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Python和C++及MATLAB低温磁态机器学习模型

🎯要点

  1. 使用小规模磁态训练模型,并在二维三维爱德华兹-安德森模型上使用四种算法测试:贪婪算法、模拟退火算法、并行回火算法和本模型。
  2. 将磁态基态搜索视为马尔可夫决策过程 (MDP),学习最优策略以累积其最大回报。
  3. 设计图神经网络式编码器表征状态和动作。
  4. 模型求解非确定性多项式时间问题。

🍁磁态分析

  • Python和MATLAB自旋玻璃投资组合神经网络广义方程
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🍪语言内容分比

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🍇Python马尔可夫决策过程策略选择

我们将要实现的马尔可夫决策过程是一个离散时间随机控制过程。它提供了一个数学框架,用于在结果部分随机、部分受决策者控制的情况下对决策进行建模。马尔可夫决策过程是一种对顺序决策问题进行建模的工具,其中决策者以顺序方式与环境进行交互。

假设问题是:我们有一个由 12 个状态组成的世界,1 个障碍初始状态(状态 5)和 2 个结束状态(状态 10、11)。对于每个状态,我们都有一个奖励,我们希望找到实施最佳奖励累积的策略。对于每个状态,奖励为 -0.04(r=-0.4)。对于状态 10,奖励为 1,对于结束状态 11,奖励为 -1。
1 4 7 10 2 5 8 11 3 6 9 12 \begin{array}{|l|l|l|l|} \hline 1 & 4 & 7 & 10 \\ \hline 2 & 5 & 8 & 11 \\ \hline 3 & 6 & 9 & 12 \\ \hline \end{array} 123456789101112
对于所处的每一个状态,我们都想找到最好的行动,是向北(N)、向南(S)、向东(E)还是向西(W)走。基本上,我们希望以最短的路到达状态 10。首先,让我们创建一个世界类,它将在我们的世界中用于解决问题。这个对象可以帮助我们声明世界、绘制世界并绘制我们发现的移动世界的策略。方法 plot_world 绘制了我们上面看到的图像,将使用值迭代方法代替策略迭代。

使用策略迭代求解马尔可夫决策过程,其中 γ = 0.9 和 r = 0.04。我们使用均匀随机策略初始化了策略迭代算法。并且分别使用 World 类的绘制值和绘制策略函数,将每次迭代步骤后的值函数和策略绘制成网格世界的两个不同图形,我们得到了以下结果:

初次迭代:网格世界值
− 0.287 − 0.169 0.05 1.0 − 0.354 − 0.479 − 1.0 − 0.402 − 0.451 − 0.524 − 0.696 \begin{array}{|l|l|l|l|} \hline-0.287 & -0.169 & 0.05 & 1.0 \\ \hline-0.354 & & -0.479 & -1.0 \\ \hline-0.402 & -0.451 & -0.524 & -0.696 \\ \hline \end{array} 0.2870.3540.4020.1690.4510.050.4790.5241.01.00.696

二次迭代:

0.509 0.649 0.795 1.0 0.398 0.486 − 1.0 0.291 0.207 0.168 − 0.009 \begin{array}{|l|l|l|c|} \hline 0.509 & 0.649 & 0.795 & 1.0 \\ \hline 0.398 & & 0.486 & -1.0 \\ \hline 0.291 & 0.207 & 0.168 & -0.009 \\ \hline \end{array} 0.5090.3980.2910.6490.2070.7950.4860.1681.01.00.009

三次迭代:
0.509 0.649 0.795 1.0 0.398 0.486 − 1.0 0.291 0.207 0.34 0.126 \begin{array}{|l|l|l|l|} \hline 0.509 & 0.649 & 0.795 & 1.0 \\ \hline 0.398 & & 0.486 & -1.0 \\ \hline 0.291 & 0.207 & 0.34 & 0.126 \\ \hline \end{array} 0.5090.3980.2910.6490.2070.7950.4860.341.01.00.126
四次迭代:
0.509 0.649 0.795 1.0 0.398 0.486 − 1.0 0.296 0.253 0.344 0.129 \begin{array}{|l|l|l|l|} \hline 0.509 & 0.649 & 0.795 & 1.0 \\ \hline 0.398 & & 0.486 & -1.0 \\ \hline 0.296 & 0.253 & 0.344 & 0.129 \\ \hline \end{array} 0.5090.3980.2960.6490.2530.7950.4860.3441.01.00.129

