From 2eb0b2e01396838ac99e728a23b96bf662829932 Mon Sep 17 00:00:00 2001 From: Teruki Tada Date: Sun, 12 Jan 2025 00:31:21 +0900 Subject: [PATCH] pinns --- .../PINNs-part1/img/boundary.webp" | Bin 0 -> 9504 bytes .../PINNs-part1/img/burgers-025.png" | Bin 0 -> 12198 bytes .../PINNs-part1/img/burgers-05.png" | Bin 0 -> 11522 bytes .../PINNs-part1/img/burgers-075.png" | Bin 0 -> 10982 bytes .../PINNs-part1/img/burgers-discrete.webp" | Bin 0 -> 17546 bytes .../img/burgers-layers-neurons.webp" | Bin 0 -> 7172 bytes .../PINNs-part1/img/burgers-nu-nf.webp" | Bin 0 -> 17612 bytes .../PINNs-part1/img/burgers-result.webp" | Bin 0 -> 21560 bytes .../PINNs-part1/img/collocation.webp" | Bin 0 -> 28762 bytes .../PINNs-part1/img/trains.webp" | Bin 0 -> 30670 bytes .../PINNs-part1/slide.md" | 389 ++++++++++++++++-- 11 files changed, 364 insertions(+), 25 deletions(-) create mode 100644 "slides/2025/\347\240\224\347\251\266\345\256\244\346\264\273\345\213\225/PINNs-part1/img/boundary.webp" create mode 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100644 --- "a/slides/2025/\347\240\224\347\251\266\345\256\244\346\264\273\345\213\225/PINNs-part1/slide.md" +++ "b/slides/2025/\347\240\224\347\251\266\345\256\244\346\264\273\345\213\225/PINNs-part1/slide.md" @@ -14,78 +14,417 @@ paginate: true # "Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations" -発表者: 多田 瑛貴 (公立はこだて未来大学 島内研究室) +島内研究室 2025年1月14日 輪読会 + +発表者: 多田 瑛貴 (公立はこだて未来大学 学部3年) + +![bg right:20%](img/tokyo.webp) + +--- + +## 書誌情報 + +**Physics Informed Deep Learning (Part I): +Data-driven Solutions of Nonlinear Partial Differential Equations** + +著者: Maziar Raissi, Paris Perdikaris, George Em Karniadakis +arXiv:1711.10561 (https://arxiv.org/abs/1711.10561) + +2017年発表の論文、Part IとPart IIの2部構成 + +- Part I: data-driven solution (深層学習による非線形偏微分方程式の数値解法) +- Part II: data-driven discovery (PINNsのパラメータ不定の方程式に対する適用) + --- ## 本研究の概要 "physics-informed neural networks" (PINNs) の提案 -- 非線形偏微分方程式を解くデータ駆動型のアプローチ -- 境界条件を教師データとしつつ、偏微分方程式を与えることで - 物理法則を満たすようニューラルネットワークを学習 -- 既知の物理法則に対して、少量のデータから有用な近似解を提供 -- パラメータ不定の微分方程式に対しても適用可能 (Part IIで詳説) +- 非線形偏微分方程式を解くデータ駆動のアプローチ +- ある物理現象の支配方程式を考慮した深層学習を導入 +- 既知の物理法則に対して、**少量の教師データから有用な近似解を提供** +- パラメータ不定の偏微分方程式に対しても適用可能 (Part IIの範囲) --- # 背景 -非線形写像を深層学習を使って学習するとき +- データ駆動での数値解法のアプローチとして、深層学習を用いたい +*物理法則を深層学習を用いて解こうとする意義は、研究結果を踏まえ後述する* + +- 物理法則を深層学習を用いて解くとき 現実の多くの場面では、教師データを十分な量取得できない *データが少ない場合、結果のロバスト性や、学習を通したモデルの収束を保証できなくなる* --> 学習に既知の支配方程式を利用したい +- 既知の支配方程式を使えないだろうか? --- -## 従来手法 +# 提案手法 + +- 教師データだけでなく、支配方程式を損失関数に取り入れる +*具体的には、時空間領域上に選点(collocation point)をとり、 +その点での支配方程式の残差を損失とする* +*教師データは、ディレクレ境界条件から与える* -- ルンゲ・クッタ法 -- 有限要素法 -- Gaussian process regression (GPR) -(一旦省略) +- 非線形偏微分方程式をそのまま (線形化等をせず) 適用 --- -# 提案手法 +## 導入対象: Burgers方程式 + +一般に、動的粘性率$\nu$から + +$$ u_t + uu_x - \nu u_{xx} = 0 $$ + +ナビエ-ストークス方程式から、圧力項を無視したもの +*ナビエ-ストークス方程式: 流体の運動を記述する偏微分方程式 +(非圧縮性粘性流体の運動方程式)* + +ここでは、パラメータおよび初期・境界条件を +次のように定め扱うこととする + +$$ +\begin{cases} +u_t + uu_x - (0.01/\pi)u_{xx} = 0 \\ +u(0, x) = -\sin(\pi x) \\ +u(t, -1) = u(t, 1) = 0 +\end{cases} +$$ + +![bg right:25%](img/burgers-025.png) +![bg right:25% vertical](img/burgers-05.png) +![bg right:25% vertical](img/burgers-075.png) --- -## 学習対象 +## 学習の目標 支配方程式を次のように表現 ($u(t, x)$: 未知関数, $\lambda$: パラメータ) $$ u_t + \mathcal{N}[u; \lambda] = 0 $$ -現実の物理法則の多くは、この形式で表現可能 -*例: Burgers方程式 $u_t+\lambda_1uu_x-\lambda_2u_{xx}=0$ より $\mathcal{N}[u; \lambda] = \lambda_1uu_x-\lambda_2u_{xx}$* -この式を満たす写像 $u(t,x)$ を学習する +現実の物理法則の支配方程式の多くは、この形式で表現可能 + +*Burgers方程式においては... $u_t+\lambda_1uu_x-\lambda_2u_{xx}=0$ より $\mathcal{N}[u; \lambda] = \lambda_1uu_x-\lambda_2u_{xx}$* --- -## 学習の方針 + +本論文 Part I ではパラメータ$\lambda$の不定性を考慮せず、次のように表現 + +$$ u_t + \mathcal{N}[u] = 0 $$ + +**目標: 未知関数 $u(t,x)$ を近似するニューラルネットワークを作る** +*ただし、$x \in \Omega, t \in [0, T]$ とし、$\Omega$は$\mathbb{R}$の部分集合* + +``` +def u(t, x): + u = neural_net(tf.concat([t,x],1), weights, biases) + return u +``` --- -# 実験 +## 学習のおおまかな方針 + +- 教師データに対する誤差の損失関数 $MSE_u$ +- 支配方程式に対する誤差の損失関数 $MSE_f$ + +この和である + +$$ MSE = MSE_u + MSE_f $$ + +を全体の損失関数として最小化する --- -# まとめ +### 教師データの設定 + +ディレクレ境界条件から +$$ +\begin{cases} +u(0, x) = -\sin(\pi x) \\ +u(t, -1) = u(t, 1) = 0 +\end{cases} +$$ + +この条件を満たす時空間領域上の点を設け +教師データ$(t_u^i, x_u^i, u^i)$とする +*$(t_u^i, x_u^i)$: 時空間上の点、$u^i$: その点における$u$の値* + +損失関数は、この平均二乗誤差$MSE_u$ +*$N_u$: 教師データの数* + +$$ MSE_u = \frac{1}{N_u} \sum_{i=1}^{N_u} |u(t_u^i, x_u^i) - u^i|^2 $$ + +![bg right:40% w:500](img/boundary.webp) + +--- + +### 支配方程式の損失関数 + +支配方程式 +$$ u_t + uu_x - (0.01/\pi)u_{xx} = 0 $$ + +![bg right:40% w:500](img/collocation.webp) + +時空間領域からランダムにおいた選点 +$(t_f^i, x_f^i)$ をおき、残差を計算 + +$$ f(t_f^i, x_f^i) = u_t + uu_x - (0.01/\pi)u_{xx} $$ + +$f$を0に近づける *=方程式を満たす* よう学習 +*具体的な$u$の値は求めない* +損失関数は、平均二乗誤差$MSE_f$ + +$$ MSE_f = \frac{1}{N_f} \sum_{i=1}^{N_f} |f(t_f^i, x_f^i)|^2 $$ + +--- + +### 補足: $MSE_f$の計算 + +$$ f(t_f^i, x_f^i) = u_t + uu_x - (0.01/\pi)u_{xx} $$ + +ニューラルネットワークに対する自動微分から得る + +``` +def f(t, x): + u = u(t, x) + u_t = tf.gradients(u, t)[0] + u_x = tf.gradients(u, x)[0] + u_xx = tf.gradients(u_x, x)[0] + f = u_t + u*u_x - (0.01/tf.pi)*u_xx + return f +``` + +--- + +## 損失関数の設定 + +全体の損失関数は + +$$ MSE = MSE_u + MSE_f $$ + +![bg right:40% w:500](img/trains.webp) + +右図は、$N_u=100, N_f=10000$ + +*参考: $N_f=0$であれば単に教師データのみに基づく* +*深層学習本来のアプローチと同じとなる* + +--- + +## 実験 + +- $N_u=100, N_f=10000$ +- 9層(中間7層)のニューラルネットワーク +- 中間層ニューロン数20 +- 活性化関数: tanh +- 最適化手法: L-BFGS +- 選点の生成手法: Latin hypercube sampling + +--- + +### 支配方程式・境界条件へのあてはまり + + +![w:780](img/burgers-result.webp) + +相対二乗誤差(RSE): $6.7 \times 10^{-4}$ +*上図: $tx$ 平面上の予測値$u$の値, 下図: $t=0.25, 0.50, 0.75$における$x$に対する$u$の値* + +--- + +### $N_u$・$N_f$の値と性能の関係 + +相対二乗誤差による比較、9層(中間7層)、中間層ニューロン数20 + +![w:900](img/burgers-nu-nf.webp) + +選点$N_f$を与えることによって精度が向上していることがわかる + +--- + +### レイヤ数・中間層ニューロン数と性能の関係 + +相対二乗誤差による比較、$N_u=100, N_f=10000$ + +![w:900](img/burgers-layers-neurons.webp) + +概ね、レイヤ・ニューロン数が多いほど精度が高まる + +--- + +## 複素関数への適用 + +Burgers方程式を近似するPINNでは、2入力1出力の実数関数をNNで表現していた +-> 複素関数の場合、実部と虚部を分けた2出力を用いる + +--- + +## 導入例: Schrödinger方程式 + +$$ +\begin{cases} +ih_t + 0.5h_{xx} + |h|^2h = 0 \\ +h(0, x) = 2\mathrm{sech}(x) \\ +h(t, -5) = h(t, 5) = 0 \\ +h_x(t, -5) = h_x(t, 5) = 0 +\end{cases} +$$ +$h = u + iv$とすると、出力は$h(t,x) = [u(t,x), v(t,x)]$となる +この場合のPINNの損失関数は + +$$ MSE = MSE_0 + MSE_b + MSE_f $$ + +$$ +\begin{cases} +MSE_0 = \frac{1}{N_0} \sum_{i=1}^{N_0} |h(t_0^i, x_0^i) - h^i|^2 \\ +MSE_b = \frac{1}{N_b} \sum_{i=1}^{N_b} (|h^i(t_b^i, -5) - h^i(t_b^i, 5)|^2 + |h_x^i(t_b^i, -5) - h_x^i(t_b^i, 5)|^2) \\ +MSE_f = \frac{1}{N_f} \sum_{i=1}^{N_f} |h(t_f^i, x_f^i)|^2 +\end{cases} +$$ + + +--- + +# 離散化した支配方程式への適用 + +特定の時間での値を教師データとして与え (例えば初期条件 $u(t=0)$) +支配方程式を離散化した上で、時間発展を計算し解を得ることを考えたい + +- 本来は、ルンゲ=クッタ法などを用いて数値的に解く +- PINNsでは、ルンゲ=クッタ法に基づき +その計算過程の値を含めて損失を計算し最小化する + +--- + +## 前提: ルンゲ=クッタ法の一般形 + +段数$q$のルンゲ=クッタ法は、$u^n$から$u^{n+1}$の時間刻みを$\Delta t$とし、次のように表される + +$$u^{n+c_j(x)} = u(t_n + c_j\Delta t, x)$$ + +$$ +\begin{cases} +u^{n+c_i} = u^n - \Delta t \sum_{j=1}^{s} a_{ij} \mathcal{N}[u^{n+c_j}] \\ +u^{n+1} = u^n - \Delta t \sum_{j=1}^{s} b_{j} \mathcal{N}[u^{n+c_j}] +\end{cases} +$$ + +*$a_{ij}, b_i, c_i$はルンゲ=クッタ法の係数であり、精度が高くなるように選ばれる* +古典的なルンゲ=クッタ法は、$q=4$ であり、係数は次の通り + +$$ +a = \begin{bmatrix} +0 & 0 & 0 & 0 \\ +0.5 & 0 & 0 & 0 \\ +0 & 0.5 & 0 & 0 \\ +0 & 0 & 1 & 0 +\end{bmatrix}, \quad +b = \begin{bmatrix} +1/6 & 1/3 & 1/3 & 1/6 +\end{bmatrix}, \quad +c = \begin{bmatrix} +0 & 0.5 & 0.5 & 1 +\end{bmatrix} +$$ --- -# Part IIの内容 +これは、つぎのように変形する + +$$ +\begin{aligned} +& u^n = u^n_i, \quad i=1\dots q \\ +& u^n = u^n_{q+1} +\end{aligned} +$$ + +つまり、$u^n_1, u^n_2, \dots, u^n_q$をルンゲ=クッタ法の計算過程の値、$u^n_{q+1}$を解の値として +$[u^n_1, u^n_2, \dots, u^n_q, u^n_{q+1}]$が存在 +これらを$u^n_i$および$u^n_{q+1}$を右辺においた式として表すと + +$$ +\begin{cases} +u^n_i = u^{n+{c_i}} - \Delta t \sum_{j=1}^{s} a_{ij} \mathcal{N}[u^{n+c_j}] \\ +u^n_{q+1} = u^{n+1} - \Delta t \sum_{j=1}^{s} b_{j} \mathcal{N}[u^{n+c_j}] +\end{cases} +$$ + +**この $[u^n_1, \dots, u^n_{q+1}]$ をニューラルネットワークの出力とし、損失を計算する** --- -# 感想 +## 離散化した支配方程式での学習の方針 + +損失関数は残差二乗和 (SSE) を用いる + +- ルンゲ=クッタ法の計算結果に対する誤差関数 $SSE_n$ +- 境界条件への当てはまりに対する損失関数 $SSE_b$ + +この和である + +$$ SSE = SSE_n + SSE_b $$ + +を全体の損失関数として最小化する --- -# 個人用: TODO +## Burgers方程式への適用 + + + +> ルンゲ=クッタ法の一般形 $\cdots +\begin{cases} +u^n_i = u^{n+{c_i}} - \Delta t \sum_{j=1}^{s} a_{ij} \mathcal{N}[u^{n+c_j}] \\ +u^n_{q+1} = u^{n+1} - \Delta t \sum_{j=1}^{s} b_{j} \mathcal{N}[u^{n+c_j}] +\end{cases} +$ + +Burgers方程式は次のように表現される + +$$ +\mathcal{N}[u^{n+{c_i}}] = u^{n+{c_i}}_t + uu^{n+{c_i}}_x - (0.01/\pi)u^{n+{c_i}}_{xx} +$$ + +--- + +ルンゲ=クッタ法の計算結果に対する誤差関数 $SSE_n$は +$$ +SSE_n = \sum_{j=1}^{q+1}\sum_{i=1}^{N_n} |u^n_j(x^{n,i}) - u^{n, i}|^2 +$$ + +*ここで、初期条件に対するデータの数は$N_n$* + +境界条件への当てはまりに対する損失関数 $SSE_b$は + +$$ +SSE_b = \sum_{i=1}^{q} \left( |u^{n+c_i}(-1)|^2 + |u^{n+c_i}(1)|^2 \right) + |u^{n+1}(-1)|^2 + |u^{n+1}(1)|^2 +$$ + +*境界条件の具体的な値は与えておらず、$N_b$にあたる値がないことに注意* + +--- + +## 実験 + +- $t=0.1$における$u$の値から、$t=0.9$での解を計算 +- $N_n = 250$ +- ルンゲ=クッタ法の段数$q=500$ + +--- + +![bg right:70% w:750](img/burgers-discrete.webp) + +--- + +--- + +# まとめ -「物理法則」「支配方程式」「微分方程式」などバラバラの言葉が混在しているので、絞る \ No newline at end of file +![bg right:20%](img/tokyo.webp)