{"id":1337,"date":"2022-01-03T00:10:37","date_gmt":"2022-01-02T15:10:37","guid":{"rendered":"https:\/\/hayashiyus.jp\/?p=1337"},"modified":"2022-04-26T18:34:35","modified_gmt":"2022-04-26T09:34:35","slug":"test","status":"publish","type":"post","link":"https:\/\/hayashiyus.jp\/?p=1337","title":{"rendered":"\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u306e\u5834\u306e\u91cf\u5b50\u8ad6\u7684\u89e3\u91c8"},"content":{"rendered":"\r\n

\u5834\u306e\u91cf\u5b50\u8ad6\u306b\u304a\u3044\u3066\uff0c\u81ea\u7531\u5834\u306f\u95a2\u6570\u7a7a\u9593\u4e0a\u306e\u30ac\u30a6\u30b9\u5206\u5e03\u3068\u3057\u3066\u8a18\u8ff0\u3055\u308c\uff0c\u76f8\u4e92\u4f5c\u7528\u5834\u306f\u81ea\u7531\u5834\u306e\u4f5c\u7528\u3078\u306e\u975e\u30ac\u30a6\u30b9\u7684\u306a\u88dc\u6b63\uff08\u6442\u52d5\uff09\u306b\u3088\u3063\u3066\u8a18\u8ff0\u3055\u308c\u308b\uff0e\u9762\u767d\u3044\u3053\u3068\u306b\uff0c\u4e2d\u9593\u5c64\u306e\u5e45\u3092\u7121\u9650\u306b\u5e83\u304f\u3068\u3063\u305f\u6df1\u5c64\u5b66\u7fd2\u30e2\u30c7\u30eb\u306e\u4e00\u90e8\u304c\u3084\u306f\u308a\u30ac\u30a6\u30b9\u904e\u7a0b\u3067\u8a18\u8ff0\u3067\u304d\u308b\u3053\u3068\u304c\u77e5\u3089\u308c\u3066\u304a\u308a\uff0c\u305d\u306e\u3088\u3046\u306a\u6027\u8cea\u3092\u3082\u3064\u6df1\u5c64\u5b66\u7fd2\u30e2\u30c7\u30eb\u306e\u4e00\u7fa4\u306f\u307e\u3068\u3081\u3066\u6f38\u8fd1\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u3068\u547c\u3070\u308c\u3066\u3044\u308b\uff0e\u66f4\u306b\uff0c\u3053\u306e\u6f38\u8fd1\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u306e\u4e2d\u9593\u5c64\u306e\u5e45\u3092\u7121\u9650\u306e\u5e83\u304c\u308a\u3092\u3082\u3064\u6f38\u8fd1\u6975\u9650\u304b\u3089\u6709\u9650\u5e45\u306b\u72ed\u3081\u3066\u3044\u304f\u306b\u3064\u308c\u3066\uff0c\u305d\u306e\u632f\u308b\u821e\u3044\u306f\u975e\u30ac\u30a6\u30b9\u904e\u7a0b\u306b\u3088\u3063\u3066\u8a18\u8ff0\u3055\u308c\u308b\u3088\u3046\u306b\u306a\u3063\u3066\u3044\u304f\uff0e\u6700\u8fd1\uff0c\u3053\u306e2\u3064\u306e\u7406\u8ad6\uff08Wilson\u6d41\u306e\u6709\u52b9\u5834\u306e\u7406\u8ad6\u3068\u6f38\u8fd1\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u306e\u7406\u8ad6\uff09\u306e\u9593\u306bNN-QFT\u5bfe\u5fdc\u3068\u547c\u3070\u308c\u308b\u81ea\u7136\u306a\u5bfe\u5fdc\u95a2\u4fc2\u3092\u3064\u3051\u3089\u308c\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u3066\u304d\u305f1<\/a><\/sup>J. Halverson, A. Maiti and K. Stoner, “Neural networks and quantum field theory”, Machine Learning: Science and Technology, Volume 2, Number 3 (2021)<\/span>2<\/a><\/sup>A. Maiti, K. Stoner and J. Halverson, “Symmetry-via-Duality: Invariant Neural Network Densities from Parameter-Space Correlators”, arXiv:2106.00694 [hep-th]. (2021)<\/span>3<\/a><\/sup>H. Erbin, V. Lahoche and D. O. Samary, “Nonperturbative renormalization for the neural network-QFT correspondence”, arXiv: 2108.01403 [hep-th]. (2021)<\/span>\uff0e\u305d\u3053\u3067\u3053\u306e\u8a18\u4e8b\u3067\u306f\uff0c\u540c\u8ad6\u6587\u3092\u984c\u6750\u3068\u3057\u3066\u300c\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u306e\u5834\u306e\u91cf\u5b50\u8ad6\u7684\u89e3\u91c8\u300d\u3092\u7d39\u4ecb\u3057\u305f\u3044\uff0e<\/p>\r\n\r\n\r\n\r\n

\uff11. \u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u306e\u5834\u306e\u91cf\u5b50\u8ad6\u7684\u89e3\u91c8<\/h4>\r\n\r\n\r\n
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\uff11\u2212\uff11. \u6f38\u8fd1\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u3068\u30ac\u30a6\u30b9\u904e\u7a0b\uff0c\u81ea\u7531\u5834\u306e\u95a2\u4fc2<\/h4>\r\n

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\u307e\u305a\uff0c\u3042\u308b\u78ba\u7387\u5206\u5e03\u304b\u3089\u5f97\u3089\u308c\u308b\u30d1\u30e9\u30e1\u30fc\u30bf \"\theta \sim P(\theta)\" \u3068\u4e2d\u9593\u5c64\u306e\u5e45 \"N\" \u306e\u81ea\u7531\u5ea6\u3092\u6301\u3064\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u3092<\/p>\r\n

(1) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle f_{\theta, N}: \mathbb{R}^{d_{\text {in }}} \rightarrow \mathbb{R}^{d_{\text {out }}} \end{eqnarray*}\"<\/p><\/p>\r\n

