利用PINN求解一维Burgers 方程代码
时间: 2024-10-23 21:01:09 浏览: 41
PINN(Physics-Informed Neural Networks)是一种结合了物理约束和机器学习的技术,用于解决偏微分方程等科学计算问题。以下是使用Python和TensorFlow库,利用Keras构建一个简单的例子,演示如何用PINN来求解一维Burgers方程:
```python
import tensorflow as tf
from tensorflow.keras import layers
# 定义 Burgers 方程
def burgers_equation(x, t, u):
return u * u - (0.01 / 2) * tf.gradients(u, x)[0]
# 假设我们有边界条件 u(0,t) = 0 和 u(L,t) = 0
L = 1
t_domain = [0., 1.] # 时间范围
x_domain = [0., L] # 空间范围
# 定义神经网络结构
model = tf.keras.Sequential([
layers.Dense(64, activation='relu', input_shape=(2,)),
layers.Dense(64, activation='relu'),
layers.Dense(1)
])
# 损失函数包含数据拟合、物理方程和边界条件
def pinn_loss(model, u_pred, x, t):
data_loss = model.trainable_variables[0].losses[0]
eqn_loss = tf.reduce_mean(tf.square(burgers_equation(x, t, u_pred)))
bc_loss = tf.reduce_sum(tf.square(u_pred[:, 0]) + tf.square(u_pred[:, -1]))
return data_loss + eqn_loss + bc_loss
# 编译模型并设置优化器
optimizer = tf.optimizers.Adam()
model.compile(optimizer=optimizer, loss=pinn_loss)
# 训练过程
num_epochs = 5000
history = model.fit((tf.stack([x_domain, t_domain], axis=-1).numpy(),),
tf.zeros_like(x_domain), epochs=num_epochs,
batch_size=128, verbose=0)
# 输出结果
u_pred = model.predict((tf.stack([x_domain, t_domain], axis=-1).numpy(),))
print("Predicted solution after training:")
print(u_pred)
阅读全文