250902_収縮破壊の確率過程と
物理情報深層学習による破片サイズ分布推定

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September 12, 25

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1.

収縮破壊の確率過程と 物理情報深層学習による破片サイズ分布推定 Fragment size density estimation for shrinkage-induced fracture based on physics-informed deep learning 伊藤伸一 東京大学 地震研究所 S. Ito, Journal of the Physical Society of Japan, 94, 104100, 2025 arXiv:2507.11799 1

2.

Outline ・導入:収縮破壊の統計則 - 破片サイズ分布の動的スケーリング則 ・モデル:収縮破壊の確率過程 - とある積分微分方程式とその求解困難性 ・手法:物理情報深層学習による解法の提案 ・結果とまとめ:分布パラメータのベイズ推定 2

3.

Outline ・導入:収縮破壊の統計則 - 破片サイズ分布の動的スケーリング則 ・モデル:収縮破壊の確率過程 - とある積分微分方程式とその求解困難性 ・手法:物理情報深層学習による解法の提案 ・結果とまとめ:分布パラメータのベイズ推定 3

4.

導入 研究対象:厚さの薄い準2次元系の収縮亀裂パターン 実験系 e.g., 泥のひび割れ、溶岩亀裂、化粧品 … ・過去の環境変動は亀裂パターンの変化に埋め込まれている。 ・破片の統計的性質からダイナミクスに関連する情報を抽出したい。 4

5.
[beta]
破片サイズ分布の動的スケーリング則
破片

・乾燥亀裂パターンのインターバル撮影
・炭酸水酸化マグネシウム粉末+蒸留水
・初期粉水体積比 6:94,
・初期厚さ20mm, 室温25℃

・画像解析による破片サイズ抽出
生の破片サイズ分布

スケールした破片サイズ分布
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p(x; t)

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P (s; t)
時間発展

ρは全体質量から換算した

動的スケーリング則

P (s; t)ds ! p(x)dx
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x = s/ hsit

粉の体積分率
(系の経過時間に対応)

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hsit :平均破片サイズ
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5

6.

Outline ・導入:収縮破壊の統計則 - 破片サイズ分布の動的スケーリング則 ・モデル:収縮破壊の確率過程 - とある積分微分方程式とその求解困難性 ・手法:物理情報深層学習による解法の提案 ・結果とまとめ:分布パラメータのベイズ推定 6

7.
[beta]
破片サイズ分布時間発展のマスター方程式
マスター方程式
% →

"
P
(s;
t)
P
(s
; t)
@P
(s;
t)
"


=
+
ds ws! →s
 @t
T (s)
T (s" )
0
% 1


 ws! →s =
dr (rs" s)q(r)

・破片の寿命

<latexit sha1_base64="1T9GiWtGGS5uZUwV4pFZ2Auxnns=">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</latexit>

T (s) / s
<latexit sha1_base64="FCrcxRF7rm207dT/n8pXo/C59P0=">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</latexit>

