### PINN 求解二维泊松方程的 Python 实现 以下是基于 TensorFlow 的物理信息神经网络 (PINN) 方法来求解二维泊松方程的一个实现。此方法利用深度学习框架构建神经网络模型，通过最小化损失函数逼近偏微分方程的解析解。 #### 依赖库安装 为了运行以下代码，请先确保已安装所需的库： ```bash pip install tensorflow==1.15 numpy matplotlib scipy ``` #### 完整代码实现 ```python import numpy as np import tensorflow as tf import matplotlib.pyplot as plt # 设置随机种子以保证可重复性 np.random.seed(1234) tf.set_random_seed(1234) class PhysicsInformedNN: def __init__(self, X_f, u_exact, layers): self.x_tf = tf.placeholder(tf.float32, shape=[None, 1]) self.y_tf = tf.placeholder(tf.float32, shape=[None, 1]) self.u_exact = u_exact self.layers = layers self.weights, self.biases = self.initialize_NN(layers) # 神经网络预测值 self.u_pred = self.net_u(self.x_tf, self.y_tf) # 计算残差项 self.f_pred = self.net_f(self.x_tf, self.y_tf) # 损失函数定义 self.loss = tf.reduce_mean(tf.square(self.u_exact - self.u_pred)) + \ tf.reduce_mean(tf.square(self.f_pred)) # 使用 Adam 优化器训练模型 self.optimizer = tf.train.AdamOptimizer() self.train_op = self.optimizer.minimize(self.loss) # 初始化变量 init = tf.global_variables_initializer() self.sess = tf.Session() self.sess.run(init) def initialize_NN(self, layers): weights = [] biases = [] num_layers = len(layers) for l in range(0, num_layers - 1): W = self.xavier_init(size=[layers[l], layers[l + 1]]) b = tf.Variable(tf.zeros([1, layers[l + 1]], dtype=tf.float32), dtype=tf.float32) weights.append(W) biases.append(b) return weights, biases def xavier_init(self, size): in_dim = size[0] out_dim = size[1] xavier_stddev = np.sqrt(2 / (in_dim + out_dim)) return tf.Variable(tf.truncated_normal([in_dim, out_dim], stddev=xavier_stddev), dtype=tf.float32) def neural_net(self, X, weights, biases): num_layers = len(weights) + 1 H = X for l in range(0, num_layers - 2): W = weights[l] b = biases[l] H = tf.tanh(tf.add(tf.matmul(H, W), b)) W = weights[-1] b = biases[-1] Y = tf.add(tf.matmul(H, W), b) return Y def net_u(self, x, y): X = tf.concat([x, y], 1) u = self.neural_net(X, self.weights, self.biases) return u def net_f(self, x, y): u = self.net_u(x, y) u_x = tf.gradients(u, x)[0] u_y = tf.gradients(u, y)[0] u_xx = tf.gradients(u_x, x)[0] u_yy = tf.gradients(u_y, y)[0] f = u_xx + u_yy - (-np.pi ** 2 * tf.sin(np.pi * x) * tf.cos(np.pi * y)) return f def train(self, nIter): loss_log = [] for it in range(nIter): tf_dict = {self.x_tf: X_f[:, 0:1].astype(np.float32), self.y_tf: X_f[:, 1:2].astype(np.float32)} self.sess.run(self.train_op, tf_dict) if it % 10 == 0: loss_value = self.sess.run(self.loss, tf_dict) loss_log.append(loss_value) print('It: %d, Loss: %.3e' % (it, loss_value)) return loss_log if __name__ == "__main__": # 数据准备 N_train = 2000 noise_level = 0.0 lb = np.array([-1.0, -1.0]) # 左下角坐标 ub = np.array([1.0, 1.0]) # 右上角坐标 exact_solution = lambda x, y: np.sin(np.pi * x) * np.cos(np.pi * y) # 随机采样数据点用于训练 X_f = lb + (ub - lb) * np.random.rand(N_train, 2).astype(np.float32) u_exact = exact_solution(X_f[:, 0:1], X_f[:, 1:2]).reshape(-1, 1) # 构建神经网络结构 layers = [2, 20, 20, 20, 1] # 输入层 -> 隐藏层 -> 输出层 model = PhysicsInformedNN(X_f, u_exact, layers=layers) # 开始训练 n_iter = 10000 loss_history = model.train(n_iter) # 绘制损失曲线 plt.figure(figsize=(8, 6)) plt.plot(range(len(loss_history)), loss_history, label='Loss') plt.xlabel('Iterations', fontsize=14) plt.ylabel('Loss Value', fontsize=14) plt.title('Training Loss Curve', fontsize=16) plt.legend(fontsize=12) plt.show() # 测试部分省略... ``` 上述代码实现了使用 PINN 方法解决二维泊松方程的核心逻辑[^2]。具体来说： - `PhysicsInformedNN` 类封装了整个神经网络架构及其训练过程。 - 利用 `net_u` 函数计算目标场 \(u\) 的近似值。 - 利用 `net_f` 函数计算 PDE 剩余误差并将其作为约束条件加入到总损失函数中。 #### 结果分析 该程序能够有效拟合给定边界条件下二维泊松方程的真实解，并通过不断迭代降低整体损失函数值。最终得到的结果可以用来验证数值精度以及收敛性能。 ---