This module implements the Physics Informed Neural Network (PINN) model for Burgers' equation. Burgers' equation is given by du/dt + u(du/dx) = nu (d^2 u)/(dx^2), where nu is the kinematic viscosity. This is a good simplification of Navier-Stokes equation where the velocity is in one spatial dimension and the external force is neglected and without any pressure gradient. We give the initial condition u(t=0, x) = -sin(x) and the boundary condition u(t, x=-1,+1) = 0. The PINN model predicts u(t, x) for the input (t, x).

The PINN is a deep learning approach to solve partial differential equations. Well-known finite difference, volume and element methods are formulated on discrete meshes to approximate derivatives. Meanwhile, the automatic differentiation using neural networks provides differential operations directly. The PINN is the automatic differentiation based solver and has an advantage of being meshless.

The effectiveness of PINNs is validated in the following works.

• M. Raissi, et al., Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations, arXiv: 1711.10561 (2017).
• M. Raissi, et al., Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations, arXiv: 1711.10566 (2017).

In addition, an effective convergent optimizer is required to solve the differential equations accurately using PINNs. The stochastic gradient dicent is generally used in deep learnigs, but it only depends on the primary gradient (Jacobian). In contrast, the quasi-Newton based approach such as the limited-memory Broyden-Fletcher-Goldfarb-Shanno method for bound constraints (L-BFGS-B) incorporates the quadratic gradient (Hessian), and gives a more accurate convergence.

Here we implement a PINN model with the L-BFGS-B optimization for Burgers' equation. Scripts is given as follows.

• lib : libraries to implement the PINN model for a projectile motion.
  • layer.py : computing 1st and 2nd derivatives as a custom layer.
  • network.py : building a keras network model.
  • optimizer.py : implementing the L-BFGS-B optimization.
  • pinn.py : building a PINN model.
  • tf_silent.py : suppressing tensorflow warnings
• main.py : main routine to run and test the PINN solver.

You need Python 3.6 and the following packages.

GPU acceleration is recommended in the following environments.

An example of PINN solver for Burgers' equation is implemented in main.py. The PINN is trained by the following procedure.

1. Building the keras network model

   from lib.network import Network
   network = Network.build().
   network.summary()

   The following table depicts layers in the default network.

   _________________________________________________________________
   Layer (type)                 Output Shape              Param #
   =================================================================
   input_1 (InputLayer)         [(None, 2)]               0
   _________________________________________________________________
   dense (Dense)                (None, 32)                96
   _________________________________________________________________
   dense_1 (Dense)              (None, 16)                528
   _________________________________________________________________
   dense_2 (Dense)              (None, 32)                544
   _________________________________________________________________
   dense_3 (Dense)              (None, 1)                 33
   =================================================================
   Total params: 1,201
   Trainable params: 1,201
   Non-trainable params: 0
   _________________________________________________________________
   

2. Building the PINN model (nu is the kinematic viscosity).

   from lib.pinn import PINN
   pinn = PINN(network, nu).build()

3. Building training data.

   # create training input
   tx_eqn = np.random.rand(num_train_samples, 2)          # t_eqn =  0 ~ +1
   tx_eqn[..., 1] = 2 * tx_eqn[..., 1] - 1                # x_eqn = -1 ~ +1
   tx_ini = 2 * np.random.rand(num_train_samples, 2) - 1  # x_ini = -1 ~ +1
   tx_ini[..., 0] = 0                                     # t_ini =  0
   tx_bnd = np.random.rand(num_train_samples, 2)          # t_bnd =  0 ~ +1
   tx_bnd[..., 1] = 2 * np.round(tx_bnd[..., 1]) - 1      # x_bnd = -1 or +1
   # create training output
   u_eqn = np.zeros((num_train_samples, 1))               # u_eqn = 0
   u_ini = np.sin(-np.pi * tx_ini[..., 1, np.newaxis])    # u_ini = -sin(pi*x_ini)
   u_bnd = np.zeros((num_train_samples, 1))               # u_bnd = 0

4. Optimizing the PINN model for the training data.

   from lib.optimizer import L_BFGS_B
   x_train = [tx_eqn, tx_ini, tx_bnd]
   y_train = [ u_eqn,  u_ini,  u_bnd]
   lbfgs = L_BFGS_B(model=pinn, x_train=x_train, y_train=y_train)
   lbfgs.fit()

   The progress is printed as follows. The optimization is terminated for loss ~ 6.7e-5.

   Optimizer: L-BFGS-B (maxiter=5000)
   4635/5000 [==========================>...] - ETA: 17s - loss: 6.7799e-05
   

An example result of main.py is shown below.