・分割比 r 2 (0, 1) の分布
1
q(r) =
r↵ 1 (1
Beta(↵, ↵)
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<latexit sha1_base64="7HhLMXCv0JuCRkkFdB0SyqwvP7w=">AAAOy3ictZdBb9xEFMdfC5QSYJvCBYnLiiUokchqNiWlQkJqw4FWKVWbNmmrbBPZzmzWrNc2trPJrtkjF74AB04gcUCIK1+AC3fEoRJ8AMSxSFw48J83dmKnuzvugbXWnnnzfm/eezOeGduh58aJEI/PnH3u+RfOvXj+pbmXX3m1dmH+4mtbcXAQOXLTCbwgemBbsfRcX24mbuLJB2Ekrb7tyft27yPVfn8go9gN/HvJMJSP+ta+73Zcx0og2p3/+LPFaOnDersTWU7aGqftRB4l6ZpMrPFi2/LCrvWufiyN69FOqsvLrfFiazlaKtR35xuiuSKurK6u1p8utJqCfw3KfreDi+d+ojbtUUAOHVCfJPmUoOyRRTGubWqRoBCyR5RCFqHkcrukMc2BPYCWhIYFaQ/3fdS2M6mPurIZM+2gFw//CGSdFsRv4nvxRPwifhB/in+n2krZhvJliKetWRnuXvjyjbv/GKk+ngl1Tygj0YGHKgcxx7cwVTu3rHLTpyPuYW5mRhLYvsKZcNFDyBKVI0d7Nxh99eTuBxsL6TviW/EXsvONeCx+Rn78wd/Od3fkxtf/u/Xp0SaIsJyZaZo2spHnQtnzIT3kse9zznzMthTyCLU9aKpynkslS1k6znI5jbY540XWhn8pS8cGcjiRHBpJs8ez2dxjlaE610zEqESMKhB2ibANxB5i0Pod0Gocp8U2WZ4Y435Y8uehQXtr4shsGXu5NpG7ZuTWJnJrRm49IybR60a6yytizCtokY+Zv27kb+JSpGZttB9lOe4gxzeZn23h1gz+VgV+fQa/XoHv4ZpuoVfBwo0Z/I0KfMAt8VMjaPPcbhn5EeQRpNO9EBW8GHCvEncr86RYN5EWVuCc0mXT6pBg/y2+j2147FbgLJQ/PUV6vF/aaDH5GvG+GhzHmFuoEqdihyVuaBxZm3sLeYTUuOoTx4mF6xX6PCwRhxWITonoVCC6JaJbgRiUiIGBcHHl+1UAbZWPmLOu54zE6Cd8Xmjy/2Slj1HvV9jTJGr5jCq+QyltGOk97kPviX4lXT0P+pV08wj0PjVLP8lmTDkKhyNL6R7kO6fyleubIozxVL1v0yU+lxV32ZQavMrs0AqXV7Lc53L1vJQ9i/WdY5/Dyh6o86DHd1nwZpIHitAn/+4zezL7vGZBK+QVM2Tm9IlCcrbVuuZk3yBqtqawWee1IkDuOhyDbsnjibFbuLCpdlWLpfuQ611W6Xv0Nvxt4N6G5Sizp20W/dGtyv/glJ0e9+cjPyHv+PnbYSaWj5n8msP3Wf4RVp9e2Fppti43L995r3F1LftSO09v0lu0iBF4n65iBbtNm+j/R/qVfqc/ap/U4tqo9rlWPXsmY16n0q/2xX8LpBZl</latexit>

0

r)↵ 1

γ : 収縮履歴に関わるパラメータ、α : 割れ方に関わるパラメータ

3
例えば、q(r) / r (1
<latexit sha1_base64="yekrudQtYFTYcZsrDSscEQ25pBk=">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</latexit>

50

生の破片サイズ分布

1.2

40
30
20

t/τ = 5
t/τ = 10
t/τ = 20
t/τ = 40
t/τ = 80

p(x; t)

10
0
0

スケールした破片サイズ分布

<latexit sha1_base64="6t2VdqwUZMZuOGq6M/OtyF4D9VE=">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</latexit>

PDF[S/ ⟨S⟩]

PDF[S/θ]

<latexit sha1_base64="gYN+FBcAKZFOPnY/tbr3OTQtbBs=">AAADY3icxVJPSxtBFP8l26qNrYm2hYIUQkMkgoRZES2FgthLj/5LFBIJs+uYLM7+YXcSa5d+gX6BHnpS6KH1Y/TSL9CDH6AH8WjBiwffTraIBu3RWXbmvd97vzdv3ntWIJ1IMXacyRoPHg4NjzzKjT5+MpYvjE/UI78b2qJm+9IPNy0eCel4oqYcJcVmEAruWlJsWLvvEvtGT4SR43vraj8QWy5ve86OY3NFUKvwvNLkMujwmWabuy6ffluZmzGnW4USqzK9ioOCmQolpGvZH89Moolt+LDRhQsBD4pkCY6IvgZMMASEbSEmLCTJ0XaBT8gRt0tegjw4obu0t0lrpKhHehIz0mybbpH0h8Qsosx+s+/sjP1iR+yEXdwaK9Yxklz26bT6XBG08p9frJ3/l+XSqdC5Yt2Zs8IOXutcHco90EjyCrvP7338crb2ZrUcT7FDdkr5H7Bj9pNe4PX+2t9WxOrXe4oeU11c3Y/yrR4KH+h+X/c3SnvnkWVP98XVlfJoEmLCQ9K2ySeR/1UwwWKNJtwcTZp5c64Ghfps1Zyvzq/MlRaX0pkbwSReoUJztYBFvMcyapRBjAP8wFH2jzFqTBjP+q7ZTMp5imvLeHkJkDDGNA==</latexit>

