---
title: 231121_Bayesian inference of grain growth prediction via multi-phase-field models
tags: 
author: [Shin-ichi Ito](https://www.docswell.com/user/7488391717)
site: [Docswell](https://www.docswell.com/)
thumbnail: https://bcdn.docswell.com/page/87DDNLZN7G.jpg?width=480
description: 231121_Bayesian inference of grain growth prediction via multi-phase-field models by Shin-ichi Ito
published: November 21, 23
canonical: https://www.docswell.com/s/7488391717/5YWLV7-2023-11-21-151700
---
# Page. 1

![Page Image](https://bcdn.docswell.com/page/87DDNLZN7G.jpg)

Bayesian inference of grain growth prediction
via multi-phase- eld models
伊藤伸一
東京大学地震研究所
fi
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
01/35


# Page. 2

![Page Image](https://bcdn.docswell.com/page/VJPX52DN78.jpg)

自己紹介
伊藤 伸一
2015.3 大阪大学 大学院理学研究科 宇宙地球科学専攻 博士（理学）
2015.4—2018.8 東京大学 地震研究所 特任研究員
2018.8—現在 東京大学 地震研究所 助教
2018.12—現在 東京大学 大学院情報理工学系研究科 助教（兼務）
対象： 破壊・亀裂進展, 摩擦, 地震,
粒成長, パターン形成, …
道具： データ同化, Adjoint法, Neural network,
有効モデル化, 確率過程, …
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
02/35


# Page. 3

![Page Image](https://bcdn.docswell.com/page/2EVLPD1YJQ.jpg)

粒成長予測とデータ科学
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
03/35


# Page. 4

![Page Image](https://bcdn.docswell.com/page/57GXPY2WEL.jpg)

拙著
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
04/35


# Page. 5

![Page Image](https://bcdn.docswell.com/page/4EQ82MPQJP.jpg)

金属粒成長
Fe0.27C-0.18Si-0.45Mn-0.014P-0.003S
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
05/35


# Page. 6

![Page Image](https://bcdn.docswell.com/page/KJ42LVYYE1.jpg)

金属粒成長
Fe0.27C-0.18Si-0.45Mn-0.014P-0.003S
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
06/35


# Page. 7

![Page Image](https://bcdn.docswell.com/page/LE1ML36N7G.jpg)

金属粒成長
核成長過程
結晶方向の食い違いによって粒が形成される。
材料としての性能は粒構造に大きく依存する。
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
07/35


# Page. 8

![Page Image](https://bcdn.docswell.com/page/GEW3VP4ME2.jpg)

金属粒成長
界面移動過程（高温状態で保持）
Grain ID
結晶方向の食い違いによって粒が形成される。
材料としての性能は粒構造に大きく依存する。
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
08/35


# Page. 9

![Page Image](https://bcdn.docswell.com/page/47ZV4K2ME3.jpg)

粒成長予測×データ科学
時間発展
発展
?
?
予測
推定
過去
現在（観測）
未来
現在得られた観測が実現するために過去どうだったのか？
現在から発展させた場合に将来どうなっているべきたっだのか？
現在の観測データだけからこれらを推定・予測できないだろうか？
「職人さんの匠の業を定量化」が究極の目標
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
10/35


# Page. 10

![Page Image](https://bcdn.docswell.com/page/YJ653QV57V.jpg)

Multi-phase- eld models for grain growth
fi
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
11/35


# Page. 11

![Page Image](https://bcdn.docswell.com/page/GJ5YLRDGJ4.jpg)