\u3068\u5165\u529b\u3068\u51fa\u529b\u3092\u7d50\u3073\u3064\u3051\u308b\u95a2\u6570 \"f\" \u3068\u3057\u3066\u8868\u3059\u3053\u3068\u306b\u3059\u308b\uff0e\u307e\u305f\uff0c\u4e2d\u9593\u5c64\u306e\u5e45 \"N \rightarrow \infty\" \u306e\u6975\u9650\u3092\u3068\u3063\u305f\u3082\u306e\u3092\u6f38\u8fd1\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u3068\u547c\u3076\u3053\u3068\u306b\u3059\u308b\uff0e\u3053\u306e\u3068\u304d\u5165\u529b \"\left\{x_{1}, \ldots, x_{k}\right\}\" \u306b\u5bfe\u3059\u308b\u6f38\u8fd1\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u306e\u51fa\u529b\u306f\u30ac\u30a6\u30b9\u904e\u7a0b\u306b\u5f93\u3046\u3053\u3068\u304c\u77e5\u3089\u308c\u3066\u3044\u308b\uff08\u4e0b\u56f3\u53c2\u7167\uff09\uff0e<\/p>\r\n

(2) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle \left\{f\left(x_{1}\right), \ldots, f\left(x_{k}\right)\right\} \sim \mathcal{N}\left(\mu, \ \Xi^{-1}\right) \end{eqnarray*}\"<\/p><\/p>\r\n

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\u00a0<\/p>\r\n

\u3064\u307e\u308a\uff0c\u4e2d\u9593\u5c64\u306e\u5e45 \"N \rightarrow \infty\" \u306e\u6975\u9650\u3092\u3068\u3063\u305f\u6f38\u8fd1\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u3067\u306f\uff0c\u78ba\u7387\u7684\u306b\u3086\u3089\u3050\u4e2d\u9593\u5c64\u306e\u5404\u30cb\u30e5\u30fc\u30ed\u30f3\u304b\u3089\u306e\u4fe1\u53f7\u306e\u548c\u306e\u5206\u5e03\u304c\u4e2d\u5fc3\u6975\u9650\u5b9a\u7406\u306b\u3088\u3063\u3066\u30ac\u30a6\u30b9\u5206\u5e03\u306b\u306a\u308b\u3053\u3068\u304c\u5c0e\u304b\u308c\u308b\uff0e\u3053\u306e\u95a2\u4fc2\u304c\u4e2d\u9593\u5c64\u306e\u5404\u5c64\u3067\u6210\u308a\u7acb\u3064\u306e\u3067\u6f38\u8fd1\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u306e\u4fe1\u53f7\u4f1d\u64ad\u306f\u30ac\u30a6\u30b9\u904e\u7a0b\u306b\u5f93\u3046\uff0e\u3053\u3053\u3067\u9006\u5171\u5206\u6563\u884c\u5217 \"\Xi\left(x, x^{\prime}\right)\" \u3092\u30ac\u30a6\u30b9\u904e\u7a0b\u306e\u30ab\u30fc\u30cd\u30eb\u95a2\u6570 \"K\left(x, x^{\prime}\right)\" \u3092\u7528\u3044\u3066 \"\left(\Xi^{-1}\right)_{i j}=K_{i j} := K\left(x_{i}, x_{j}\right)\" \u3068\u5b9a\u7fa9\u3057\u3066\u304a\u304f\uff0e\"\Xi\left(x, x^{\prime}\right) = K^{-1}\left(x, x^{\prime}\right)\" \u3088\u308a<\/p>\r\n

(3) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle \int d^{d_{\mathrm{in}}} x^{\prime} K\left(x, x^{\prime}\right) \Xi \left(x^{\prime}, y\right) = \delta^{\left(d_{\mathrm{in}}\right)}\left(x-y\right) \end{eqnarray*}\"<\/p><\/p>\r\n

\u304c\u6210\u308a\u7acb\u3064\uff0e\u30ac\u30a6\u30b9\u904e\u7a0b\u306e\u4f5c\u7528\u7a4d\u5206\u3092<\/p>\r\n

(4) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle S_{\mathrm{GP}} = \frac{1}{2} \int d^{d_{\mathrm{in}}} x d^{d_{\mathrm{in}}} x^{\prime} f(x) \Xi\left(x, x^{\prime}\right) f\left(x^{\prime}\right) <\/p><\/p>\r\n

\u3067\u3042\u3089\u308f\u3059\u3053\u3068\u306b\u3059\u308b\u3068\uff0c\"\left\{f\left(x_{1}\right), \ldots, f\left(x_{k}\right)\right\} \sim \mathcal{N}\left(\mu, \ \Xi^{-1}\right)\" \u306e\u3068\u304d<\/p>\r\n

(5) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle \hspace{-280pt} -\log p\left(f | \theta\right) = -\log \int d^{d_{\mathrm{in}}} x \ p\left(f(x) | \theta, x\right) p(x) \\<\/p><\/p>\r\n

\u3068\u306a\u308b\u306e\u3067\uff0c\u5b9a\u6570\u9805\u3092\u9664\u3051\u3070\u4f5c\u7528\u7a4d\u5206\u304c\u5bfe\u6570\u5c24\u5ea6\u3068\u4e00\u81f4\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308b\uff0e\u3053\u306e\u3068\u304d\u30ac\u30a6\u30b9\u904e\u7a0b\u306e\u5206\u914d\u95a2\u6570\u3092 \"Z_{\mathrm{GP}} := \int d f e^{-S_{\mathrm{GP}}}\" \u3068\u5b9a\u3081\u308b\u3068\uff0c\u30ac\u30a6\u30b9\u904e\u7a0b\u306e\u78ba\u7387\u5206\u5e03\uff08\u95a2\u6570\u7a7a\u9593\u4e0a\u306e\u6f38\u8fd1\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8 \"f\" \u306e\u5206\u5e03\uff09<\/p>\r\n

(6) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle p(f) = \frac{e^{-S_{\mathrm{GP}}}}{Z_{\mathrm{GP}}} \\ & \displaystyle \int \mathcal{D} f p(f) = 1 \end{eqnarray*}\"<\/p><\/p>\r\n

