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| import numpy as np
import matplotlib.pyplot as plt
plt.rcParams['font.sans-serif'] = ['Microsoft YaHei', 'SimHei', 'Arial Unicode MS']
plt.rcParams['axes.unicode_minus'] = False
def train(patterns_1d):
"""
根据给定的1D模式集训练霍普菲尔德网络权重矩阵。
patterns_1d: 列表，每个元素是一个一维numpy数组（扁平化的模式）。
"""
num_neurons = patterns_1d[0].shape[0]
w = np.zeros((num_neurons, num_neurons))
for pattern in patterns_1d:
pattern_col = pattern.reshape(-1, 1)
w += np.dot(pattern_col, pattern_col.T)
np.fill_diagonal(w, 0)
return w
def E(state_1d, w):
state_col = state_1d.reshape(-1, 1)
e = -0.5 * np.dot(state_col.T, np.dot(w, state_col))
return e.item()
def update(state_1d, w, index):
current_state_1d = np.copy(state_1d)
net_input = np.dot(w[index, :], current_state_1d)
new_state_val = current_state_1d[index]
changed = False
if net_input > 0:
new_state_val = 1
elif net_input < 0:
new_state_val = -1
if new_state_val != current_state_1d[index]:
changed = True
current_state_1d[index] = new_state_val
return current_state_1d, changed
def work(initial_state_1d, w, n_iterations=1000, max_converge_sweeps=5):
current_state_1d = np.copy(initial_state_1d)
e_history = [E(initial_state_1d, w)]
num_neuron = initial_state_1d.shape[0]
no_change_count = 0
convergence_threshold = max_converge_sweeps * num_neuron
for i in range(n_iterations):
index = np.random.randint(num_neuron)
current_state_1d, changed = update(current_state_1d, w, index)
e_history.append(E(current_state_1d, w))
if not changed:
no_change_count += 1
else:
no_change_count = 0
if no_change_count >= convergence_threshold:
break
return current_state_1d, e_history
def _flatten_pattern(pattern):
return pattern.flatten()
def _reshape_pattern(pattern_1d, original_shape):
return pattern_1d.reshape(original_shape)
def create_random_pattern(shape):
return np.random.choice([-1, 1], size=shape)
def add_noise(pattern, noise_level=0.05):
original_shape = pattern.shape
flat_pattern = _flatten_pattern(pattern)
noisy_flat_pattern = np.copy(flat_pattern)
num_to_flip = int(np.round(len(flat_pattern) * noise_level))
if num_to_flip == 0 and noise_level > 0 and len(flat_pattern) > 0:
num_to_flip = 1
if num_to_flip > 0:
flip_indices = np.random.choice(len(flat_pattern), num_to_flip, replace=False)
noisy_flat_pattern[flip_indices] *= -1
return _reshape_pattern(noisy_flat_pattern, original_shape)
def get_overlap(state1, state2):
flat_state1 = _flatten_pattern(state1)
flat_state2 = _flatten_pattern(state2)
if len(flat_state1) == 0: return 0
return np.dot(flat_state1, flat_state2) / len(flat_state1)
def test_storage_capacity(pattern_shape, max_patterns_to_test, num_trials_per_pattern_count=50, noise_level=0.05):
if isinstance(pattern_shape, int):
num_neurons = pattern_shape
elif isinstance(pattern_shape, tuple):
num_neurons = np.prod(pattern_shape)
else:
raise ValueError("pattern_shape 必须是整数（1D）或元组（2D）。")
pattern_counts = np.unique(np.linspace(1, max_patterns_to_test, 15).astype(int))
pattern_counts = pattern_counts[pattern_counts <= num_neurons]
if len(pattern_counts) == 0:
print("错误：模式数量范围无效，请检查 pattern_shape 和 max_patterns_to_test 参数。")
return
success_rates = []
theoretical_capacity_estimate = 0.138 * num_neurons
print(f"\n--- 评估霍普菲尔德网络存储能力 ---")
print(f"模式形状: {pattern_shape}")
print(f"总神经元数量 (N): {num_neurons}")
print(f"理论容量参考 (0.138 * N): {theoretical_capacity_estimate:.2f} 个模式")
print(f"噪声水平: {noise_level * 100:.0f}%")
print(f"每个模式数量重复试验次数: {num_trials_per_pattern_count}\n")
print("正在运行测试，请稍候...")