Multi-phase- eld models for grain growth
N個の粒が領域Ωに存在している状況を考える。
粒 i の存在確率を場の関数
&lt;latexit sha1_base64=&quot;V9XXIzxntnB6Et1pjtFZGXO9OGw=&quot;&gt;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&lt;/latexit&gt;
i (x, t)
2 G で表現する。
G = {y 2 R | 0  y  1}
&lt;latexit sha1_base64=&quot;sdcLeuR++Q/5KYDxMj7aOh/RkA8=&quot;&gt;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&lt;/latexit&gt;
&lt;latexit sha1_base64=&quot;rCN2qGJyaiuRMHUdSGlAQwQf97E=&quot;&gt;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&lt;/latexit&gt;
F
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系のエネルギー勾配に従って時間変化
i
エネルギー関数のデザイン
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&lt;latexit sha1_base64=&quot;DgqxC9Qz4vy130NzIOyRNFMNiGE=&quot;&gt;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&lt;/latexit&gt;
+
&lt;latexit sha1_base64=&quot;jBgCqLqAd3mBxVZfIuN7cdRbGV0=&quot;&gt;AAAWiXicldjNbxtFFADw1/JVXKApSFYlLqu6RYlQrHFFSlRUqU2afjffX223WLvrsT3NfnV37Ti23D+APwAOnEDigPpncOHKgUPhL0Aci8SFA2/HbtaNn+0XR7FnZ+f33sx6kt0ZO3RVnAjx8sTJt95+5933Tr2fO/3Bhx+dmTr78XYcNCJHbjmBG0S7thVLV/lyK1GJK3fDSFqe7code28xPb/TlFGsAn8zOQjlE8+q+aqqHCvBqvLU0ueGqfyk3DFXPFmzuhWz2TK/MkwXQ1QsAwuymkyXZs244ZU76mqp+3VnuWuGdYVHXcOMVK2ezJSnCqJ4SczPzc0Zw4VSUehXAfqv1eBs7lswoQIBONAADyT4kGDZBQti/HkMJRAQYt0T6GBdhCWlz0voQg5tA1tJbGFh7R6+1/Docb/Wx+M0Zqy1g1lc/I1QGnBR/C5+Fq/Er+KF+Ev8NzJWR8dI+3KAn3bPyrB85ptzG/9OVAa+W9jneubG9jqBKszr3irsfahr0nE4Pd9sf/dq48r6xc5n4kfxN47gB/FS/IJj8Jv/OD+tyfXvx/QogRZGD/QVjt/oT2p8bLOvr5Gn++zjt9LB+vSa13RNC01a83pEAWZIjyOsMfrtnh+2NPFaKzxS/WycHOmYeTl6LYdz5PRskjhOE5r46cA0FHAOzfSj2hixpcu2ng2Vw0wGXNAtL2Cc4UgWy1uHetDaLGuT1mFZh7QVlq2QVrKsJG2VZaukrbFsjbR1lq2TVrGsIu1Tln1K2j2W3SOty7IuaT2W9Ujrs6xP2oBlA9KGLBuS9hnLPiNtxLIRaWOWjUmbsGxC2gbLNkjbZNkmafdZdp+0LZZtkfaAZQ9I22bZ9og7w3WWvk5mXmDZBdIusuwiaW+w7A3SLrHsEmlvsuxN0t5i2Vukvc2yt0l7h2XvkPYuy94l7T2WvUfa+yx7n7QPWPYBaZdZdpm0Kyy7QtpVll0l7RrLrpF2nWXXSbvBshuk3WTZTdJusewWabdZdpu0Oyy7Q9pdlt0l7UOWfUjaRyz7aMSdoY2liPnEI0Y8LaWr1xpz9WEOtKefRVR/JceLlrWnn07qWE7Y0bL29LOspWNU2PEGBX1XP07v2mP61mKuDkzdkl5N8Xsyuh8e86nO1C3p1VV6zfgzIGtPjyrUcyTdU+GtCcwjhv7fls2rXpwqWnOgnvpfms3sQZPVZ3+fo/Yg0h2GQJ/tHh5P42cd+9vbSUnXxEUowTyWPGylxp6dgfF7Hmn8BM8EE/KJgXxURnHMjHX8lEdGORx39kjmSec5udO9p4befeNnbo44f/kYY65ilubE7zWLOZzzuBnTGd4am292Qkb6+ubKU4XXm6jG6ML2pWLpcnFu7YvCtYX+Tusp+BTOYz9K8CVcwyfdVbzDO/ACfoM/4M/86XwpP5+/0mt68kTffAJvvPKL/wN3RmC8&lt;/latexit&gt;
+
dx
Z
⌦
界面が存在することによるエネルギー
0
1
n
X CC
BBB
CCC 粘弾塑性変形など
dx BB@化学組成変化,
1
iA
&lt;latexit sha1_base64=&quot;XysoovbplG33Pg+Rb4I01vm1akg=&quot;&gt;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&lt;/latexit&gt;
⌦
Z
⌦
BBB
dx BB@1
i=1
N
X
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&gt;tixetal/&lt;==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&gt;&quot;=gka1mv10I4bR+gP33GlpbvoosyX&quot;=46esab_1ahs tixetal&lt;
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F=
Grain ID
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1
CCC
CA
iC
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&lt;latexit sha1_base64=&quot;UssoEJ4RgljaM3kqRQyiMW8lIVs=&quot;&gt;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&lt;/latexit&gt;
1
&lt;latexit sha1_base64=&quot;BU/AsobXCzC2DTjmytGtECP0izQ=&quot;&gt;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&lt;/latexit&gt;
2
3…
0
さまざまな相互作用を
比較的自由にモデリングできる
fi
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
12/35


# Page. 12

![Page Image](https://bcdn.docswell.com/page/9E2GLD8DER.jpg)

Multi-phase- eld models for grain growth
・Steinbach and Pezzolla (1999) の
multi-phase- eld model (等方バージョン)
Grain ID
&lt;latexit sha1_base64=&quot;sRsxuBCLB42tJyMGvk6xULtNh5Y=&quot;&gt;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&lt;/latexit&gt;
n(x, t) =
N
X
si (x, t)
(i=1
0
si (x, t) =
1
&lt;latexit sha1_base64=&quot;/6NcIn+hoSZaT0c55KbplzUaOYI=&quot;&gt;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&lt;/latexit&gt;
when
when
i (x, t) = 0
i (x, t) &gt; 0
&lt;latexit sha1_base64=&quot;tYo7YQ73ik5H5Yp63ildkcUaA2c=&quot;&gt;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&lt;/latexit&gt;
1
・粒同士の成長競合による界面移動, 粒消失のみを記述
&lt;latexit sha1_base64=&quot;UssoEJ4RgljaM3kqRQyiMW8lIVs=&quot;&gt;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&lt;/latexit&gt;
1
&lt;latexit sha1_base64=&quot;BU/AsobXCzC2DTjmytGtECP0izQ=&quot;&gt;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&lt;/latexit&gt;
2
3…
0
・ 界面情報のみをメモリに乗せ
界面の運動のみを計算する効率的なシミュレーションアルゴリズムが存在する。
Kim, Kim, Kim, and Park, 2006
fi
fi
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
13/35


