This module implements the Physics Informed Neural Network (PINN) model for a projectile motion. The differential equation is given by (d^2 r)/(dt^2) = -g, where r = (x, z) is reduced to two dimensions for simplicity, and g is the gravity acceleration. The initial positions are fixed to x(0) = z(0) = 0. The PINN model predicts x(t), z(t) for t, v0_x, v0_z, where v0_x, v0_z are initial velocities at t=0.

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.

We implement a PINN model with the L-BFGS-B optimization for a projectile motion as the simple example.
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 solving the projectile motion by the PINN is demonstraned 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, 3)]               0
   _________________________________________________________________
   dense (Dense)                (None, 32)                128
   _________________________________________________________________
   dense_1 (Dense)              (None, 32)                1056
   _________________________________________________________________
   dense_2 (Dense)              (None, 2)                 66
   =================================================================
   Total params: 1,250
   Trainable params: 1,250
   Non-trainable params: 0
   _________________________________________________________________
   

2. Building the PINN model (g is the gravity acceleration).

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

3. Optimizing the PINN model for training samples.

   from lib.optimizer import L_BFGS_B
   samples = np.random.rand(num_train_samples, 3)
   lbfgs = L_BFGS_B(model=pinn, samples=samples)
   lbfgs.fit()

   The progress is printed as follows. The optimization is terminated for loss ~ 3e-6.

   Optimizer: L-BFGS-B (maxiter=3000)
   2619/3000 [=========================>....] - ETA: 19s - loss: 3.5697e-06
   

An example result of main.py is shown below.