PDESolverByDeepLearning 2.1.3

This package is suitable for solving the problem of one-dimensional n-order differential equation with Dirichlet boundary conditions.

2. Input parameter description:

import tensorflow as tf

import PDESolverByDeepLearning.hanzuliang as PDESolver

def PDESolver(domain, n, realSolution, StructureOfNeuralNetwork, ImplicitSchemeOfEquation,

DirichletBCPoint, DirichletBCValue, numBatches):

‘’’

param domain:

The domain of the definition of the equation.

param n:

Discretize the domain into n grid points.

param realSolution:

The true solution of the equation. If we do not know the true solution, please input the parameter as ‘None’.

param StructureOfNeuralNetwork = [n1,…,ni,…,no]:

Number of layers of neural network is len(StructureOfNeuralNetwork)

n1: Number of neurons in input layer, whose value is equal to the number of variables in the equation

ni: The number of neurons in the ith hidden layer, whose value is selected according to the

complexity and oscillation of the equation.

no: The number of neurons in the output layer must be 1.

param ImplicitSchemeOfEquation:

Implicit scheme of differential equation.

:param DirichletBCPoint = [x1,x2,x3,…,xn]

:param DirichletBCValue = [u1,u2,u3,…,un]

param numBatches:

Number of training iterations.

return y_output:

The numerical solution predicted by Deep Learning is returned in the form of row vector.

‘’’

3. Case:

Case1. First order differential equation

real solution: u(x)= 5*x**3 + x**2 + 2*x + 1

u’(x) = 15*x**2 + 2*x + 2; u(-1) = -5; x¡Ê[-1,1]

#Code of Case1

import tensorflow as tf

import PDESolverByDeepLearning.hanzuliang as PDESolver

domain = [-1, 1] #Domain

realSolution = lambda x: 5*x**3 + x**2 + 2*x + 1 #Real solution

#If we do not know the true solution, please input the parameter as ‘None’.

#realSolution = None

n = 100 #Divide the domain into n sample points

#If there is an exception of ‘Fail rename;Input/output error’,please delete the last saved model parameter file ‘CKPT’ and train again.

StructureOfNeuralNetwork = [1, 10, 1] #Neural network structure

ImplicitSchemeOfEquation = lambda x, u: tf.gradients(u, x)[0] - 15*x**2 - 2*x - 2 #It must be the implicit scheme of the equation

DirichletBCPoint = [-1] #Dirichlet boundary conditions

DirichletBCValue = [-5]

numBatches = 30000 #Number of iterations

y_output = PDESolver.PDESolver(domain, n, realSolution, StructureOfNeuralNetwork,

ImplicitSchemeOfEquation, DirichletBCPoint, DirichletBCValue, numBatches)

print(‘The discrete solution predicted by Deep Learning is:’)

print(y_output)

print(y_output[0])

Case2. Second order differential equation

real solution: u(x)=x**5

u’’(x)=20*x**3; u(-1)=-1; u(1)=1; x¡Ê[-1,1]

#Code of Case2

import tensorflow as tf

import PDESolverByDeepLearning.hanzuliang as PDESolver

domain = [-1, 1] #Domain

realSolution = lambda x: x**5 #Real solution

#If we do not know the true solution, please input the parameter as ‘None’.

#realSolution = None

n = 100 #Divide the domain into n sample points

#If there is an exception of ‘Fail rename;Input/output error’,please delete the last saved model parameter file ‘CKPT’ and train again.

StructureOfNeuralNetwork = [1, 10, 5, 2, 1] #Neural network structure

ImplicitSchemeOfEquation = lambda x, u: tf.gradients(tf.gradients(u, x)[0], x)[0] - 20*x**3 #It must be the implicit scheme of the equation

DirichletBCPoint = [-1, 1]

DirichletBCValue = [-1, 1]

numBatches = 1000 #Number of iterations

y_output = PDESolver.PDESolver(domain, n, realSolution, StructureOfNeuralNetwork,

ImplicitSchemeOfEquation, DirichletBCPoint, DirichletBCValue, numBatches)

print(‘The discrete solution predicted by Deep Learning is:’)

print(y_output)

Case3. Third order differential equation

real solution: u(x)=x**7 + 2*x**5 + 3*x**3 + x**2

u’’’(x) = 210*x**4 + 120*x**2 + 18; u(-1)=-5; u(0)=0; u(1)=7; x¡Ê[-1,1]

#Code of Case3

import tensorflow as tf

import PDESolverByDeepLearning.hanzuliang as PDESolver

domain = [-1, 1] #Domain

realSolution = lambda x: x**7 + 2*x**5 + 3*x**3 + x**2 #Real solution

#If we do not know the true solution, please input the parameter as ‘None’.

#realSolution = None

n = 100 #Divide the domain into n sample points

#If there is an exception of ‘Fail rename;Input/output error’,please delete the last saved model parameter file ‘CKPT’ and train again.

StructureOfNeuralNetwork = [1, 50, 30, 10, 1] #Neural network structure

ImplicitSchemeOfEquation = lambda x, u: tf.gradients(tf.gradients(tf.gradients(u, x)[0], x)[0], x)[0] - 210*x**4 - 120*x**2 - 18

DirichletBCPoint = [-1, 0, 1]

DirichletBCValue = [-5, 0, 7]

numBatches = 5000 #Number of iterations

y_output = PDESolver.PDESolver(domain, n, realSolution, StructureOfNeuralNetwork,

ImplicitSchemeOfEquation, DirichletBCPoint, DirichletBCValue, numBatches)

print(‘The discrete solution predicted by Deep Learning is:’)

print(y_output)

Keywords: Deep Learning,Machine Learning,Neural Networks,Scientific computing,Differential equations,PDE solver Platform: UNKNOWN Classifier: License :: OSI Approved :: MIT License Classifier: Programming Language :: Python Classifier: Intended Audience :: Developers Classifier: Operating System :: OS Independent