Solving a Second Order ODE Initial Value Problem with Soft Constraints

Problem

We will look at solving the ODE

Over \(t\in[0,1]\), with inital values

Implementation

First import package. We will only import ode_Solvers from PinnDE as that is all that is needed. We will also import numpy to plot the exact solution

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 - ut**2"
order = 2
inits = [1, 0.5]
t_bdry = [0,1]
N_pde = 100
epochs = 1000

To solve, we simply call the corresponding solving function to our problem, and we will leave constraint undeclared as it defaults to soft

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

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()

We can also plot the predicted solution against an exact solution if we have one with

exact_eqn = "-(np.log(abs(t-2))) + np.log(2) + 1"
mymodel.plot_predicted_exact(exact_eqn)

All Code

import pinnde.ode_Solvers as ode_Solvers
import numpy as np

eqn = "utt - ut**2"
order = 2
inits = [1, 0.5]
t_bdry = [0,1]
N_pde = 100
epochs = 1000

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

mymodel.plot_epoch_loss()

mymodel.plot_solution_prediction()

exact_eqn = "-(np.log(abs(t-2))) + np.log(2) + 1"
mymodel.plot_predicted_exact(exact_eqn)