\u3092\u5f97\u308b\uff08\u305f\u3060\u3057\uff0c\"\mathcal{D} f\" \u306f\u95a2\u6570\u7a7a\u9593\u306b\u304a\u3051\u308b\u7d4c\u8def\u7a4d\u5206\u6e2c\u5ea6\u3092\u3042\u3089\u308f\u3057\u3066\u3044\u308b\uff09\uff0e\u3053\u306e\u30ac\u30a6\u30b9\u904e\u7a0b\u306e\u78ba\u7387\u5206\u5e03\u3092\u4f7f\u3046\u3068\u30ab\u30fc\u30cd\u30eb\u95a2\u6570\u306f<\/p>\r\n

(7) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle K(x, y) := \int \mathcal{D} f p(f) f(x) f(y) \end{eqnarray*}\"<\/p><\/p>\r\n

\u3068\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\uff0e\u3053\u306e\u3068\u304d\u30ac\u30a6\u30b9\u904e\u7a0b\u306e \"n\" \u70b9\u30b0\u30ea\u30fc\u30f3\u95a2\u6570\u306f<\/p>\r\n

(8) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle G^{(n)}_{\mathrm{GP}} \left(x_{1}, \ldots, x_{n}\right) = \frac{\int \mathcal{D} f f\left(x_{1}\right) \ldots f\left(x_{n}\right) e^{-S_{\mathrm{GP}}}}{Z_{\mathrm{GP}}} \end{eqnarray*}\"<\/p><\/p>\r\n

\u3067\u4e0e\u3048\u3089\u308c\u308b\uff0e\u4e00\u65b9\uff0c\u3088\u304f\u77e5\u3089\u308c\u3066\u3044\u308b\u3088\u3046\u306b\u5834\u306e\u91cf\u5b50\u8ad6\u3068\u3044\u3046\u79d1\u5b66\u7406\u8ad6\u306b\u304a\u3051\u308b\u81ea\u7531\u30b9\u30ab\u30e9\u30fc\u5834\u306e\u4f5c\u7528\u7a4d\u5206\u306f<\/p>\r\n

(9) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle S_{\mathrm{QFT}}[\phi] = \int d^{d} x \phi(x) \left(\square+m^{2}\right) \phi(x) \end{eqnarray*}\"<\/p><\/p>\r\n

\u3068\u66f8\u3051\u308b\uff0e\u305f\u3060\u3057\uff0c\"\square := \partial_{\mu} \partial^{\mu}\"\uff0e\u81ea\u7531\u5834\u306e\u5206\u914d\u95a2\u6570\u306f \"Z_{\mathrm{QFT}} = \int \mathcal{D} \phi e^{-S_{\mathrm{QFT}}[\phi(x)]}\" \u3068\u306a\u308b\u306e\u3067\uff0c\u81ea\u7531\u5834\u306e\u78ba\u7387\u5206\u5e03<\/p>\r\n

(10) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle p(\phi) = \frac{e^{-S_{\mathrm{QFT}}}}{Z_{\mathrm{QFT}}} \\ & \displaystyle \int \mathcal{D} \phi p(\phi) = 1 \end{eqnarray*}\"<\/p><\/p>\r\n

\u3092\u5f97\u308b\uff08\u305f\u3060\u3057\uff0c\"\mathcal{D} \phi\" \u306f\u5834\u306e\u7a7a\u9593\u306b\u304a\u3051\u308b\u7d4c\u8def\u7a4d\u5206\u6e2c\u5ea6\u3092\u3042\u3089\u308f\u3057\u3066\u3044\u308b\uff09\uff0e\u307e\u305f\u81ea\u7531\u5834\u306e\"n\"\u70b9\u30b0\u30ea\u30fc\u30f3\u95a2\u6570\u306f<\/p>\r\n

(11) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle G^{(n)}_{\mathrm{QFT}} \left(x_{1}, \ldots, x_{n}\right) = \frac{\int \mathcal{D} \phi \phi\left(x_{1}\right) \ldots \phi\left(x_{n}\right) e^{-S_{\mathrm{QFT}}}}{Z_{\mathrm{QFT}}} \end{eqnarray*}\"<\/p><\/p>\r\n

\u3068\u66f8\u3051\u308b\u306e\u3067 \"(8)\" \u5f0f\u3068\u306e\u5bfe\u5fdc\u95a2\u4fc2\u304c\u307f\u3066\u3068\u308c\u308b\uff0e\u3053\u3046\u3057\u3066\u6f38\u8fd1\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u3068\u81ea\u7531\u5834\u306e\u5834\u306e\u91cf\u5b50\u8ad6\u306e\u5bfe\u5fdc\u95a2\u4fc2\uff08NN-QFT\u5bfe\u5fdc<\/strong>\uff09\u3092\u78ba\u8a8d\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u305f\uff0e<\/p>\r\n<\/div>\r\n\r\n

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\uff11\u2212\uff12. \u6709\u9650\u5e45\u306e\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u3068\u975e\u30ac\u30a6\u30b9\u904e\u7a0b\uff0c\u6442\u52d5\u8ad6\uff08\u5834\u306e\u91cf\u5b50\u8ad6\uff09\u306e\u95a2\u4fc2<\/h4>\r\n

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\u7121\u9650\u5e45\u306e\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\uff08\u6f38\u8fd1\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\uff09\u306e\u4e2d\u9593\u5c64\u306e\u5e45\u3092\u5f90\u3005\u306b\u72ed\u3081\u3066\u6709\u9650\u5e45\u306e\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u306b\u79fb\u884c\u3057\u3066\u3044\u304f\u3068\uff0c\u306f\u3058\u3081\u30ac\u30a6\u30b9\u904e\u7a0b\u306b\u5f93\u3063\u3066\u3044\u305f\u51fa\u529b\u306e\u5206\u5e03\u306f\u975e\u30ac\u30a6\u30b9\u904e\u7a0b\u306b\u5f93\u3046\u3088\u3046\u306b\u306a\u3063\u3066\u3044\u304f\uff0e\u5192\u982d\u3067\u7d39\u4ecb\u3057\u305f\u8ad6\u6587\u304c\u660e\u3089\u304b\u306b\u3057\u305f\u3053\u3068\u306f\uff0c\u5165\u529b\u30c7\u30fc\u30bf\u306e\u7a7a\u9593\u3092\u6642\u7a7a\u3068\u307f\u306a\u3057\u305f\u3068\u304d\uff0c\u6f38\u8fd1\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u304c\u81ea\u7531\u5834\u306b\u5bfe\u5fdc\u3057\u305f\u3088\u3046\u306b\uff0c\u6709\u9650\u5e45\u306e\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u304c\u76f8\u4e92\u4f5c\u7528\u5834\u306b\u5bfe\u5fdc\u3059\u308b\u3068\u3044\u3046\u3053\u3068\u3060\u3063\u305f\uff0e<\/p>\r\n