t/τ = 5
t/τ = 10
t/τ = 20
t/τ = 40
t/τ = 80

<latexit sha1_base64="6j7Gs6fqahNveonlOUQ4535x+8I=">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</latexit>

P (s; t)

r)3 , T (s) = ⌧ (s/s0 ) 1 , i.e., (↵, ) = (4, 1) を設定して数値計算すると…

0.8
0.4
0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0

1

Area S/θ

2

3

4

Scaled area S/ ⟨S⟩

動的スケーリング則

P (s; t)ds ! p(x)dx
<latexit sha1_base64="RrNgwtZ3K6fFs62D30kVVAQjvno=">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</latexit>

x = s/ hsit
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hsit :平均破片サイズ
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7

8.

積分微分方程式の導出 動的スケーリング下での不変分布の形が知りたいため、 先ほどのマスター方程式を x = s/ hsit で変数変換し、長時間極限を取ると以下の積分微分方程式を得る。 <latexit sha1_base64="/kiI/kk/OfQC1ETcbUfPGS+MBpc=">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</latexit> 積分微分方程式 <latexit sha1_base64="dy7cHIQ7Z2iaXQePgM+auf5zkkg=">AAAPXnictZdPb9xEFMBnC4Q20G4Kl0pcLJZURSir2ZSUqhJSGi6tUlVt2qStsk1ke8e7Zv0P29ns2vIX4Atw4NRKPVTc+QJc+AIcyokr4likXjjw5o0nsbdrjzmw1tozb97vzXtvxjNjI3DsKKb0VevMO+++t/T+2XPLH3x4/kJ75eJHe5F/FJps1/QdP3xs6BFzbI/txnbssMdByHTXcNgjY/wNb380YWFk+97DeBawp64+9GzLNvUYRIcrv6/1rVA3016W9oe66+qZqA+ydDDN+g6z4ivaNBCFaT+0h6P4c00+1zTRIJhplgIumg6kuXn0C61ve/EhKIOK7VnxTPaYZGmSfTdnMJEGi/KkpqMkl39ND1c6tLtOr29sbGhvF3pdir8OyX/3/ItLP5M+GRCfmOSIuIQRj8RQdohOIrj2SY9QEoDsKUlBFkLJxnZGMrIM7BFoMdDQQTqG+xBq+7nUgzq3GSFtQi8O/EMgNbJKf6Mv6Wv6K/2J/kn/qbSVog3uywyehmBZcNj+/tKDN0rKhWdMRqeUkrDAQ56DCONbrdSWlnluXDLFHpZrMxKD7euYCRt6CFDCc2QK7ybJD68f3NhZTS/T5/QvyM4z+or+AvnxJn+bL+6znR//d+vV0cYQYTkzVZoGZEPmgtvzQHqMY+9izjyYbSnIQ6gNQJOXZS65LEVplueyijYw40XWAP9SlGYKcraQnClJtcf1rPSYZ0jDmopISkTSgDBKhKEgBhCD0LeA5uNYFdtieayM+0nJnycK7b2FI7On7OXmQu6mkttayG0pue2cWERvK+kRrogRrqBFPkL+lpK/AxcnBWtA+zTPsQU5voN8vYW7NfzdBvx2Db/dgB/DVW1h3MDC7Rr+dgPex5borRE0cG73lHwC8hCk1V7QBl5MsFcGdz33pFhXkTqswJISZdXqEMP+W3wf++Cx3YDTofztHOngfmlAi8rXEPdV/yRGaaFJnJydlbiZcmQN7C3AEeLjKk4cpxZuNejzuEQcNyCsEmE1IEYlYtSAmJSIiYKw4ZL7lQ/aPB8RZl3MGQajH+N5oYv/05U+grrbYE9jUJMzqvgOpWRHSQ+wD7Eneo10xTxwG+nKCMQ+Vacf5zOmHIWJkaXkIcgP5vIl9VURRvDkve+Tq3guK+6yKengKnNA1rG8nudeyvnzav4s1g9OfA4ae8DPgw7eWcGbRR5wQpz8R//Zk/rzmg5aAa6YATLzJwqG2ebrmpl/g/DZmoJNDdcKH3JnYQyiRcYTwW5hg02+q+ooHYJc7LJc3yGfgb8duPfBcpjbEzaL/ohW7r8/Z2eM/XmQnwB3fPl2qIm1E0Zey/B9Jj/CtOrC3nq3d6177f6Xnc2t/EvtLPmEfEquwAh8RTZhBbtHdonZ2mxZLb8VXPijvdQ+324L1TOtnPmYlH7tS/8CUoNP2Q==</latexit> 