣΕ΋τʔΫϯ΁෼ղɻ 4/*1
    ͷߏஙʹ༻͍Δσʔλͷදݱํ๏ 24
26. • ਺ࣜɾ਺஋ྻΤϯίʔμɿ5SBOTGPSNFS
    ϕʔεͷΞʔΩςΫνϟ • ࣄલֶश໨ඪɿ $-*1
    ܕରরֶशɻ • ϑΝΠϯνϡʔχϯά໨ඪɿ ԼྲྀλεΫ͝ͱʹઃఆɻ 4/*1
    ͷֶशํ๏ 25
27. 4/*1
    ͷϑΝΠϯνϡʔχϯάɿ4ZNCPMJD
    3FHSFTTJPO 26
28. 4/*1
    ͷධՁɿϑΝΠϯνϡʔχϯάલޙͷຒΊࠐΈ 27
29. 4/*1
    ͷධՁɿ਺ࣜຒΊࠐΈͷ U4/&
    ϓϩοτ 28
30. ʮ਺ࣜͷҙຯʯΛϕΫτϧͰ࣋ͨͤΔํ๏Λɺ̏؍఺Ͱ੔ཧ͢Δɻ  पลςΩετ͔ΒͳΔจ຺ [Krstovski&Blei, arXiv, 2018]  ୅਺త౳Ձͳ਺ࣜϖΞ [Gangwar&Kani, TMLR, 2023]
    
     ೖग़ྗͷ਺஋తৼΔ෣͍ [Meidani+, ICLR, 2024] ਺ࣜຒΊࠐΈʹඞཁͳͦΕҎ֎ͷݕ౼߲໨  ܇࿅ɾධՁσʔλɿจݙ͔Βऩूɺ౳Ձू߹ͷੜ੒  σʔλͷදݱํ๏ɿςΩετܗࣜɺτʔΫφΠζɺߏ଄Խ  Ϟσϧͱֶशํ๏ɿXPSEWFDɺTFRTFRɺରরֶशʢDPOTUSBTUJWF
    MFBSOJOHʣ  ධՁํ๏ɺԼྲྀλεΫɺԠ༻ઌ ࠓ೔ͷΰʔϧ 29
31. ੡଄ϓϩηεɿੴ༉Խֶɺҩༀ඼ɺΤωϧΪʔ࢈ۀͳͲʢ㱠
    ૊Έཱͯʣ ϓϩηεͷ࠷దԽ΍ӡసͷͨΊʹϓϩηεͷਫ਼៛ͳϞσϧʢϞσϦϯάʣ͕ඞཁɻ ݚڀ঺հ c
    ੡଄ϓϩηεͷϞσϦϯά 30 ৠཹϓϩηεʢϘ΢ϞΞৠཹॴʣ ੴ༉Խֶϓϩηε https://www.ipc-ihi.co.jp/business/oil_plant/index.html
32. ݚڀ঺հ cϞσϧͷछྨ 31 ౷ܭϞσϧ ʢϒϥοΫϘοΫεϞσϧʣ σʔλͱ౷ܭతͳख๏ʹ ج͍ͮͯߏஙͨ͠Ϟσϧ • ઢܗճؼϞσϧ • Ψ΢εաఔճؼ • χϡʔϥϧωοτϫʔΫ • ϥϯμϜϑΥϨετ • ޯ഑ϒʔεςΟϯάܾఆ໦ ෺ཧϞσϧ ʢϗϫΠτϘοΫεϞσϧʣ ෺ཧతɾԽֶతݪཧʹ ج͍ͮͯߏஙͨ͠Ϟσϧ • ӡಈํఔࣜ • φϏΤ-ετʔΫεํఔࣜ • ೤఻ಋํఔࣜ • ൓Ԡ଎౓ํఔࣜ • ঢ়ଶํఔࣜ े෼ͳσʔλऔಘ͕೉͍͠ϓϩηε࢈ۀͰ͸ɺ෺ཧϞσϧߏங͕ඇৗʹॏཁ σʔλɾ౷ܭతͳख๏ͱ ՊֶݪཧΛ౷߹ͯ͠ߏஙͨ͠Ϟσϧ ϋΠϒϦουϞσϧ ʢάϨʔϘοΫεϞσϧʣ PINN (physics-informed neural network) ͰಘΒΕΔχϡʔϥϧωοτϫʔΫ SINDy (sparse identification for nonlinear dynamics)ͰಘΒΕΔ ඍ෼ํఔࣜ
33. ݚڀ঺հ c