# Page. 13

![Page Image](https://bcdn.docswell.com/page/D7YVQ26M7M.jpg)

Multi-phase- eld models for grain growth
・Steinbach and Pezzolla (1999) の
multi-phase- eld model (等方バージョン)
Grain ID
Phase- eld のパラメータ⇆物理”的&quot;パラメータ
&lt;latexit sha1_base64=&quot;tYo7YQ73ik5H5Yp63ildkcUaA2c=&quot;&gt;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&lt;/latexit&gt;
界面幅
1
L
易動度
W
&lt;latexit sha1_base64=&quot;UssoEJ4RgljaM3kqRQyiMW8lIVs=&quot;&gt;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&lt;/latexit&gt;
1
&lt;latexit sha1_base64=&quot;BU/AsobXCzC2DTjmytGtECP0izQ=&quot;&gt;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&lt;/latexit&gt;
2
3…
0
fi
fi
fi
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
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# Page. 14

![Page Image](https://bcdn.docswell.com/page/VENMK8Z3J8.jpg)

Multi-phase- eld models for grain growth
・Steinbach and Pezzolla (1999) の
multi-phase- eld model (等方バージョン)
Grain ID
パラメータと初期場が与えられれば将来予測が可能
&lt;latexit sha1_base64=&quot;tYo7YQ73ik5H5Yp63ildkcUaA2c=&quot;&gt;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&lt;/latexit&gt;
興味のある（データから推定したい）もの
&lt;latexit sha1_base64=&quot;1oY5jbo9LEepfV4QP10RViJqdsg=&quot;&gt;AAAWRXicldjLbttGFAbgk/SWqpc4LQQU6EaokiIBGmEU1KnRVWznfrUt35IoMEhqJDGmSIakZFlG+gAFuk0WXbVAF0Ufo4v2BbrIIxRFVwnaTYHmcKSYinUk/ZZhkRzOd84MNRI5Y4eeGydKPTty9I0333r7nWPv5t57/4MPj8+c+Gg9DtqRo9ecwAuiTduKtef6ei1xE09vhpG2WranN+ztxfT8RkdHsRv4q8luqB+0rIbv1l3HSrioMv/FwtZMUZXOqbnZ2dnC6E65pMyrSIPXUnAi94SqVKOAHGpTizT5lPC+RxbF/HefyqQo5LIHtMdlEe+55rymx5Rj2+ZammtYXLrN7w0+uj8o9fk4jRkb7XAWj/8jlgU6pf5QP6vn6nf1i/pT/Tc21p6JkbZll7d23+pw6/i3n1T+naoK/G5xm5uZm9jqhOo0Z1rrcutDU5L2w+n7Tu/p88rXK6f2Plc/qr+4Bz+oZ+pX7oPfeeH8tKxXvp/QooS6HD0wVzh+rT2p8bnOjrlGLdNmnz+VPS5Pr3nDlHTZpCWvehRwhvQ44pLCoN43+zWrfK1dPnIH2ZAcaZ+xHP2aozlyZjRp7meVOrx16DQVeQydGUS1OWLX7NtmNNT2MxXopKl5kuOMRrIgb+3rYWtD1hatA1lHtDXI1kSrIatFW4dsXbQNyDZE24RsU7QuZF3RPoTsQ9FuQ3ZbtB5kPdG2INsSrQ9ZX7QBZAPRhpANRfsIso9EG0E2Em0M2Vi0CWQT0bYh2xZtB7Id0e5Adke0Xch2RbsL2V3R9iDbG3NnmIf0vJh5AbILol2E7KJoL0L2omgvQfaSaC9D9rJor0D2imivQvaqaK9B9ppor0P2umhvQPaGaG9C9qZob0H2lmhvQ/a2aO9A9o5olyC7JNplyC6LdgWyK6KtQLYi2lXIrop2DbJrol2H7LpoNyC7IdpNyG6K9i5k74r2HmTvjbkz9HgvAp941JinpXT22gBnH9Wh+vKziDuYyWHRsvry00mT9xM4WlZffpa1TIwaHG9YyHf1w7SuN6FtXXB2UDU15dkU3pLx7WiBT3VVU1OeXaXXDB8BWX25V6EZI+maCjYnqB4w8m9bNq76cepsq0Pl0m9pNrKHTVaefT/HrUGkKwyBOft4//g0b5vc3v5KSjonLlGZ5nivxbXciWfP0OQ1jzR+wmeCKfnUUD4pozpkxiZv9YFejsY9eyDztPNI7nTtqW1W3/DMnTHnzx+iz3XO0pn6uWYxR3MeNmM6wrsT852dklG+vrmtmeKrRdTC+J31c6Xy+dLs8pfFCwuDldZj9Cl9xu0o01d0gZ90l/gO7/B34zt6Qk/zv+X/zr/I/9OvevTIwHxMr73y/78ERMNKgQ==&lt;/latexit&gt;
モデルパラメータ
初期場
&lt;latexit sha1_base64=&quot;thOELD67SkM/seWQVeS8JREqdqA=&quot;&gt;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&lt;/latexit&gt;
A, B
= { 1 (x, t0 ),
1
&lt;latexit sha1_base64=&quot;UssoEJ4RgljaM3kqRQyiMW8lIVs=&quot;&gt;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&lt;/latexit&gt;
1
&lt;latexit sha1_base64=&quot;BU/AsobXCzC2DTjmytGtECP0izQ=&quot;&gt;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&lt;/latexit&gt;
2
3…
0
2 (x, t0 ), ...,
N (x, t0 )}
fi
fi
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
15/35