1 d (xp (x)) dx <latexit sha1_base64="VuUOnwSscq7+dJfT9RzeZ72p6Kg=">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</latexit> s.t. Z 1 0 dz p(z) = 1, 確率の正規化 Z 1 0 ⇣x⌘ a p (x) + Z 1 x dz ⇣ x ⌘ ⇣ z ⌘ p (z) = 0 q z z a r↵ 1 (1 r)↵ 1 dz zp(z) = 1, and q(r) = Beta(↵, ↵) 平均の正規化 求めたいもの (α, γ) が与えられた時の <latexit sha1_base64="VNVKgdHf1UM4FnJ3YD7wH7zUvoo=">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</latexit> p(x):不変分布 a :スケール因子 <latexit sha1_base64="S1eQDYogNT3VsKhE4zZFN7vaNGY=">AAAOhHictZfNbtRIEMeLjw0huwxfFyQuo50N4gCjTkJCxAEl4QIKQhBIACUE2U5PxozHtmxnkomVF0Bc2T3sCaQ9IO68AJd9gT3wCKs9shKXPey/y3Zih/G0OTDW2N3V9auuqm53t03fscNIiE9Hjh47/sPIidGTYz/+dKp2+szZcyuhtxVYctnyHC94YhqhdGxXLkd25MgnfiCNrunIx2bnlmp/3JNBaHvuo6jvy2ddY9O1W7ZlRBAtGWPPzzREc1LMTk9P178uTDQF/xqU/u57Z0c+0BptkEcWbVGXJLkUoeyQQSGuVZogQT5kzyiGLEDJ5nZJezQGdgtaEhoGpB3cN1FbTaUu6spmyLSFXhz8A5B1Ghd/iXfis/hTvBd/i/9KbcVsQ/nSx9NMWOk/P/3ywsMvWqqLZ0TtA0pLtOChykHI8Y2XameWVW66tMM9jA3NSATbs5wJGz34LFE5shLveru/fX54Y2k8viTein+QnTfik/iI/Li9f60/Hsil37+79fJoI0RYzEyZpolsZLlQ9lxIt3nsu5wzF7MthjxAbQOaqpzlUslilu6luSyjTc54njXhX8zSPQ3ZH0j2taTe4+Fs5rHKUJ1rOmK3QOxWIMwCYWqIDcSQ6LdAq3Esi22wPNLG/bTgz1ON9srAkVnR9jI/kJvXcgsDuQUtt5gSg+hFLd3mFTHkFTTPh8zf1vJ3cSkyYU2076Q5biHHd5kfbuHeEP5eBX5xCL9Yge/gKrfQqWDhzhD+TgXe45bwqxE0eW5PaPldyANIy70QFbzoca8SdyP1JF/XkQZW4IxKyrrVIcL+m38f1+CxXYEzUH5xiHR4vzTRovM14H3V248xs1AlTsX2C1xfO7Im9+bzCKlxTU4cBxZuV+hzu0BsVyBaBaJVgWgXiHYFolcgehrCxpXtVx60VT5CznoyZyRGP+LzQpP/Byt9iHq3wp4mUctmVP4dimlJS29wH8me6FbSTeZBt5JuFkGyTw3Tj9IZU4zC4shiegT5+qF8Zfq6CEM8Ve+rNMXnsvwuG1ODV5l1muTyZJr7TK6eU+kzX1/f99mv7IE6Dzp8lzlvBnmgiOTk3/5mT4af1wxo+bxi+swcPlFIzrZa16z0G0TN1hg267xWeMhdi2NIWrJ4QuwWNmyqXdVg6SbkyS6r9B36Bf42cF+D5SC1l9jM+5O0Kv+9Q3Y63J+L/Pi842dvh564us9kl/o+yz7C6uWFlcnmxExz5sG1xtxC+qU2ShfpZ7qMEbhOc1jB7tMy+m/RK3pNv9ZGaldqU7XpRPXokZQ5T4Vf7eb/puf6fw==</latexit> ※ いくつかのパラメータセットについては p(x) の厳密解が知られている。 初等関数による系統化された 解表示は無さそう。 8