    ෺ཧϞσϧߏஙͷޮ཰Խ 32 γϯϘϦοΫ ճؼ ݱ࣮ͷڍಈΛද͢ Պֶ๏ଇʹجͮ͘ ਺ࣜ܈ ෺ཧϞσϧ ୈҰݪཧϞσϧ モデル候補の 作成と検証 情報の 抽出と統合 対象プロセス 関連文献の収集 Ϟσϧߏஙʹඞཁͳ৘ใ ͷࣗಈநग़ɺಉٛੑ൑ఆɺ දه౷Ұ ʁ Ϟσϧީิͷ ࣗಈ࡞੒ɾݕূ 1/# ln ' O ࿈ଓ૧ܕ൓Ԡث จݙσʔλϕʔε͔Β ର৅ϓϩηεؔ࿈จݙ Λࣗಈऩूɺܗࣜ౷Ұ ๲େͳ࡞ۀͱߴ౓ͳ஌ࣝͱεΩϧ ࣌ؒΛ͔͚ͨࢼߦࡨޡ ैདྷख๏ɿख࡞ۀ "VUP1.P#ɿ"*ʹΑΔࣗಈԽ ߏங ߏங Ϟσϧީิ̍ Ϟσϧީิ̎ Ϟσϧީิ̏ 記号 定義 ! 温度 "! ⼊⼝流量 ⁝ ⁝ #" 反応速度 記号 定義 ! 温度 "! ⼊⼝流量 ⁝ ⁝ #" 反応速度 d" d# = %! − % ︙ ' = '! exp(− , -. ) −0" = '1" d1" d# = % " 1"! − 1" + 0" ֧፩ػ ݪྉ ੡඼ จݙσʔλى఺ͷϞσϦϯά จݙ৘ใ͔Β෺ཧϞσϧΛࣗಈߏங͢Δ࿮૊ΈΛఏҊɻ [Kato&Kano, Comput. Aided. Chem. Eng., 2022] ਺஋σʔλى఺ͷϞσϦϯά γϯϘϦοΫճؼ (Symbolic regression) ʹΑΓ࣌ܥྻσʔλ͔Β෺ཧϞσϧΛ ಋग़͢Δख๏ΛϨΦϩδʔ෼໺Ͱ࣮ূɻ [Sato+, J. Rheol., 2025] ࣌ؒ ೱ౓ ੡଄ϓϩηεΛର৅ͱͨ͠ɺ ෺ཧϞσϦϯάޮ཰Խख๏ ʢจݙ͔Βͷ৘ใநग़ख๏΍ ֫ಘͨ͠ϞσϧͷධՁख๏ͳͲʣ Λݚڀɻ
34. • ෳ਺ཁૉͷ଍͠߹Θͤ΍ҟͳΔϓϩηεͷ஌ࣝసҠʹΑΔϞσϧൃݟ ޻ֶʢϞσϦϯάʣʹ͓͚Δ਺ࣜදݱֶशͷԠ༻ 33 1 2 3 ੡଄ϓϩηε" ʢྫɿ੡ༀϓϩηεʣ ೤ͷ ӨڹΛ௥Ճ ୯७ͳϞσϧ ؀ڥมԽͷ ӨڹΛ௥Ճ ? Ճ೤ͱ؀ڥมԽͷ ӨڹΛ௥Ճ ੡଄ϓϩηε#ʢྫɿόΠΦϓϩηεʣ 1 ? 3 ೤ͷ ӨڹΛ௥Ճ ୯७ͳϞσϧ
35. ʮ਺ࣜͷҙຯʯΛϕΫτϧͰ࣋ͨͤΔํ๏Λɺ̏؍఺Ͱ੔ཧɻ  पลςΩετ͔ΒͳΔจ຺ [Krstovski&Blei, arXiv, 2018]  ୅਺త౳Ձͳ਺ࣜϖΞ [Gangwar&Kani, TMLR, 2023]
    
     ೖग़ྗͷ਺஋తৼΔ෣͍ [Meidani+, ICLR, 2024] ্هͷΑ͏ʹ·ͱΊ͕ͨɺάϥϑߏ଄Λར༻ͨ͠ݚڀͳͲ΋͋Δɻ • ޻ֶͰͷԠ༻ల๬Λ঺հ͠·͕ͨ͠ɺ୳ͤ͹·ͩ·ͩ͋Δ͸ͣʜ • ͨͩɺݱࡏ஍ͱԠ༻ʢଟม਺ɺه߸ͷදه༳Εɺෳ਺ͷ਺͕ࣜ̍୯Ґʣʹ͸Ϊϟοϓ͕͋Δɻ • ਺ࣜΛ׆༻ͨ͠ݚڀʹڵຯ͕͋ΔํɺͥͻҰॹʹٞ࿦͠·͠ΐ͏ʂ ·ͱΊ 34
36. [Valentino+, NAACL, 2024] Valentino, M. et al., “Multi-Operational Mathematical Derivations in Latent Space,” NAACL, 1446–1458, 2024. [Mansouri+, ICTIR, 2019] Mansouri, B. et al., “Tangent-CFT: An Embedding Model for Mathematical Formulas,” ICTIR, 11–18, 2019. [Krstovski&Blei, arXiv, 2018] Krstovski, K. and Blei, D.M., “Equation Embeddings,” arXiv, 2018. [Gangwar&Kani, TMLR, 2023] Gangwar, N. and