# Page. 15

![Page Image](https://bcdn.docswell.com/page/Y79NVKZP73.jpg)

実験データのデザイン
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
16/35


# Page. 16

![Page Image](https://bcdn.docswell.com/page/G78KPR1XJD.jpg)

実験データのデザイン
&lt;latexit sha1_base64=&quot;sRsxuBCLB42tJyMGvk6xULtNh5Y=&quot;&gt;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&lt;/latexit&gt;
n(x, t) =
1 [mm]
si (x, t)
(i=1
1 [mm]
0
si (x, t) =
1
&lt;latexit sha1_base64=&quot;/6NcIn+hoSZaT0c55KbplzUaOYI=&quot;&gt;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&lt;/latexit&gt;
1 [mm]
N
X
1 [mm]
when
when
i (x, t)
=0
i (x, t) &gt; 0
1 [mm]
モデルは準備OK. データはどうするか。
成長途中の表面の直接観測は技術的に困難（高温の炉の中で撮影, 表面処理）
→ 粒構造の”なま”の時間発展が得られない。
→ 1つの実験サンプルから1つの静画像のみしか得られない。
→ 直接的にモデル時系列vsデータ時系列の比較ができない。
アイデア💡: データを”時系列的なもの”に変換
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
17/35


# Page. 17

![Page Image](https://bcdn.docswell.com/page/L7LZ1KQNER.jpg)

実験データのデザイン
アイデア💡: データを”時系列的なもの”に変換
粒成長する時間（保持時間: th）を変えて
Heating Holding
複数のサンプルを作成。
パターンに”時間発展の長さ”が
∼
反映されていると期待する。
1 [mm]
th = 1sec
1 [mm]
th = 2sec
1 [mm]
th = 5sec
Cooling
th
1 [mm]
th = 10sec
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPFs
1 [mm]
th = 20sec
18/35


# Page. 18

![Page Image](https://bcdn.docswell.com/page/4EMVW21QEW.jpg)

実験データのデザイン
アイデア💡: データを”時系列的なもの”に変換
画像から粒1個1個の面積 {Si} をカウント。
系全体の特徴的な粒の大きさ
&lt;latexit sha1_base64=&quot;oVMUBcJ28qPvOQJEkaks+HNdjAA=&quot;&gt;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&lt;/latexit&gt;
P 2
i Si
K= P
i Si
1 [mm]
粒ごとの形状の詳細に対して頑健性を持つことを期待。
画像中の粒の個数がそれほど多くないため、
単純な統計量（例えば算術平均）だと外れ値の影響が大きくなりすぎる。
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
19/35


# Page. 19

![Page Image](https://bcdn.docswell.com/page/PER2YMLK79.jpg)

実験データのデザイン
アイデア💡: データを”時系列的なもの”に変換
K (mm2 )
0.8
0.6
0.4
&lt;latexit sha1_base64=&quot;oVMUBcJ28qPvOQJEkaks+HNdjAA=&quot;&gt;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&lt;/latexit&gt;
0.2
0.0
1 [mm]
012
5
10
P 2
i Si
K= P
i Si
15
Holding time th (s)
1 [mm]
1 [mm]
良い感じに単調増加
特徴的な大きさの
20
時間発展を表現でき
ていると思われる。
1 [mm]
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
1 [mm]
20/35


# Page. 20

![Page Image](https://bcdn.docswell.com/page/P7XDGVW5JX.jpg)

実験データのデザイン
アイデア💡: データを”時系列的なもの”に変換
シミュレーションにおいても同様な対応物を定義。
系全体の特徴的な粒の大きさ
&lt;latexit sha1_base64=&quot;QO+GRgmqhtsBXCm8U/0/fXSDZ9Y=&quot;&gt;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&lt;/latexit&gt;
P 2
i Si (t)
K(t) = P
i Si (t)
where
&lt;latexit sha1_base64=&quot;SltnON01TBPw9YB9tQWVJI7x1RQ=&quot;&gt;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&lt;/latexit&gt;
Si (t) =
Z
dx
↵+1
(x, t)
i
α は界面を強調するパラメータ。
論文では α = 4
Time t (s) or Holding time th (s)
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
21/35