9.

数値計算の困難さ 積分微分方程式 <latexit sha1_base64="dy7cHIQ7Z2iaXQePgM+auf5zkkg=">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</latexit> 1 d (xp (x)) dx <latexit sha1_base64="VuUOnwSscq7+dJfT9RzeZ72p6Kg=">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</latexit> s.t. Z 1 0 dz p(z) = 1, 確率の正規化 Z 1 0 ⇣x⌘ a p (x) + Z 1 x dz ⇣ x ⌘ ⇣ z ⌘ p (z) = 0 q z z a 求めたいもの r↵ 1 (1 r)↵ 1 dz zp(z) = 1, and q(r) = Beta(↵, ↵) 平均の正規化 (α, γ) が与えられた時の <latexit sha1_base64="VNVKgdHf1UM4FnJ3YD7wH7zUvoo=">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</latexit> p(x):不変分布 a :スケール因子 <latexit sha1_base64="S1eQDYogNT3VsKhE4zZFN7vaNGY=">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</latexit> 有限差分法や有限要素法などの離散解法で解く際には... 困難さ① :無限遠方までの積分を含み、その計算精度は分布全体の精度に大きく影響する。 困難さ② :左辺第1項と第2項の効果より非常に硬く不安定化しやすい。 困難さ③ :スケール因子 a の値は数値計算における事実上の x の範囲を決めるが p(x) を求めて初めて a は計算される。 離散化設計の試行錯誤が必要。 決め打ちのパラメータセットで1回の計算ならば問題にならないが、 パラメータの最尤推定・ベイズ推定など、多数回の p(x) の評価が必要な場面では致命的。 ニューラルネットの力を借りてこれら計算困難さの解消を試みる。 9

10.

Outline ・導入:収縮破壊の統計則 - 破片サイズ分布の動的スケーリング則 ・モデル:収縮破壊の確率過程 - とある積分微分方程式とその求解困難性 ・手法:物理情報深層学習による解法の提案 ・結果とまとめ:分布パラメータのベイズ推定 10

11.