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\"\"<\/figure>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

<\/p>\r\n

\u76f8\u4e92\u4f5c\u7528\u3092\u6301\u305f\u306a\u3044\u81ea\u7531\u7c92\u5b50\u306f\uff0c\u304a\u4e92\u3044\u7d20\u901a\u308a\u3059\u308b\u3060\u3051\u3067\u89b3\u6e2c\u3059\u308b\u3053\u3068\u3082\u89e6\u308c\u308b\u3053\u3068\u3082\u3067\u304d\u306a\u3044\uff0e\u305d\u306e\u305f\u3081\uff0c\u7269\u7406\u5b66\u8005\u306e\u4e3b\u8981\u306a\u95a2\u5fc3\u306f\u81ea\u7531\u5834\u306e\u5834\u306e\u91cf\u5b50\u8ad6\u3067\u306f\u306a\u304f\u76f8\u4e92\u4f5c\u7528\u5834\u306e\u5834\u306e\u91cf\u5b50\u8ad6\u306b\u96c6\u307e\u308b\uff0e\u3067\u306f\uff0c\u3069\u3046\u3084\u3063\u3066\u76f8\u4e92\u4f5c\u7528\u5834\u306e\u5834\u306e\u91cf\u5b50\u8ad6\u3092\u69cb\u7bc9\u3059\u308b\u304b\u3068\u8a00\u3046\u3068\uff0c\u7d20\u7c92\u5b50\u3092\u52a0\u901f\u3057\u3066\u9ad8\u30a8\u30cd\u30eb\u30ae\u30fc\u72b6\u614b\u306b\u3057\u305f\u3046\u3048\u3067\u885d\u7a81\u30fb\u6563\u4e71\u3055\u305b\uff0c\u305d\u306e\u6563\u4e71\u554f\u984c\u306e\u9077\u79fb\u78ba\u7387\u3092\u7cbe\u78ba\u306b\u4e88\u6e2c\u3067\u304d\u308b\u304b\u3069\u3046\u304b\u306b\u3088\u3063\u3066\u7406\u8ad6\u306e\u6b63\u3057\u3055\u3092\u691c\u8a3c\u3057\u3066\u3044\u304f\uff0e\u3053\u306e\u6563\u4e71\u554f\u984c\u306e\u9077\u79fb\u78ba\u7387\u3092\u8a08\u7b97\u3059\u308b\u969b\uff0c\u91cd\u8981\u306b\u306a\u308b\u306e\u304c\uff11\u2212\uff11\u3067\u767b\u5834\u3057\u305f\"n\"\u70b9\u30b0\u30ea\u30fc\u30f3\u95a2\u6570\u306b\u306a\u308b\uff0e<\/p>\r\n

\"n\"\u70b9\u30b0\u30ea\u30fc\u30f3\u95a2\u6570\u3092\u8a08\u7b97\u3059\u308b\u305f\u3081\u306b\uff0c\u307e\u305a1\u5909\u6570\u30ac\u30a6\u30b9\u7a4d\u5206\u306e\u516c\u5f0f\u306b\u3064\u3044\u3066\u601d\u3044\u51fa\u305d\u3046\uff0e<\/p>\r\n

(12) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle \int_{-\infty}^{\infty} d q e^{-\frac{a}{2} q^{2}-J q} = \sqrt{\frac{2 \pi}{a}} e^{\frac{1}{2 a} J^{2}} \quad(a \in \mathbb{C}, \operatorname{Re}(a)>0) \end{eqnarray*}\"<\/p><\/p>\r\n

\u3053\u308c\u306f\u3059\u3050\u306b\u591a\u5909\u6570\u30ac\u30a6\u30b9\u7a4d\u5206\u306e\u516c\u5f0f\u306b\u62e1\u5f35\u3067\u304d\u308b\uff0e<\/p>\r\n

(13) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle \int_{-\infty}^{\infty} d q_{1} \cdots \int_{-\infty}^{\infty} d q_{N} \exp \left[-\frac{1}{2} \sum_{i, j=1}^{N} q_{i} A_{i j} q_{j} - \sum_{i=1}^{N} J_{i} q_{i}\right] = \sqrt{\frac{(2 \pi)^{N}}{\operatorname{det} A}} \exp \left[\frac{1}{2} \sum_{i, j=1}^{N} J_{i}\left(A^{-1}\right)_{i j} J_{j}\right] \end{eqnarray*}\"<\/p><\/p>\r\n

\u3053\u308c\u3092\u884c\u5217\u8868\u793a\u3067\u66f8\u304d\u76f4\u3059\u3068<\/p>\r\n

(14) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle \int_{-\infty}^{\infty} d^{N} \vec{q} \exp \left[-\frac{1}{2} \vec{q}^{T} A \vec{q}-\vec{J}^{T} \vec{q}\right] = \sqrt{\frac{(2 \pi)^{N}}{\operatorname{det} A}} \exp \left[\frac{1}{2} \vec{J}^{T} A^{-1} \vec{J}\right] \end{eqnarray*}\"<\/p><\/p>\r\n

\u3068\u66f8\u3051\u308b\uff0e\u307e\u305f\uff0c\"S[\vec{q}] := \frac{1}{2} \vec{q}^{T} A \vec{q}+\vec{J}^{T} \vec{q}\" \u3068\u5b9a\u7fa9\u3059\u308b\u3068\uff0c\u3053\u306e\u516c\u5f0f\u306f<\/p>\r\n