Kani, N., “Semantic Representations of Mathematical Expressions in a Continuous Vector Space,” TMLR, 2023. [Meidani+, ICLR, 2024] Meidani, K. et al., “SNIP: Bridging mathematical symbolic and numeric realms with unified pre-training,” ICLR, 2024. [Bird+, 2009] Bird, S., Klein, E., & Loper, E., “Natural Language Processing with Python: Analyzing Text with the Natural Language Toolkit,” O’Reilly Media, Inc., 2009. [Rudolph+, NIPS, 2016] Rudolph, M. et al., “Exponential Family Embeddings,” NIPS, 2016. [Zanibbi+, SIGIR, 2016] Zanibbi, R. et al., “Multi-stage math formula search: Using appearance-based similarity metrics at scale,” SIGIR, 145–154, 2016. [La Cava+, NeurIPS, 2021] La Cava, W. et al., “Contemporary Symbolic Regression Methods and their Relative Performance,” NeurIPS, 2021. [Charton, arXiv, 2022] Charton, F., “Linear algebra with transformers.” arXiv, 2022. [Kamienny+, NeurIPS, 2022] Kamienny, P.-A. et al., “End-to-end Symbolic Regression with Transformers,” NeurIPS, 2022. [Kato&Kano, Comput. Aided Chem. Eng., 2022] Kato, S. and Kano, M., “Towards an automated physical model builder: CSTR case study,” Comput. Aided Chem. Eng., 1669–1674, 2022. [Sato+, J. Rheol., 2025] Sato, T. et al., “Rheo-SINDy: Finding a constitutive model from rheological data for complex fluids using sparse identification for nonlinear dynamics,” Journal of Rheology, 69, 15–34, 2025. 3FGFSFODFT 35