# Page. 21

![Page Image](https://bcdn.docswell.com/page/37KX4QVRJD.jpg)

実験データのデザイン
&lt;latexit sha1_base64=&quot;QO+GRgmqhtsBXCm8U/0/fXSDZ9Y=&quot;&gt;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&lt;/latexit&gt;
パターン（モデル）
パターン（データ）
統計量（モデル）
P 2
i Si (t)
統計量（データ）
K(t) = P
i
Si (t)
P 2
i Si
K(th ) = P
i Si
&lt;latexit sha1_base64=&quot;XVXqogO+bbhE4JDG0H1XL9xt+1Y=&quot;&gt;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&lt;/latexit&gt;
比較可能な特徴量の時系列を構成できた。
→ ベイズ推論によるパラメータ推定へ
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
22/35


# Page. 22

![Page Image](https://bcdn.docswell.com/page/LJ35LPMG75.jpg)

ベイズ推論の枠組み
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
23/35


# Page. 23

![Page Image](https://bcdn.docswell.com/page/8JDDNL6NJG.jpg)

ベイズ推論
ベイズ推論の枠組みでパラメータ・初期場を推定する。
事後分布
データに乗っているノイズの仮定
&lt;latexit sha1_base64=&quot;dBCZbcMIYR8nKYklK/8rCzQrlr4=&quot;&gt;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&lt;/latexit&gt;
K(t j ) = K(t j ) + ! j
尤度
&lt;latexit sha1_base64=&quot;QkXT0vN8OWpt84+Ir1mNvp223BU=&quot;&gt;AAAWynicldhLbxtVFADg61KgNY+mRbKQ2oWFW5RIiXUdkRJVQmoefSZtnXfa3GCNx9eeSeaVmbFjezArxIIfAAtWILFA/Aw2/AEW/QkIdkXqhgVnrp2MGx/bJ45iz9w53zn3jieZubfsWWYQcv4ydeGti2+/8+6ly+n33v/gwysTV69tB27d1+WW7lquv1vWAmmZjtwKzdCSu54vNbtsyZ3y4VJ8fKch/cB0nc2w5cl9W6s5ZtXUtRCaShPfepOisfzVwvTitAjMmq1Ni6JhTn0hPN+tlKKDTlpUfU2PCp1IBEd+GM0Kz+xGfjnbgaOy6QlLVsO9mWw3UkTZlckwplMzwtZCQ9esaKXTaxIdYJDkJIPwzZoR7pcmcjw/y+fn5uaygxuFPFevHOu9iu7V9PdMsApzmc7qzGaSOSyEbYtpLICfPVZgnHnQts8iaPNhy1THJeuwNNg6REmI0KD1EN5rsLfXa3VgP84ZKK1DFQt+fZBZdov/yX/lr/gf/Df+F/9vaK5I5Yj70oLPctdKr3Tlu483Xo9VWXjXoM9G4kb2OmRVNq96a0LvPdUSj0Pv+kb7h1cbd9ZvRZ/yn/nfMIKf+Ev+O4zBafyr/7Im138c0aOQNSG7q85w8EZ/YuNAzLE6R7bqswPfSgTt8TmvqZYmmLjlZEQuVIj3fWjJ9uK+Po0UcK5N2DN71Sg14jHTanQjB2uk1dUkYZyCNeBTZ5MsB9fQVC9rGTI21XZZXQ2V00pZdlNF3oQ8g5k0ktdOdb8tk2wZtTrJ6qitkGwFtZJkJWqrJFtFbY1ka6g1SNZArUmyJmoPSPYAtYcke4hai2Qt1Noka6PWIVkHtS7Juqj1SNZD7RHJHqHWJ1kftQHJBqgNSTZEbZ1k66htkGwDtccke4zaJsk2Udsi2RZq2yTbHnJnWCDpBbTyIskuonaJZJdQu0yyy6i9R7L3UHufZO+j9gHJPkDtQ5J9iNpHJPsItY9J9jFqV0h2BbWrJLuK2ick+wS1T0n2KWqfkewz1BZJtojaNZJdQ+06ya6jdoNkN1C7SbKbqN0i2S3UbpPsNmp3SHYHtbsku4va5yT7HLUvSPbFkDtDG7Z84hMPH/K0FM9ea8TZh+iLx59FzN5MjpYticefTgzYDsnZknj8WVZTOSrkfP0Cv6ufp3ftEX1rEmcHQkXisyl6T4b3wyY+1QkVic+u4nNGvwKSeHxUnrpG4jUV2pxAnDH4/7bkuurmqYIVfe3Y/9Lkyu43SXvy9zlsDSJeYXDV0c7p/iR8GtDf7kpKPCfOswKbhy0bosyRR6fY6DWPOH8IR9wx9XhfPawiP2dFAz7lmVEO5p05U3nccUrteO2prlbf6JUbQ47fPseYq1ClMfZ7TXIO1jxvxfgKb46sNzOmIn5+06WJ3Mkianb4xvZsvnA7P7f2We7uYm+l9RK7zj6BfhTY5+wuPOkW4Q6vs39S11LXUzcyqxk/08pE3dALqZ75iL3xynzzP4T4eM4=&lt;/latexit&gt;
p( D|A, B, , ) =
Y
j
2
! j ⇠ Normal(0,
&lt;latexit sha1_base64=&quot;clGYTqV8fx61fwRmB76UOBQxdwY=&quot;&gt;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&lt;/latexit&gt;
p
1
2⇡
2
exp
&quot;
{K(t j )
2
)
尤度
2#
K(t j )}
2
論文では
パラメータ事後分布
事前分布
&lt;latexit sha1_base64=&quot;JHl88VSXPLwyvNzsuCZPMJviL2o=&quot;&gt;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&lt;/latexit&gt;
事前分布
尤度
一様分布の事前分布を仮定
p(A, B)p( D|A, B, , )
p(A, B|D, , ) =
p( D| , )
規格化定数
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
24/35


