Solving a System of 2 ODEs as Initial Value Problem with soft Constraints with a DeepONet

Problem

We will look at solving the ODEs

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

Implementation

First import package. We will only import ode_Solvers from PinnDE as that is all that is needed

import pinnde.ode_Solvers as ode_Solvers

Next, we declare our equations, orders, inital values, t boundary, number of points, sensor_range, num_sensors, and epochs. Equation must be in form eqn = 0

eqn1 = "utt - ut**2"
eqn2 = "xt + ut - x"
eqns = [eqn1, eqn2]
orders = [2, 1]
inits = [[1, 0.5], [1]]
t_bdry = [0,1]
N_pde = 100
sensor_range = [-3, 3]
num_sensors = 3000
epochs = 1500

To solve, we simply call the corresponding solving function to our problem

mymodel = ode_Solvers.solveODE_DeepONetSystem_IVP(eqns, orders, inits, t_bdry, 
                                    N_pde, sensor_range, num_sensors, 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()

All Code

import pinnde.ode_Solvers as ode_Solvers

eqn1 = "utt - ut**2"
eqn2 = "xt + ut - x"
eqns = [eqn1, eqn2]
orders = [2, 1]
inits = [[1, 0.5], [1]]
t_bdry = [0,1]
N_pde = 100
sensor_range = [-3, 3]
num_sensors = 3000
epochs = 1500

mymodel = ode_Solvers.solveODE_DeepONetSystem_IVP(eqns, orders, inits, t_bdry, 
                                    N_pde, sensor_range, num_sensors, epochs)

mymodel.plot_epoch_loss()

mymodel.plot_solution_prediction()