for p_count in pattern_counts:
successful_recalls = 0
for _ in range(num_trials_per_pattern_count):
if p_count == 0: continue
stored_patterns_original_shape = [create_random_pattern(pattern_shape) for _ in range(p_count)]
stored_patterns_1d = [_flatten_pattern(p) for p in stored_patterns_original_shape]
w = train(stored_patterns_1d)
for original_pattern_original_shape in stored_patterns_original_shape:
initial_state_original_shape = add_noise(original_pattern_original_shape, noise_level)
initial_state_1d = _flatten_pattern(initial_state_original_shape)
final_state_1d, _ = work(initial_state_1d, w, n_iterations=num_neurons * 100, max_converge_sweeps=5)
overlap = get_overlap(final_state_1d, _flatten_pattern(original_pattern_original_shape))
if abs(overlap) > 0.95:
successful_recalls += 1
total_attempts = p_count * num_trials_per_pattern_count
if total_attempts > 0:
success_rate = successful_recalls / total_attempts
else:
success_rate = 0
success_rates.append(success_rate)
print(f" 模式数量 P = {p_count:2d}: 召回成功率 = {success_rate:.3f}")
plt.figure(figsize=(10, 6))
plt.plot(pattern_counts, success_rates, marker='o', linestyle='-', color='blue')
if theoretical_capacity_estimate > 0:
plt.axvline(x=theoretical_capacity_estimate, color='red', linestyle='--',
label=f'理论容量参考 (≈{theoretical_capacity_estimate:.1f})')
plt.title('霍普菲尔德网络模式存储能力', fontsize=16)
plt.xlabel('存储模式数量 (P)', fontsize=12)
plt.ylabel('平均召回成功率', fontsize=12)
plt.grid(True, linestyle='--', alpha=0.7)
plt.ylim(-0.05, 1.05)
plt.legend()
plt.show()
print("\n评估完成。")
if __name__ == '__main__':
pattern_shape_param = 30
if isinstance(pattern_shape_param, int):
num_total_neurons = pattern_shape_param
else:
num_total_neurons = np.prod(pattern_shape_param)
max_patterns_to_test_ratio = 0.5
max_patterns_to_test = int(num_total_neurons * max_patterns_to_test_ratio)
if max_patterns_to_test == 0: max_patterns_to_test = 1
num_trials = 100
noise_level_param = 0.05
test_storage_capacity(
pattern_shape=pattern_shape_param,
max_patterns_to_test=max_patterns_to_test,
num_trials_per_pattern_count=num_trials,
noise_level=noise_level_param
)
print("\n--- 2D 模式召回演示 ---")
if isinstance(pattern_shape_param, tuple):
demo_shape = pattern_shape_param
num_demo_neurons = np.prod(demo_shape)
original_2d_pattern = create_random_pattern(demo_shape)
w_demo = train([_flatten_pattern(original_2d_pattern)])
noisy_2d_input = add_noise(original_2d_pattern, noise_level=0.1)
noisy_1d_input = _flatten_pattern(noisy_2d_input)
recalled_1d_pattern, _ = work(noisy_1d_input, w_demo, n_iterations=num_demo_neurons * 100, max_converge_sweeps=5)
recalled_2d_pattern = _reshape_pattern(recalled_1d_pattern, demo_shape)
fig, axes = plt.subplots(1, 3, figsize=(12, 4))
axes[0].imshow(original_2d_pattern, cmap='binary')
axes[0].set_title('原始模式')
axes[0].axis('off')
axes[1].imshow(noisy_2d_input, cmap='binary')
axes[1].set_title('带噪声输入')
axes[1].axis('off')
axes[2].imshow(recalled_2d_pattern, cmap='binary')
axes[2].set_title('召回模式')
axes[2].axis('off')
plt.suptitle(f"2D模式召回演示 (重叠度: {get_overlap(original_2d_pattern, recalled_2d_pattern):.3f})")
plt.tight_layout()
plt.show()
else:
print("当前模式形状为1D，跳过2D模式召回演示。")
|
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