物理情報深層学習による求解の定式化 物理情報深層学習 = 物理モデル + ニューラルネットによる解近似 Physics-informed neural networks (PINNs) による定式化 <latexit sha1_base64="kbUvetmXurkEFNTmdaqrM+gM0vM=">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</latexit> p✓ (x; ↵, ), a (↵, ): p, a のNN表現 ✓, :NNのパラメータ <latexit sha1_base64="6lQrVdiMGjYIodYaj/GuGC4rvco=">AAAOjXictZdBb9RGFMcftNCQtpukXJB6WXUJ6gFWk6QJUVVVIRwKCopCIIEoSyLbmc2667Ut29mwsfIFekc9VGrVSj1UvfMFeukX6IGPgDiCxKWH/ufZTuyw3jGHrrX2zJv3e/Pem/HM2PQdO4yEeHHu/AcfXrj40dil8Y8/+bQ2MTn12WboHQSW3LA8xwsem0YoHduVG5EdOfKxH0ijZzrykdm9rdof9WUQ2p77MBr48knP2Hfttm0ZEUTbragjI+N6y+/Yu5MN0ZwVi/Pz8/V3CzNNwb8Gpb81b+ric2rRHnlk0QH1SJJLEcoOGRTi2qYZEuRD9oRiyAKUbG6XdEzjYA+gJaFhQNrFfR+17VTqoq5shkxb6MXBPwBZp2nxj/hDvBZ/iz/FS/Fvqa2YbShfBniaCSv93Ykfrjx4q6V6eEbUOaW0RBseqhyEHN90qXZmWeWmR0+5h/GRGYlge5EzYaMHnyUqR1biXf/ox9cPvl6fjq+J38QrZOdX8UL8hfy4/TfW7/fl+k//u/XyaCNEWMxMmaaJbGS5UPZcSA957HucMxezLYY8QG0Pmqqc5VLJYpYep7kso03OeJ414V/M0mMNORhKDrSk3uPRbOaxylCdazriqEAcVSDMAmFqiD3EkOi3QatxLIttuDzSxr1V8GdLo705dGQ2tb3cGsrd0nLLQ7llLbeSEsPoFS3d4RUx5BU0z4fM39Hy93ApMmFNtD9Nc9xGju8xP9rC6gh+tQK/MoJfqcB3cZVb6FawcHcEf7cC73FL+M4Imjy3Z7T8EeQBpOVeiApe9LlXibuRepKv60gDK3BGJWXd6hBh/82/jy14bFfgDJS/P0M6vF+aaNH5GvC+6p3EmFmoEqdiBwVuoB1Zk3vzeYTUuCYnjlMLdyr0eVggDisQ7QLRrkB0CkSnAtEvEH0NYePK9isP2iofIWc9mTMSox/xeaHJ/9OVPkS9V2FPk6hlMyr/DsW0rqX3uI9kT3Qr6SbzoFdJN4sg2adG6UfpjClGYXFkMT2EfOdMvjJ9XYQhnqr3bZrjc1l+l42pwavMDs1yeTbNfSZXz7n0ma/vnPjsV/ZAnQcdvsucN8M8UERy8u+8tyejz2sGtHxeMX1mzp4oJGdbrWtW+g2iZmsMm3VeKzzkrs0xJC1ZPCF2Cxs21a5qsHQf8mSXVfoOXYW/DdxbsByk9hKbeX+SVuW/d8ZOl/tzkR+fd/zs7dATN06Y7BrH91n2EVYvL2zONmcWmgv3v2osLadfamP0OX1BX2IEbtISVrA12uAsP6Of6ZfaRG2+9k3t20T1/LmUuUyFX+27/wBTJ/6p</latexit> NN x <latexit sha1_base64="FzfrPsmaUfMWIopPCA8Wid3G3t4=">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</latexit> <latexit sha1_base64="912DJY7pd4iiFi4fahZX+NyE9V4=">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</latexit> p✓ (x; ↵, ) ↵ 確率正規化条件 の2乗 平均正規化条件 の2乗 <latexit sha1_base64="VVlOSGYdG+LIW2+tqoT7fyxBqNQ=">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</latexit> <latexit sha1_base64="wYxZuh7wXqcwZJ+e/J+nKUKfyRY=">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</latexit> (α, γ) 所与の 損失関数 <latexit sha1_base64="OGL1NvkZb33o4kaN3Q9MONv3dXY=">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</latexit> l(✓, ; ↵, ) (α, γ) 空間で <latexit sha1_base64="NyYdWcvLX4WCPGz3/GD1/Bcc2JM=">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</latexit> 重み付け積分 <latexit sha1_base64="IejVbVIYYqoabkRSYSaz3L19kbQ=">AAAOnHictZfPb9RGFMcflFK6bZdQLkiVqhXbVFSC1SQ0gHqC0ANoKYJAFlCWRLYzm3XXv2Q7GzZW+AP4B3rgBGoPVdUe+w/00n+gB/6Eqkcq9dJDv/NsJ3ZY75hD11p75s37vHnvzXhmbAaOHcVCvDpy9J1j7x5/78T7jQ8+/Kh5cu7Ux73I3w4tuWr5jh8+NI1IOrYnV2M7duTDIJSGazrygTm6rtofjGUY2b53P54E8rFrbHn2wLaMGKKNuTMNYyPpB0N771zfcIKhcb6/Zbiu8cXGXFt0FsWVpaWl1puFhY7gX5uy3x3/1PFfqU+b5JNF2+SSJI9ilB0yKMK1RgskKIDsMSWQhSjZ3C5pjxpgt6EloWFAOsJ9C7W1TOqhrmxGTFvoxcE/BNmiefGH+FG8Fr+Ln8Sf4t9KWwnbUL5M8DRTVgYbJ5+dufePlnLxjGl4QGmJATxUOYg4vvlK7dyyyo1LT7iHxsyMxLB9hTNho4eAJSpHVurdePe71/e+WplPPhcvxV/IzgvxSvyG/Hjjv60f7sqV5/+79epoY0RYzkyVpols5LlQ9jxId3jsXc6Zh9mWQB6itglNVc5zqWQJS/eyXFbRJme8yJrwL2HpnoacTCUnWlLv8Ww291hlqMU1HbFbInZrEGaJMDXEJmJI9Qeg1ThWxTZdHmvjflTy55FGuzd1ZHraXq5N5a5pueWp3LKW62bENLqrpYe8Ika8ghb5iPkbWv4WLkWmrIn2J1mOB8jxLeZnW7g9g79dg+/O4Ls1+BGuagujGhZuzuBv1uB9boneGEGT5/aClt+FPIS02gtRw4sx9ypxNzJPinUdaWAFzqm0rFsdYuy/xfexD4/tGpyB8reHSIf3SxMtOl9D3lf9/RhzC3XiVOykxE20I2tybwGPkBrX9MRxYOFGjT53SsRODWJQIgY1iGGJGNYgxiVirCFsXPl+5UNb5SPirKdzRmL0Yz4vdPh/sNJHqLs19jSJWj6jiu9QQitaepP7SPdEr5ZuOg/cWrp5BOk+NUs/zmZMOQqLI0voPuTrh/KV6+sijPBUva/RRT6XFXfZhNq8yqzTIpcXs9zncvW8mD2L9fV9n4PaHqjzoMN3WfBmmgeKSE/+w7f2ZPZ5zYBWwCtmwMzhE4XkbKt1zcq+QdRsTWCzxWuFj9wNOIa0JY8nwm5hw6baVQ2WbkGe7rJK36HP4G8b9z4sh5m91GbRn7RV+e8fsjPi/jzkJ+AdP3879MSFfSa/Gvg+yz/CWtWF3mJn4VLn0t0v21eXsy+1E/QJnaVzGIHLdBUr2B1aRf9P6Xv6mX5pftr8utltfpOqHj2SMaep9Gv2/gMM4QPN</latexit> a (↵, ) 積分微分方程式 のL2ノルム2乗 訓練損失関数 <latexit sha1_base64="pGlUG05EXLK3XJM68iwei32DWAg=">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</latexit> L(✓, ) 誤差逆伝播 11