(15) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle \int_{-\infty}^{\infty} d^{N} \vec{q} e^{-S[\vec{q}]} = \sqrt{\frac{(2 \pi)^{N}}{\operatorname{det} A}} e^{-\left.S[\vec{q}]\right|_{\delta S[\vec{q}]=0}} \end{eqnarray*}\"<\/p><\/p>\r\n

\u3068\u66f8\u304d\u76f4\u305b\u308b\uff0e\u3053\u3053\u3067\u53f3\u8fba\u306e \"\left.S[\vec{q}]\right|_{\delta S[\vec{q}]=0}\" \u306f\uff08\"S[\vec{q}]\" \u3092\u4f5c\u7528\u7a4d\u5206\u3068\u307f\u306a\u3057\u305f\u3068\u304d\u306e\uff09\u4f5c\u7528\u539f\u7406 \"\delta S[\vec{q}]=0 \Leftrightarrow A \vec{q}+\vec{J}=0\" \u306e\u89e3\u3092 \"S[\vec{q}]\" \u306b\u4ee3\u5165\u3057\u305f\u3082\u306e\u306b\u306a\u3063\u3066\u3044\u308b\uff0e\u3064\u307e\u308a\uff0c\"S[\vec{q}]\" \u304c \"\vec{q}\" \u306e2\u6b21\u5f62\u5f0f\uff08\u81ea\u7531\u7c92\u5b50\u306b\u5bfe\u5fdc\uff09\u3067\u4e0e\u3048\u3089\u308c\u308b\u3068\u304d\u306f\uff0c\u30ac\u30a6\u30b9\u7a4d\u5206\u3092\u5b9f\u884c\u3059\u308b\u3053\u3068\u3068\uff0c\u904b\u52d5\u65b9\u7a0b\u5f0f\uff08\"\Leftrightarrow\" \u4f5c\u7528\u539f\u7406 \"\delta S[\vec{q}]=0\"\uff09\u3092\u89e3\u3044\u3066\u305d\u306e\u89e3\u3092 \"S[\vec{q}]\" \u306b\u4ee3\u5165\u3059\u308b\u3053\u3068\u306f\u7b49\u4fa1\u306b\u306a\u308b\uff0e\u30ac\u30a6\u30b9\u7a4d\u5206\u306e\u516c\u5f0f\u306f\u7269\u7406\u5b66\u8005\u306b\u3068\u3063\u3066\u306f\u3053\u306e\u3088\u3046\u306a\u30e1\u30c3\u30bb\u30fc\u30b8\u3092\u6301\u3063\u3066\u3044\u308b\uff0e<\/p>\r\n

\u3053\u306e\u591a\u5909\u6570\u30ac\u30a6\u30b9\u7a4d\u5206\u306e\u516c\u5f0f\u3092\u4f7f\u3048\u3070\uff0c\u30ac\u30a6\u30b9\u904e\u7a0b\u306e\u5206\u914d\u95a2\u6570\u306f<\/p>\r\n

(16) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle Z_{\mathrm{GP}}[J] = \frac{\int \mathcal{D} f e^{-S_{\mathrm{GP}}-\frac{1}{2} \int d^{d_{\mathrm{in}}} x J(x) f(x)-\frac{1}{2} \int d^{d_{\mathrm{in}}} x^{\prime} J(x^{\prime}) f(x^{\prime})}}{Z_{\mathrm{GP}}} \\ & \displaystyle \hspace{20pt} = \exp \left(\frac{1}{2} \int d^{d_{\mathrm{in}}} x d^{d_{\mathrm{in}}} x^{\prime} J(x) K(x, x^{\prime}) J(x^{\prime})\right) \end{eqnarray*}\"<\/p><\/p>\r\n

\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b\uff0e\u3053\u306e\u5206\u914d\u95a2\u6570\u3092\u4f7f\u3048\u3070\uff0c\u30ac\u30a6\u30b9\u904e\u7a0b\u306e\"n\"\u70b9\u30b0\u30ea\u30fc\u30f3\u95a2\u6570\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\u7c21\u5358\u306b\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\uff0e<\/p>\r\n

(17) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle G_{\mathrm{GP}}^{(n)}\left(x_{1}, \ldots, x_{n}\right) = \left.\left[\left(-\frac{\delta}{\delta J\left(x_{1}\right)}\right) \ldots\left(-\frac{\delta}{\delta J\left(x_{n}\right)}\right) Z_{\mathrm{GP}}[J]\right]\right|_{J=0} \\ & \displaystyle \hspace{3pt} = \frac{\int \mathcal{D} f f\left(x_{1}\right) \ldots f\left(x_{n}\right) e^{-S_{\mathrm{GP}}}}{Z_{\mathrm{GP}}} \end{eqnarray*}\"<\/p><\/p>\r\n

\u3053\u306e\u3068\u304d\uff0c\"(16)-(17)\"\u5f0f\u304b\u3089\uff0c\"n\"\u304c\u5947\u6570\u306e\u3068\u304d\u306f\"n\"\u70b9\u30b0\u30ea\u30fc\u30f3\u95a2\u6570\u306e\u5024\u304c0\u306b\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b\uff0e<\/p>\r\n

\u3053\u3053\u3067\u6709\u9650\u5e45\u306e\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u306e\uff08\u672a\u77e5\u306e\uff09\u4f5c\u7528\u7a4d\u5206\u3092 \"S\" \uff0c\"n\"\u70b9\u30b0\u30ea\u30fc\u30f3\u95a2\u6570\u3092 \"G^{(n)}\left(x_{1}, \ldots x_{n}\right)\" \u3068\u7f6e\u304d\uff0c\uff08\u65e2\u77e5\u306e\uff09\u7121\u9650\u5e45\u306e\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\uff08\u6f38\u8fd1\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\uff09\u306e\u5834\u5408\u306e\u305d\u308c\u3068\u306e\u30ba\u30ec\u3092\u305d\u308c\u305e\u308c \"\Delta S\"\uff0c\"\Delta G^{(n)}\" \u3068\u8868\u3059\u3053\u3068\u306b\u3059\u308b\uff0e<\/p>\r\n

