Solving a Second Order ODE Boundary Value Problem with Hard Constraints

Problem

We will look at solving the ODE

Over \(t\in[0,\pi/4]\), with boundary values

Implementation

First import package. We will only import ode_Solvers from PinnDE as that is all that is needed. Since we need to represent pi for the boundary, we also import numpy.

import pinnde.ode_Solvers as ode_Solvers
import numpy as np

Next, we declare our equation, order, inital values, t boundary, number of points, and epochs. Equation must be in form eqn = 0

eqn = "utt + 4*u"
order = 2
inits = [-2, 10]
t_bdry = [0,np.pi/4]
N_pde = 100
epochs = 1000

To solve, we simply call the corresponding solving function to our problem, and we will declare a hard constraint on boundaries

mymodel = ode_Solvers.solveODE_BVP(eqn, order, inits, t_bdry, N_pde, epochs, constraint = "hard")

If we want to quickly vizualize our data from training we can add after the solving function

mymodel.plot_epoch_loss()

mymodel.plot_solution_prediction()

All Code

import pinnde.ode_Solvers as ode_Solvers
import numpy as np

eqn = "utt + 4*u"
order = 2
inits = [-2, 10]
t_bdry = [0,np.pi/4]
N_pde = 100
epochs = 1000

mymodel = ode_Solvers.solveODE_BVP(eqn, order, inits, t_bdry, N_pde, epochs)

mymodel.plot_epoch_loss()

mymodel.plot_solution_prediction()