12.

NN構造の工夫 NNの設計 NNp (x, ↵, ; ✓), NNa (↵, ; ) をニューラルネットワークとする。 ✓, <latexit sha1_base64="VB1AoqZi66FoFlxtlZDoNWASonE=">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</latexit> <latexit sha1_base64="6lQrVdiMGjYIodYaj/GuGC4rvco=">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</latexit> ( :NNのパラメータ <latexit sha1_base64="usw+fXFmZHUPDEQrAVbqMy8zpr8=">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</latexit> p✓ (x; ↵, ) = exp [(↵ a (↵, ) = exp [(↵ 1)NNp (x, ↵, ; ✓)] p0 (x; ↵, ) 1)NNa (↵, ; )] c (↵, ) ただし、p0, c は x → 0,∞ での理論的な漸近挙動を再現する「近似解」とそのスケール因子 ✓ ◆↵ 1  ✓ ◆ x x (↵/ ) p0 (x; ↵, ) = exp , where c (↵, ) = c (↵, ) (↵/ ) c (↵, ) c (↵, ) ((↵ + 1)/ ) <latexit sha1_base64="m/9ap9bUfpNtUZnJaG2QfqCAIPM=">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</latexit> 工夫のポイント ・exp にNNを通すことで p, a ともに非負性を担保 積分微分方程式の積分項 Z 1 <latexit sha1_base64="QBlxXQvAhxZwxW4NLJgEkK3HOLY=">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</latexit> x dz ⇣ x ⌘ ⇣ z ⌘ p (z) q z z a ・p0 を乗算することで、x → ∞ での p の上昇を抑え積分の発散を回避 ・「p0, c からの補正」という形式にすることで α = 1での厳密解を保証 & 最適化の収束スピードを向上 12

13.