(18) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle \Delta S = S - S_{\mathrm{GP}} \\ & \displaystyle \Delta G^{(n)}\left(x_{1}, \ldots, x_{n}\right) = G^{(n)}\left(x_{1}, \ldots x_{n}\right) - G_{\mathrm{GP}}^{(n)}\left(x_{1}, \ldots, x_{n}\right) \end{eqnarray*}\"<\/p><\/p>\r\n

\u3053\u3053\u3067\"n\"\u70b9\u30b0\u30ea\u30fc\u30f3\u95a2\u6570 \"G^{(n)}\left(x_{1}, \ldots x_{n}\right)\" \u306f<\/p>\r\n

(19) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle G^{(n)}\left(x_{1}, x_{2}, \cdots, x_{n}\right) := \left\langle 0\left|\mathrm{T}\left[\phi\left(x_{1}\right) \phi\left(x_{2}\right) \cdots \phi\left(x_{n}\right)\right]\right| 0\right\rangle \end{eqnarray*}\"<\/p><\/p>\r\n

(20) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle \mathrm{T}\left[\phi\left(x_{1}\right) \phi\left(x_{2}\right)\right] = \theta\left(x_{1}^{0}-x_{2}^{0}\right) \phi\left(x_{1}\right) \phi\left(x_{2}\right) + \theta\left(x_{2}^{0}-x_{1}^{0}\right)\phi\left(x_{2}\right) \phi\left(x_{1}\right) \\ & \displaystyle \hspace{-28pt} = \left\{\begin{array}{cl} \phi\left(x_{1}\right) \phi\left(x_{2}\right) & \left(x_{1}^{0}>x_{2}^{0}\right) \\ \phi\left(x_{2}\right) \phi\left(x_{1}\right) & \left(x_{2}^{0}>x_{1}^{0}\right) \end{array}\right. \end{eqnarray*}\"<\/p><\/p>\r\n

\u307e\u305f\uff0c\u30ac\u30a6\u30b9\u904e\u7a0b\u306e\"n\"\u70b9\u30b0\u30ea\u30fc\u30f3\u95a2\u6570\u306f\uff0c\u30a6\u30a3\u30c3\u30af\u306e\u5b9a\u7406\u304b\u3089\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a08\u7b97\u3067\u304d\u308b\u306e\u3067<\/p>\r\n

(21) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle \operatorname{Wick}\left(x_{1}, \ldots, x_{n}\right) = \left\{P \in \operatorname{Partitions}\left(x_{1}, \ldots, x_{n}\right)\mid \ | p |=2 \forall p \in P\right\} \end{eqnarray*}\"<\/p><\/p>\r\n

(22) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle G_{\mathrm{GP}}^{(n)}\left(x_{1}, \ldots, x_{n}\right) = \sum_{p \in \operatorname{Wick}\left(x_{1}, \ldots, x_{n}\right)} K\left(a_{1}, b_{1}\right) \ldots K\left(a_{n / 2}, b_{n / 2}\right) \end{eqnarray*}\"<\/p><\/p>\r\n

\"\Delta G^{(n)}\" \u306f\u6b21\u306e\u3088\u3046\u306b\u5f62\u5f0f\u7684\u306b\u66f8\u304d\u4e0b\u3059\u3053\u3068\u304c\u3067\u304d\u308b\uff0e<\/p>\r\n

(23) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle \hspace{0pt} \Delta G^{(2)} = G^{(2)}\left(x_{1}, x_{2}\right) - G_{\mathrm{GP}}^{(2)}\left(x_{1}, x_{2}\right) \\ <\/p><\/p>\r\n

(24) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle \hspace{0pt} \Delta G^{(4)} = G^{(4)}\left(x_{1}, x_{2}, x_{3}, x_{4}\right) - G_{\mathrm{GP}}^{(4)}\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \\ <\/p><\/p>\r\n

\u3053\u308c\u3067\u3088\u3046\u3084\u304f\u6e96\u5099\u304c\u6574\u3063\u305f\uff0e\u6b21\u7bc0\u3067\u306f\u3053\u306e \"\Delta G^{(n)}\" \u3092\u6709\u9650\u5e45\u306e\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u3092\u4f7f\u3063\u3066\u5177\u4f53\u7684\u306b\u8a08\u7b97\u3057\uff0c\u7406\u8ad6\u304b\u3089\u4e88\u6e2c\u3055\u308c\u308b\u6027\u8cea\u3068\u6bd4\u8f03\u3057\u3066\u307f\u3088\u3046\uff0e<\/p>\r\n<\/div>\r\n\r\n

<\/p>\r\n

\r\n

\uff11\u2212\uff13. \u6709\u9650\u5e45\u306e\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u3092\u4f7f\u3063\u305f\u5b9f\u9a13\u7d50\u679c<\/h4>\r\n

<\/p>\r\n

<\/p>\r\n

\u81ea\u7531\u5834\uff08\u30ac\u30a6\u30b9\u904e\u7a0b=\u6f38\u8fd1\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\uff09\u306e\"n\"\u70b9\u30b0\u30ea\u30fc\u30f3\u95a2\u6570\u306f\uff0c\u76f8\u4e92\u4f5c\u7528\u3092\u3082\u305f\u306a\u3044\uff08\uff1d\u4ea4\u70b9\u3092\u542b\u307e\u306a\u3044\uff09\u76f4\u7dda\u304b\u3089\u306a\u308b\u30b7\u30f3\u30d7\u30eb\u306a\u30d5\u30a1\u30a4\u30f3\u30de\u30f3\u56f3\u3067\u3042\u3089\u308f\u305b\u308b\uff0e<\/p>\r\n

\u81ea\u7531\u5834\u306e2\u70b9\u30b0\u30ea\u30fc\u30f3\u95a2\u6570\u306a\u3089<\/p>\r\n

\"Rendered<\/p><\/p>\r\n

\u81ea\u7531\u5834\u306e4\u70b9\u30b0\u30ea\u30fc\u30f3\u95a2\u6570\u306a\u3089<\/p>\r\n