我们可以看到,经过 4 次迭代后我们得到了解,并且我们可以看到与上一次迭代相比,每次迭代的进展情况。如果使用值迭代,应该得到与策略迭代算法相同的结果。

世界类:

import numpy as np
import matplotlib.pyplot as pltclass World:def __init__(self):self.nRows = 3self.nCols = 4self.stateObstacles = [5]self.stateTerminals = [10, 11]self.nStates = 12self.nActions = 4def _plot_world(self):nStates = self.nStatesnRows = self.nRowsnCols = self.nColsstateObstacles = self.stateObstaclesstateTerminals = self.stateTerminalscoord = [[0, 0], [nCols, 0], [nCols, nRows], [0, nRows], [0, 0]]xs, ys = zip(*coord)plt.plot(xs, ys, "black")for i in stateObstacles:(I, J) = np.unravel_index(i, shape=(nRows, nCols), order='F')coord = [[J, nRows - I],[J + 1, nRows - I],[J + 1, nRows - I + 1],[J, nRows - I + 1],[J, nRows - I]]xs, ys = zip(*coord)plt.fill(xs, ys, "0.5")plt.plot(xs, ys, "black")for ind, i in enumerate(stateTerminals):(I, J) = np.unravel_index(i, shape=(nRows, nCols), order='F')coord = [[J, nRows - I],[J + 1, nRows - I],[J + 1, nRows - I + 1],[J, nRows - I + 1],[J, nRows - I]]xs, ys = zip(*coord)plt.fill(xs, ys, "0.8")plt.plot(xs, ys, "black")plt.plot(xs, ys, "black")X, Y = np.meshgrid(range(nCols + 1), range(nRows + 1))plt.plot(X, Y, 'k-')plt.plot(X.transpose(), Y.transpose(), 'k-')@staticmethoddef _truncate(n, decimals=0):multiplier = 10 ** decimalsreturn int(n * multiplier) / multiplierdef plot(self):nStates = self.nStatesnRows = self.nRowsnCols = self.nColsself._plot_world()states = range(1, nStates + 1)k = 0for i in range(nCols):for j in range(nRows, 0, -1):plt.text(i + 0.5, j - 0.5, str(states[k]), fontsize=26, horizontalalignment='center', verticalalignment='center')k += 1plt.title('MDP gridworld', size=16)plt.axis("equal")plt.axis("off")plt.show()def plot_value(self, valueFunction):nRows = self.nRowsnCols = self.nColsstateObstacles = self.stateObstaclesself._plot_world()k = 0for i in range(nCols):for j in range(nRows, 0, -1):if k + 1 not in stateObstacles:plt.text(i + 0.5, j - 0.5, str(self._truncate(valueFunction[k], 3)), fontsize=26, horizontalalignment='center', verticalalignment='center')k += 1plt.title('MDP gridworld', size=16)plt.axis("equal")plt.axis("off")plt.show()def plot_policy(self, policy):nStates = self.nStatesnActions = self.nActionsnRows = self.nRowsnCols = self.nColsstateObstacles = self.stateObstaclesstateTerminals = self.stateTerminalspolicy = policy.reshape(nRows, nCols, order="F").reshape(-1, 1)X, Y = np.meshgrid(range(nCols + 1), range(nRows + 1))X1 = X[:-1, :-1]Y1 = Y[:-1, :-1]X2 = X1.reshape(-1, 1) + 0.5Y2 = np.flip(Y1.reshape(-1, 1)) + 0.5X2 = np.kron(np.ones((1, nActions)), X2)Y2 = np.kron(np.ones((1, nActions)), Y2)mat = np.cumsum(np.ones((nStates, nActions)), axis=1).astype("int64")if policy.shape[1] == 1:policy = (np.kron(np.ones((1, nActions)), policy) == mat)index_no_policy = stateObstacles + stateTerminalsindex_policy = [item - 1 for item in range(1, nStates + 1) if item not in index_no_policy]mask = policy.astype("int64") * matmask = mask.reshape(nRows, nCols, nCols)X3 = X2.reshape(nRows, nCols, nActions)Y3 = Y2.reshape(nRows, nCols, nActions)alpha = np.pi - np.pi / 2 * maskself._plot_world()for ii in index_policy:ax = plt.gca()j = int(ii / nRows)i = (ii + 1 - j * nRows) % nCols - 1index = np.where(mask[i, j] > 0)[0]h = ax.quiver(X3[i, j, index], Y3[i, j, index], np.cos(alpha[i, j, index]), np.sin(alpha[i, j, index]), 0.3)states = range(1, nStates + 1)k = 0for i in range(nCols):for j in range(nRows, 0, -1):plt.text(i + 0.25, j - 0.25, str(states[k]), fontsize=16, horizontalalignment='right', verticalalignment='bottom')k += 1plt.axis("equal")plt.axis("off")plt.show()