計算設定 • 乱択フーリエ特徴+スキップ接続+Swish活性化 • 2重指数関数積分変換による数値積分の高精度化 • L-BFGS 法+Levenberg–-Marquardt 法の ハイブリッド戦略による最適化 • (α, γ) 空間のlog-uniform 分布による重み付け積分 • 比較用の有限差分法 • 1次風上差分+台形積分で離散化、 擬似時間発展を exponential integrator で解く 13

14.

Outline ・導入:収縮破壊の統計則 - 破片サイズ分布の動的スケーリング則 ・モデル:収縮破壊の確率過程 - とある積分微分方程式とその求解困難性 ・手法:物理情報深層学習による解法の提案 ・結果とまとめ:分布パラメータのベイズ推定 14

15.

最適化の様子 γ 厳密解 有限差分法解 PINN解 α 15

16.

°10 °10 log( °15 信頼性のある求解 1 °20 °15 °10 °15 °5 0 °20 log[° log(PDF)] log(x) あまり硬くないパラメータセット(厳密解既知) (Æ, ∞) = (2, 2) 1.00 PDF 0.75 p(x) 0.50 2 1.00 6 °15 0.75 4 0.50 2 0 6 FDでは理論的な漸近形に 全く合わない解に収束する 4 6 2 0.25 0 0 0 0.00 0 1 1 2 3 2log(x) 3 4 0.00 4 0 0 x log(PDF) °5 log(x) 非常に硬いパラメータセット(厳密解未知) (Æ, ∞) = (3, 6) 2 0.25 0 °10 FD, k = 26 FD, k = 210 °5 1 1 2 2log(x) 3 3 4 4 x 0 °5 FD, k = 213 PINN Exact k:有限差分の格子点数 °10 °10 • 有限差分(FD)による解法では、どんなに格子点を細かくしても 1 2 °15 物理的に正しい解に到達できない場合がある(ゴースト解)。 °15 °20 • °20 ゴースト解の存在は、有限差分の解を訓練データとしたNN回帰の危うさと °15 °10 °5 0 °15 °10 °5 DF)] PINNにより物理情報を入れることの有用性を示唆している。 log(x) log(x) 6 6 2 0 16

17.

計算速度比較 有限差分法 1.9 sec/call x 1,000万回 〜 5,300時間 0.00076 sec/call x 1,000万回 〜 2時間 PINN (最適化にかかった時間 〜 6時間) 約2,500倍の高速化 NVIDIA H200 SXM 1GPUによる比較 パラメータのベイズ推定 MCMCのテスト パラメータの事後分布 擬似データ <latexit sha1_base64="wYxZuh7wXqcwZJ+e/J+nKUKfyRY=">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</latexit> ↵ PINNモデルは計算が軽く、 多量回の分布評価が可能 <latexit sha1_base64="hxA17jnzhd1WcAyIqj/vH9jQCSU=">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</latexit> 17

18.

まとめと今後の展開 まとめ ・収縮破壊確率過程を解く物理情報深層学習の方法を提案 ・既存方法より約2,500倍の高速化と求解の安定化を実現 ・MCMC計算に非常に有効、高速なベイズ推定 今後の展開 ・確率過程の多くは積分微分方程式で記述 ・凝集過程や地震など様々な系への展開 ・場に依存するパラメータを含む系 (今回の場合で言えばT(s)やq(r) の具体形を与えないなど) ・作用素学習(NO)への発展 凝集過程 地震のモデル → データ駆動な物理素過程の抽出 19

19.

ありがとうございました 20

20.

最適化 ・L-BFGS法 + Levenberg–-Marquardt (LM)法 のハイブリッド戦略 21

21.

テスト損失 PINN用テスト損失 PINNの解 FD用テスト損失 厳密解 PINNテスト損失の履歴 FDの解 厳密解 PINN vs FD テスト損失の比較 22