\"Rendered<\/p><\/p>\r\n

\u3068\u3044\u3063\u305f\u5177\u5408\u306b\u306a\u308b\uff0e\u3053\u306e\u30d5\u30a1\u30a4\u30f3\u30de\u30f3\u56f3\u3092\u4f7f\u3046\u3068\uff0c\uff11\u2212\uff12\u3067\u7d39\u4ecb\u3057\u305f \"\Delta G^{(n)}\" \u306f\u6b21\u306e\u3088\u3046\u306b\u66f8\u304d\u3042\u3089\u308f\u305b\u308b\uff0e<\/p>\r\n

\"Rendered<\/p><\/p>\r\n

\"Rendered<\/p><\/p>\r\n

\u3053\u3053\u3067\"n\"\u70b9\u30b0\u30ea\u30fc\u30f3\u95a2\u6570\u306e\u6a19\u6e96\u5316\u3055\u308c\u305f\u504f\u5dee\b<\/p>\r\n

(25) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle m_{n}\left(x_{1}, \ldots, x_{n}\right) := \frac{\Delta G^{(n)}\left(x_{1}, \ldots, x_{n}\right)}{G_{\mathrm{GP}}^{(n)}\left(x_{1}, \ldots, x_{n}\right)}<\/p><\/p>\r\n

\u3068\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u306e\u4e2d\u9593\u5c64\u306e\u5e45 \"N\" \u306e\u95a2\u4fc2\u3092\u307f\u3066\u307f\u3088\u3046\uff0e\u5b9f\u9a13\u306e\u30bb\u30c3\u30c8\u30a2\u30c3\u30d7\u306f\u4ee5\u4e0b\u306e\u901a\u308a\uff0e<\/p>\r\n

\u6d3b\u6027\u5316\u95a2\u6570\u3092\uff08\u6307\u6570\u95a2\u6570\uff0b\u6b63\u898f\u5316\uff09\u306b\u8a2d\u5b9a\u3057<\/p>\r\n

(26) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle g\left(W_{0} x+b_{0}\right) = \frac{\exp \left(W_{0} x+b_{0}\right)}{\sqrt{\exp \left[2\left(\sigma_{b}^{2}+\frac{\sigma_{W}^{2}}{d_{\text {in }}} x^{2}\right)\right]}}<\/p><\/p>\r\n

\u30ab\u30fc\u30cd\u30eb\u95a2\u6570\u306f\uff08\u30ac\u30a6\u30b9\u30ab\u30fc\u30cd\u30eb\uff09\u306b\u8a2d\u5b9a\u3059\u308b\uff0e<\/p>\r\n

(27) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle K(x, y) := \sigma_{b}^{2}+K_{W}(x, y), \quad K_{W}(x, y) = \sigma_{W}^{2} \mathrm{e}^{-\frac{\sigma_{W}^{2}}{2 d_{\text {in }}}|x-y|^{2}}<\/p><\/p>\r\n

\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30fb\u30a2\u30f3\u30b5\u30f3\u30d6\u30eb\u306e\u6570\uff0c\u30c1\u30e3\u30f3\u30af\u306e\u6570\u3092<\/p>\r\n

(28) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle n_{\text {net }} = 10^{5}, \quad n_{\text {bags }} = 10^{2}<\/p><\/p>\r\n

\u5165\u529b\u30c7\u30fc\u30bf\u306e\u6b21\u5143, \u30a6\u30a7\u30a4\u30c8\u306e\u5206\u6563\uff0c\u30d0\u30a4\u30a2\u30b9\u306e\u5206\u6563\uff0c\u4e2d\u9593\u5c64\u306e\u5e45\u3092<\/p>\r\n

(29) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle d_{\text {in }} = 1,\ \sigma_{W} = 1,\ \sigma_{b} = 1,\ N \in\{2,3,4,5,10,20,50,100,500,1000\}<\/p><\/p>\r\n

\u5165\u529b\u30c7\u30fc\u30bf\u3092<\/p>\r\n

(30) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle x_{1}, \ldots, x_{6} \in\{-0.01,-0.006,-0.002,0.002,0.006,0.01\}<\/p><\/p>\r\n

\u3068\u3057\u3066\uff0c\u7406\u8ad6\u304b\u3089\u4e88\u6e2c\u3055\u308c\u308b\"n\"\u70b9\u30b0\u30ea\u30fc\u30f3\u95a2\u6570\uff0c\u504f\u5dee\u306e\u6027\u8cea<\/p>\r\n

(31) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle G^{(2)}(x_{1}, x_{2}) \approx K(x_{1}, x_{2}) \quad \Longrightarrow \quad m_{2}(x_{1}, x_{2}) \approx 0, \quad \forall N<\/p><\/p>\r\n

(32) <\/span>   <\/span>\"\begin{eqnarray*} & \displaystyle m_{4}\left(x_{1}, \ldots, x_{4}\right) = \mathcal{O}(1 / N), \quad m_{6}\left(x_{1}, \ldots, x_{6}\right) = \mathcal{O}(1 / N)<\/p><\/p>\r\n

\u304c\u78ba\u304b\u3081\u3089\u308c\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3057\u3088\u3046\uff0e<\/p>\r\n

\r\n
\"\"<\/figure>\r\n\r\n\r\n
\"\"<\/figure>\r\n\r\n\r\n
\"\"<\/figure>\r\n<\/figure>\r\n