# Page. 24

![Page Image](https://bcdn.docswell.com/page/VEPX52YNJ8.jpg)

ベイズ推論
パラメータ事後分布
事前分布
&lt;latexit sha1_base64=&quot;JHl88VSXPLwyvNzsuCZPMJviL2o=&quot;&gt;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&lt;/latexit&gt;
尤度
p(A, B)p( D|A, B, , )
p(A, B|D, , ) =
p( D| , )
事後分布
事前分布
尤度
規格化定数
規格化定数最大化
&lt;latexit sha1_base64=&quot;P1aSeqBL/V7tLOIg1Yux107HcDw=&quot;&gt;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&lt;/latexit&gt;
p( D| , ) =
Z
dA
Z
dB p(A, B)p( D|A, B, , )
= ある σ, Φを仮定した際のデータDが実現する確率
&lt;latexit sha1_base64=&quot;aNRmxgRFw/jV3YGFXLBaJx/WsBw=&quot;&gt;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&lt;/latexit&gt;
ˆ , ˆ = arg max p( D| , )
,
実際には, Φを関数空間全体から最適化するのは困難（ほぼ不可能）
→ Φの候補を与え（関数空間を絞り）、そこからベストなΦを選ぶ
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
25/35


# Page. 25

![Page Image](https://bcdn.docswell.com/page/27VLPDGYEQ.jpg)

