pinn-ics 1.0.2

Solving physics problems by using Deep Learning

pinn-ics

PINN framework

Installation

    pip install pinn-ics

Latest version: 1.0.2

- New Features: `RAR algorithm` - generate data where the loss is higher

Syntax

Var Configuration

• pinnics requires configurations for all variables.

  • VarSpec can import directly from pinnics

  • VarSpec object contains indentify character and limit of this variable.

• Example:

  • Your solution depends on 2 domains: dimension and time.

  • Format u(x, t)

from pinnics import VarSpec

x = VarSpec('x', limit=(-1, 1))

t = VarSpec('x') # which defaults the limit of var t from 0 to 1

Function Definition

• pinnics supports solving partial differential equation with the simple syntax.

  • Define a function with format:

  import numpy as np 
  
  import tensorflow as tf
  
  
  
  def pde_loss(res):
  
  
  
      # that presents u = u(x, t)
  
      u = res(x='x', t='t', num=10000)
  
  
  
      # u't + u * u'x = 3 * u''xx + sin(pi * x)
  
      return u.diff('t') + u() * u.diff('x') - 3 * u.diff('x', 'x') - tf.sin(np.pi * res.var['x'])
  

  • Whereas:

    • res is the required predefine argument, that will store all information.

    • u = res(x='x', t='t', num==10000) that represents u = u(x, t).

    • You can also use like that u = res(x=-1., t='t') that represents u = u(-1, t)

    • u() will return the value of u(x, t)

    • u.diff('x') returns the first order partial derivative of u(x, t) by x

    • u.diff('x', 'x') returns the second order partial derivative

    • res.var['x'] is the input value x for the model.

Network

• pinnics provides a Network class to help people solve PDE problems by easy way.

Example:

from pinnics import NetWork



# define network to solve pde problem. 

net = NetWork(variables=[x, t], 

    losses = [pde_loss],

    layers=[2, 20, 20, 20, 1])



net.solve(epochs=10000, show_every=1000)

• Whereas:

  • variables, losses, layers are required to create a new network.

    • variables is the list which contains all variables (VarSpec object) of model.

    • losses is the list which contains all equations (function defined in previous part).

    • layers is the list that represents network's architecture.

  • Non-required arguments:

    • activation_func: the activation function after each layer (expect result layer), default by tf.keras.activations.tanh

    • optimizer: the optimizer of model, default by tf.keras.optimizers.Adam()

    • initializer_func: the initializer for model's parameters, default by tf.keras.initializer.glorut_normal

  • net.solve(10000) will training and approximate the result.

After obtaining the model, you can get predict by using call() function.

x = np.linspace (-1, 1, 200).reshape(-1, 1)

t = np.linspace (0, 1, 200).reshape(-1, 1)

input = np.concatentate([x, t], axis=1)

y_pred = net(input)

A completely example

from pinnics import NetWork, VarSpec

import matplotlib.pyplot as plt

import numpy as np





t = VarSpec('t', limit=(0, 5))



def pde(res):

    u = res(t='t', num=1000)

    return u.diff('t', 't') + 3*u.diff('t') + 2*u()



def bound(res):

    u_pos = res(t=0, num=50)

    return u_pos() - 3



def bound_t(res):

    u_zero = res(t=0, num=50)

    return u_zero.diff('t') - 1



net = NetWork(variables=[t], losses=[pde, bound, bound_t], layers=[1, 20, 20, 1])

net.solve(epochs=1000, show_every=100)



plt.plot(net.history_loss)

plt.show()



x = np.linspace(0, 5, 300).reshape(-1, 1)

y = 7 * np.exp(-x) - 4 * np.exp(-2 * x)

y_pred = net(x)





plt.plot(x, y, label='truth', linewidth=3)

plt.plot(x, y_pred, label='predict', linewidth=3, linestyle='dashed')

plt.plot(x, y-y_pred, label='different') 

plt.show()