\u3053\u3053\u3067\u30d0\u30c3\u30af\u30b0\u30e9\u30a6\u30f3\u30c9\u306f\u504f\u5dee \"m_{n}\" \u306e\u6a19\u6e96\u504f\u5dee\u306e\u5e73\u5747\u5024\u3092\u3042\u3089\u308f\u3057\u3066\u3044\u308b\uff0e\u4e2d\u592e\u306e\"4\"\u70b9\u30b0\u30ea\u30fc\u30f3\u95a2\u6570\uff0c\u53f3\u7aef\u306e\"6\"\u70b9\u30b0\u30ea\u30fc\u30f3\u95a2\u6570\u3067\u306f\u4e2d\u9593\u5c64\u306e\u5e45 \"N\" \u304c\u5927\u304d\u304f\u306a\u308b\u306b\u3064\u308c\u3066\uff0c\u504f\u5dee \"m_{n}\" \uff0c\u30d0\u30c3\u30af\u30b0\u30e9\u30a6\u30f3\u30c9\u3068\u3082\u306b \"N^{-1}\" \u306e\u30aa\u30fc\u30c0\u30fc\u3067\u5c0f\u3055\u304f\u306a\u3063\u3066\u3044\u304f\u3053\u3068\u304c\u307f\u3066\u3068\u308c\u308b\uff0e\u3053\u3046\u3057\u3066 \"N\" \u304c\u5927\u304d\u304f\u306a\u308b\u306b\u3064\u308c\u3066\uff0c\u2460\u6709\u9650\u5e45\u306e\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\uff08\u975e\u30ac\u30a6\u30b9\u904e\u7a0b\uff09\u3068\u6f38\u8fd1\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\uff08\u30ac\u30a6\u30b9\u904e\u7a0b\uff09\u306e\u4e56\u96e2\u304c\u5f90\u3005\u306b\u72ed\u307e\u3063\u3066\u3044\u304f\u3053\u3068\uff0c\u2461\u51fa\u529b\u306e\u76f8\u95a2\u95a2\u6570\uff08\"n\"\u70b9\u30b0\u30ea\u30fc\u30f3\u95a2\u6570\uff09\u306e\u632f\u308b\u821e\u3044\u304c\uff08\u7406\u8ad6\u306e\u4e88\u6e2c\u901a\u308a\uff09\u6b21\u6570\"n\"\u306b\u3088\u3063\u3066\u7570\u306a\u308b\u3053\u3068\u304c\u5b9f\u9a13\u7684\u306b\u3082\u78ba\u304b\u3081\u3089\u308c\u305f\uff0e<\/p>\r\n

\u3064\u307e\u308a\uff0c\u6f38\u8fd1\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\uff08\u30ac\u30a6\u30b9\u904e\u7a0b\uff09\u21d4 \u81ea\u7531\u5834\uff0c\u6709\u9650\u5e45\u306e\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\uff08\u975e\u30ac\u30a6\u30b9\u904e\u7a0b\uff09\u21d4 \u76f8\u4e92\u4f5c\u7528\u5834\u306e\u5bfe\u5fdc\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3064\u3068\u3044\u3046\u4eee\u8aac\u306b\u3064\u3044\u3066\u5b9f\u9a13\u7684\u508d\u8a3c\u304c\u5f97\u3089\u308c\u305f\uff01<\/p>\r\n<\/div>\r\n

<\/p>\r\n

  • 1<\/span>
    J. Halverson, A. Maiti and K. Stoner, “Neural networks and quantum field theory”, Machine Learning: Science and Technology, Volume 2, Number 3 (2021)<\/div><\/li>
  • 2<\/span>
    A. Maiti, K. Stoner and J. Halverson, “Symmetry-via-Duality: Invariant Neural Network Densities from Parameter-Space Correlators”, arXiv:2106.00694 [hep-th]. (2021)<\/div><\/li>
  • 3<\/span>
    H. Erbin, V. Lahoche and D. O. Samary, “Nonperturbative renormalization for the neural network-QFT correspondence”, arXiv: 2108.01403 [hep-th]. (2021)<\/div><\/li><\/ul>","protected":false},"excerpt":{"rendered":"

    \u5834\u306e\u91cf\u5b50\u8ad6\u306b\u304a\u3044\u3066\uff0c\u81ea\u7531\u5834\u306f\u95a2\u6570\u7a7a\u9593\u4e0a\u306e\u30ac\u30a6\u30b9\u5206\u5e03\u3068\u3057\u3066\u8a18\u8ff0\u3055\u308c\uff0c\u76f8\u4e92\u4f5c\u7528\u5834\u306f\u81ea\u7531\u5834\u306e\u4f5c\u7528\u3078\u306e\u975e\u30ac\u30a6\u30b9\u7684\u306a\u88dc\u6b63\uff08\u6442\u52d5\uff09\u306b\u3088\u3063\u3066\u8a18\u8ff0\u3055\u308c\u308b\uff0e\u9762\u767d\u3044\u3053\u3068\u306b\uff0c\u4e2d\u9593\u5c64\u306e\u5e45\u3092\u7121\u9650\u306b\u5e83\u304f\u3068\u3063\u305f\u6df1\u5c64\u5b66\u7fd2\u30e2\u30c7\u30eb\u306e\u4e00\u90e8\u304c\u3084\u306f\u308a\u30ac\u30a6\u30b9\u904e\u7a0b\u3067 […]<\/p>\n","protected":false},"author":135576855,"featured_media":1405,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_coblocks_attr":"","_coblocks_dimensions":"","_coblocks_responsive_height":"","_coblocks_accordion_ie_support":"","advanced_seo_description":"","jetpack_seo_html_title":"","jetpack_seo_noindex":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"_wpas_customize_per_network":false,"jetpack_post_was_ever_published":false},"categories":[1],"tags":[],"class_list":["post-1337","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"https:\/\/i0.wp.com\/hayashiyus.jp\/wp-content\/uploads\/2021\/10\/AdobeStock_424836455.jpeg?fit=12000%2C6000&ssl=1","jetpack_likes_enabled":true,"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/scnxx5-test","_links":{"self":[{"href":"https:\/\/hayashiyus.jp\/index.php?rest_route=\/wp\/v2\/posts\/1337","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/hayashiyus.jp\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/hayashiyus.jp\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/hayashiyus.jp\/index.php?rest_route=\/wp\/v2\/users\/135576855"}],"replies":[{"embeddable":true,"href":"https:\/\/hayashiyus.jp\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1337"}],"version-history":[{"count":100,"href":"https:\/\/hayashiyus.jp\/index.php?rest_route=\/wp\/v2\/posts\/1337\/revisions"}],"predecessor-version":[{"id":2329,"href":"https:\/\/hayashiyus.jp\/index.php?rest_route=\/wp\/v2\/posts\/1337\/revisions\/2329"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/hayashiyus.jp\/index.php?rest_route=\/wp\/v2\/media\/1405"}],"wp:attachment":[{"href":"https:\/\/hayashiyus.jp\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1337"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/hayashiyus.jp\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1337"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/hayashiyus.jp\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1337"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}