提案手法
(×104)
1.2
1. 初期場の候補を用意する。
1,
2 , ...}
2. Qを候補それぞれで計算する。
&lt;latexit sha1_base64=&quot;i1vDi/ZM/davE2TBgNsCgNZjNYg=&quot;&gt;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&lt;/latexit&gt;
ˆ k = arg max p( D| ,
Q(
&lt;latexit sha1_base64=&quot;yosQ348gvVJfCXRb84Ba90H59Bc=&quot;&gt;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&lt;/latexit&gt;
k)
= p( D| ˆ k ,
Φ1
Φ2
Φ3
0.9
Q
V={
&lt;latexit sha1_base64=&quot;dptOo2gveBHOc70HzvlFSUCpE6o=&quot;&gt;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&lt;/latexit&gt;
Candidates
0.6
0.3
k)
0.0
(a)
(a)
(a)
ΦΦ
1Φ
1 1
111
ΦΦ
3Φ
3 3
ΦΦ
2Φ
2 2
k)
000
(b)
(b)
(b)
Candidates
Candidates
Candidates
Φ1Φ1Φ1
Φ2Φ2Φ2
3. Qが最大になる候補をベストな初期場として採用、事後分布を計算する。
9 9 9
Q
Q
Q
101010
&lt;latexit sha1_base64=&quot;Pn972eKESB+vCm7YdgMkkX3KWqk=&quot;&gt;AAAW+nicldjLbttGFADQUdJHqj7ipIBQoF0IVVLYgCOMgjo1ChSIZedpJ5HfTsJAIKmRxJivkBQti1Y/oB/QLrpqgC6CbIv+QDf9gS7yCUWXKdBNF7kzUkzZupKuZFgaDufcO0ONTc4Yvm2FEeevMmfOvvPue++f+yD74Ucff3J+5sLFndBrBabYNj3bC/YMPRS25YrtyIpssecHQncMW+wa+8vy/G4sgtDy3K3o0BdPHL3hWnXL1COoqs787s8uzZePtHhlXmvqUaKFVsPRu9VEi0Q7SgwRRt3uvFZpWier5vLf5bV6oJuJCjDnz0KIIyhNGaab9OSUqjpT4MWrfHFhYSE/XCgVuXoVWP9V8S5kf2QaqzGPmazFHCaYyyIo20xnIfw8ZiXGmQ91T1gCdQGULHVesC7Lgm1BKwEtdKjdh/cGHD3u17pwLGOGSpuQxYbfAGSeXeZ/8Rf8Nf+Tv+R/8/9HxkpUDNmXQ/g0elb41fM/fLb530SVh3cd+txM3dheR6zOFlVvLei9r2rkOMyejzs/vd78duNy8hV/zv+BEfzCX/E/YAxu/K/567rY+HlMjyLWhuieusLhif5I40KbA3WNHNVnF76VBOrlNW+omjYYWfN2RB5kkMcB1OT77b4/bqnBtbbgyOpno+SQY6bl6LUczpFVs0nAODUWw6fJZlkB5tBcP6oBEduqbKjZUDvOlGeXVMtLEGc4kk7y+rEetAbJGqg1SdZEbY1ka6gVJCtQWyfZOmobJNtAbZNkm6i1SNZC7VOSfYrafZLdR61NsjZqHZJ1UOuSrItaj2Q91Pok66P2Gck+Q21AsgFqQ5INURuRbITaFsm2UBuTbIzaA5I9QG2bZNuoPSTZQ9R2SLYz4s6wRNJLaOYyyZZRu0yyy6hdIdkV1N4g2RuovUmyN1F7i2RvofY2yd5G7R2SvYPauyR7F7WrJLuK2jWSXUPtPZK9h9r7JHsftQ9I9gFqKyRbQe06ya6jdoNkN1C7SbKbqN0i2S3UbpPsNmp3SHYHtbsku4vaPZLdQ+1Dkn2I2kck+2jEnaEDpYD4xMNHPC3J1WuDuPrQBtrjzyJWfyVHi5a2x59OmlCOyNHS9vizrK5i1MjxBgV+V5+md50xfWsTVweaaomvpug9Gd0Ph/hUp6mW+OpKXjP6DEjb46Py1RyReyq0NYF2yuD/29J51YtTB6sN1GP/S9OZPWjS+vTvc9QehNxh8NTZ7vHxLHw2ob+9nRS5Ji6yEluEkgOtrLFn59j4PQ8ZP4Iz3oR8fCAflpFPmbEJn+LUKIfjXjmVedJ5Sm6599RSu2/0zPGI89emGHMdssQTv9c05nDOaTPKGd4em+/KhIz49c1WZwpvN1Hzows7V4ula8WF9a8L18v9ndZz7HP2JfSjxL5h1+FJtwJ3eDPzRaacWc2s5Y5yz3Mvci97Tc9k+uZTduKV++0NNs2PNw==&lt;/latexit&gt;
1010
p(A, B)p( 10D|A,
B, ˆ best ,
101010
p(A, B|D, ˆ best , best ) =
101010
p( D|
ˆ bestΦΦ,Φ bestΦ)ΦΦ
6 6 6
3 3 3
Φ3Φ3Φ3
best )
0 0 0
1 1 1
2 2 2
ΦΦ
3Φ
3 3
True
initial
phase-fields
True
initial
phase-fields
True
initial
phase-fields
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
26/35


# Page. 26

![Page Image](https://bcdn.docswell.com/page/5JGXPY8W7L.jpg)

提案手法のチェック
1. とある初期場を仮定する。
(a) Φ1
Φ3
Φ2
2. 擬似データ時系列を作成する。 K(t) = K(t) + !t
!t ⇠ Normal(0,
1
2
)
&lt;latexit sha1_base64=&quot;w8aptWm03OujlYh11n9nEyklxUs=&quot;&gt;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&lt;/latexit&gt;
&lt;latexit sha1_base64=&quot;WwxSaxw2UTz3KfazKpWVUZahpe8=&quot;&gt;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&lt;/latexit&gt;
3. 最初の仮定した初期場を含む候補のQをそれぞれ計算・比較
たくさんのノイズサンプルで実行
ノイズ標準偏差の推定
0
初期場の推定
(b)
Candidates
Q
Φ1
Φ3
Φ2
109
106
103
100
ノイズ推定&amp;初期場推定
ともに正しく動作している
Φ1
Φ3
Φ2
True initial phase-fields
(a) Φ(a)
(a)
Φ1
1
Φ1 Φ2
Φ2 Φ2
Φ3
Φ3 Φ3 1
1
1
0
0
0
(b) (b) (b)
Candidates
Candidates
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction
viaCandidates
MPF
models
Φ
Φ
Φ1 Φ
Φ
Φ2 Φ
Φ
27/35
Φ3


# Page. 27

![Page Image](https://bcdn.docswell.com/page/47Q82M9QEP.jpg)

実際の粒構造データへの適用
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# Page. 28

![Page Image](https://bcdn.docswell.com/page/KE42LVXY71.jpg)