主程度:

from World import World
import numpy as np
import pandas as pd
import copydef construct_p(world, p=0.8, step=-0.04):nstates = world.get_nstates()nrows = world.get_nrows()obsacle_index = world.get_stateobstacles()terminal_index = world.get_stateterminals()bad_index = obsacle_index + terminal_indexrewards = np.array([step] * 4 + [0] + [step] * 4 + [1, -1] + [step])actions = ["N", "S", "E", "W"]transition_models = {}for action in actions:transition_model = np.zeros((nstates, nstates))for i in range(1, nstates + 1):if i not in bad_index:if action == "N":if i + nrows <= nstates and (i + nrows) not in obsacle_index:transition_model[i - 1][i + nrows - 1] += (1 - p) / 2else:transition_model[i - 1][i - 1] += (1 - p) / 2if 0 < i - nrows <= nstates and (i - nrows) not in obsacle_index:transition_model[i - 1][i - nrows - 1] += (1 - p) / 2else:transition_model[i - 1][i - 1] += (1 - p) / 2if (i - 1) % nrows > 0 and (i - 1) not in obsacle_index:transition_model[i - 1][i - 1 - 1] += pelse:transition_model[i - 1][i - 1] += pif action == "S":if i + nrows <= nstates and (i + nrows) not in obsacle_index:transition_model[i - 1][i + nrows - 1] += (1 - p) / 2else:transition_model[i - 1][i - 1] += (1 - p) / 2if 0 < i - nrows <= nstates and (i - nrows) not in obsacle_index:transition_model[i - 1][i - nrows - 1] += (1 - p) / 2else:transition_model[i - 1][i - 1] += (1 - p) / 2if 0 < i % nrows and (i + 1) not in obsacle_index and (i + 1) <= nstates:transition_model[i - 1][i + 1 - 1] += pelse:transition_model[i - 1][i - 1] += pif action == "E":if i + nrows <= nstates and (i + nrows) not in obsacle_index:transition_model[i - 1][i + nrows - 1] += pelse:transition_model[i - 1][i - 1] += pif 0 < i % nrows and (i + 1) not in obsacle_index and (i + 1) <= nstates:transition_model[i - 1][i + 1 - 1] += (1 - p) / 2else:transition_model[i - 1][i - 1] += (1 - p) / 2if (i - 1) % nrows > 0 and (i - 1) not in obsacle_index:transition_model[i - 1][i - 1 - 1] += (1 - p) / 2else:transition_model[i - 1][i - 1] += (1 - p) / 2if action == "W":if 0 < i - nrows <= nstates and (i - nrows) not in obsacle_index:transition_model[i - 1][i - nrows - 1] += pelse:transition_model[i - 1][i - 1] += pif 0 < i % nrows and (i + 1) not in obsacle_index and (i + 1) <= nstates:transition_model[i - 1][i + 1 - 1] += (1 - p) / 2else:transition_model[i - 1][i - 1] += (1 - p) / 2if (i - 1) % nrows > 0 and (i - 1) not in obsacle_index:transition_model[i - 1][i - 1 - 1] += (1 - p) / 2else:transition_model[i - 1][i - 1] += (1 - p) / 2elif i in terminal_index:transition_model[i - 1][i - 1] = 1transition_models[action] = pd.DataFrame(transition_model, index=range(1, nstates + 1), columns=range(1, nstates + 1))return transition_models, rewardsdef max_action(transition_models, rewards, gamma, s, V, actions, terminal_ind):maxs = {key: 0 for key in actions}max_a = ""action_map = {k: v for k, v in zip(actions, [1, 3, 2, 4])}for action in actions:if s not in terminal_ind:maxs[action] += rewards[s - 1] + gamma * np.dot(transition_models[action].loc[s, :].values, V)else:maxs[action] = rewards[s - 1]maxi = -10 ** 10for key in maxs:if maxs[key] > maxi:max_a = keymaxi = maxs[key]return maxi, action_map[max_a]def value_iteration(world, transition_models, rewards, gamma=1.0, theta=10 ** -4):nstates = world.get_nstates()terminal_ind = world.get_stateterminals()V = np.zeros((nstates, ))P = np.zeros((nstates, 1))actions = ["N", "S", "E", "W"]delta = theta + 1while delta > theta:delta = 0v = copy.deepcopy(V)for s in range(1, nstates + 1):V[s - 1], P[s - 1] = max_action(transition_models, rewards, gamma, s, v, actions, terminal_ind)delta = max(delta, np.abs(v[s - 1] - V[s - 1]))return V, Pdef policy_iter(policy, world, transition_models, rewards, gamma=0.9, theta=10 ** -4):nstates = world.get_nstates()terminal_ind = world.get_stateterminals()V = np.zeros((nstates,))a = ["N", "S", "E", "W"]while True:delta = 0for s in range(nstates):v = 0for action, action_prob in enumerate(policy[s]):action = a[action]if s not in terminal_ind:v += rewards[s - 1] + action_prob * gamma * np.dot(transition_models[action].loc[s, :].values, V)else:v = rewards[s - 1]delta = max(delta, np.abs(v - V[s]))V[s] = vprint (V[s])if delta < theta:breakreturn np.array(V)

👉参阅、更新:计算思维 | 亚图跨际


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