PINN Class Constructor: Initialization and Optimization Setup

This is the constructor of the PINN class. It initializes all the necessary variables and placeholders, constructs the neural network, and sets up the optimization problem. Here is a breakdown of what each line does:

• 'self.lb' and 'self.ub' are the lower and upper bounds of the domain.
• 'self.x0' and 'self.x1' are the initial and final states of the system.
• 'self.u0' is the initial condition of the system.
• 'self.layers' is a list that specifies the number of neurons in each hidden layer of the neural network.
• 'self.dt' is the time step for the time integration.
• 'self.q' is the number of stages in the implicit Runge-Kutta method.
• 'self.weights' and 'self.biases' are the weights and biases of the neural network, which are initialized using the 'initialize_NN' function.
• 'self.IRK_weights' and 'self.IRK_times' are the weights and times of the implicit Runge-Kutta method, which are loaded from a file.
• 'self.sess' is the TensorFlow session.
• 'self.x0_tf', 'self.x1_tf', and 'self.u0_tf' are TensorFlow placeholders for the initial and final states and the initial condition, respectively.
• 'self.dummy_x0_tf' and 'self.dummy_x1_tf' are dummy TensorFlow placeholders used for computing gradients.
• 'self.U0_pred' is the predicted solution at the initial state.
• 'self.U1_pred' and 'self.U1_x_pred' are the predicted solution and its derivative at the final state.
• 'self.loss' is the loss function that measures the discrepancy between the predicted and actual solutions.
• 'self.optimizer' is the optimizer that minimizes the loss function using the L-BFGS-B method.
• 'self.optimizer_Adam' is an alternative optimizer that uses the Adam method.
• 'self.train_op_Adam' is the operation that minimizes the loss function using the Adam optimizer.
• 'init' is the operation that initializes all the variables in the TensorFlow graph.
• 'self.sess.run(init)' initializes all the variables in the TensorFlow session.