実際の粒構造データへの適用
ボロノイ図を用いて作成した 3つの初期場候補に対し、提案手法を適用
The best initial grain structure
(×104)
1.2
実験データ
0.4
0.2
0.0
Φ1
Φ2
Φ3
0.9
0.6
Q
K (mm2 )
0.8
Candidates
012
5
10
15
0.3
20
Holding time th (s)
0.6
0.0
(a)
(a)
(a)
ΦΦ
1Φ
1 1
1 [mm]
1 [mm]
1 [mm]
1 [mm]
111
ΦΦ
3Φ
3 3
ΦΦ
2Φ
2 2
1 [mm]
000
(b)
(b)
(b)
Candidates
Candidates
Candidates
注）より多くの候補でやると、より良いものが見つかる可能性は大いにある。
Φ ΦΦ
Φ
Φ
Φ1Φ1
1
Φ2Φ2
2
Q
Q
Q
9 9 9
10
1010
6 6 6
10
10
10
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
3 3 3
3 3 3
29/35


# Page. 29

![Page Image](https://bcdn.docswell.com/page/L71ML3NNJG.jpg)

B [mm2
0.1
実際の粒構造データへの適用
0.0
0
200
400
600
最適解で計算されたモデル時系列とデータ
A [s−1 ]
0.8
K(t), K(t) [mm2 ]
(b)
800
15
10
5
0
0.6
0.4
K(t)
K(t)
0.2
0.0
012
5
10
20
Time t [s]
(c) (×104 )
p (W| D, σ̂1, Φ1 )
15
7.5
5.0
1 [mm]
1 [mm]
1 [mm]
1 [mm]
1 [mm]
2.5
・候補から選ばれた最良な初期場,
最適パラメータはデータ時系列をよく再現
0.0
0.00
0.05
0.10
W [mm]
0.15
0.20
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
(d)
3
30/35


# Page. 30

![Page Image](https://bcdn.docswell.com/page/G7W3VXMM72.jpg)

実際の粒構造データへの適用
パラメータA,Bの事後分布
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
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# Page. 31

![Page Image](https://bcdn.docswell.com/page/4JZV46QM73.jpg)

実際の粒構造データへの適用
パラメータA,Bの事後分布
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
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# Page. 32

![Page Image](https://bcdn.docswell.com/page/YE653265JV.jpg)

A [s−1 ]
(b)
0
200
400
600
800
A [s−1 ]
L
0.8
K(t), K(t) [mm2 ]
(b)
易動度
0.0
0.6
0.4
K(t)
K(t)
0.2
0.6
0.4
K(t)
K(t)
0.2
0.0
012
5
10
15
4
7.5
5.0
2.5
・AとBの線形的な相関から、界面幅Wの決まりが良いことが言える。実際、
0.0
0.0
5
p (W| D, σ̂1, Φ1 )
15
20
Time t [s]
(c) (×104 )
分散が小さい
5.0
2.5
0.0
0.00
0.05
0.10
W [mm]
0.00
0.05
0.15
0.20
0.10
0.15
0.20
W [mm]
(d) (×103 )
7.5
(d) (×103 )
ˆ 1 , Φ1 )
10
p (γL| D, σ̂1, Φ1 )
012
20
Time t [s]
W (c)
(×10 )
p (W| D, σ̂1, Φ1 )
B [mm2 s−1 ]
界面幅
0.1
20
15
10
5
0
K(t), K(t) [mm2 ]
(a) 0.2
25
実際の粒構造データへの適用
0.8
p (A, B| D, σ̂1 , Φ1 )
7.5
5.0
分散が大きい
2.5
0.0
0
2
4
6
8
γL [mm/s]
2023. 11.7.5
21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
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# Page. 33

![Page Image](https://bcdn.docswell.com/page/GE5YL26GE4.jpg)

実際の粒構造データへの適用
界面幅
L
易動度
W
考察
物理パラメータで書き直したMPF
&lt;latexit sha1_base64=&quot;/5E0bE6H/Hclq1KmxTZUMaE3ol4=&quot;&gt;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&lt;/latexit&gt;
2
666 2
@ i
= 4 Lsi 6664
@t
W
i
0
1
W BBBB 2
+ 2 BB@r
n
⇡
!
2
i
N
X
1
n
j=1
s jr
2
・φが”尤もらしい粒”を形作るための良い W が存在し得る。
・一方で、多数の界面が同時に移動するため、γL は決まりが悪い。
13
CCC777
C
7
C
7
jC
A75
・事前分布でγL を絞ることでより良い推定が得られる可能性がある。
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
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# Page. 34

![Page Image](https://bcdn.docswell.com/page/972GL4MD7R.jpg)

まとめ
・MPFを用いて、粒構造静止写真データから
ダイナミクスを特徴づけるパラメータを
抽出するベイズ推定の方法を提案した。
・本提案手法は重いが、効率化は可能。
・differentiable なMPF models (勾配法最適化が利用可)
・初期場のparameterization
・BOなどを使った事後分布最適化の効率化
・構造特徴量デザインの検討
・周期的な構造があるならば
波数空間で作った特徴量が良い場合もある。
・Encoder-decoder モデルなどの潜在空間で作る特徴量
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
35/35


# Page. 35

![Page Image](https://bcdn.docswell.com/page/DJYVQMXMJM.jpg)

ありがとうございました。
2023. 11. 21 セミナー 伊藤伸一 Bayesian inference of grain growth prediction via MPF models
35/35


