Second-Order ODE

The full solution can be found at basic_odes2.ipynb. DeepXDE result can be found on their website.

Problem Setup

\[y''-10y'+9y=5t, y(0)=-1, y'(0)=2\]

The solution is \(y=\frac{50}{81} + \frac{5}{9}t + \frac{31}{81}e^{9t} - 2e^t\)

Implementation

1. Import necessary packages

   import os
   import numpy as np
   import torch
   import matplotlib.pyplot as plt
   from deep_macrofin import PDEModel
   from deep_macrofin import ActivationType, Comparator, EndogVar, EndogVarConditions, EndogEquation
   

2. Define problem
   In this example, we define a second-order linear PDE model with specific initial conditions and training configurations.

   ode = PDEModel("second_order_linear", config={"num_epochs": 10000}) # define PDE model to solve
   ode.set_state(["t"], {"t": [0., 0.25]}) # set the state variable, which defines the dimensionality of the problem
   ode.add_endog("y", config={
       "device": "cuda" if torch.cuda.is_available() else "cpu",
       "hidden_units": [50, 50, 50],
       "activation_type": ActivationType.Tanh,
       "positive": False,
       "derivative_order": 2,
   }) # we use endogenous variable to represent the function we want to approximate
   ode.add_endog_equation("y_tt-10*y_t+9*y=5*t", label="base_ode", weight=0.01) # endogenous equations are used to represent the ODE
   ode.add_endog_condition("y", 
                               "y(SV)", {"SV": torch.zeros((1, 1))},
                               Comparator.EQ,
                               "-1", {},
                               label="ic1") # define initial condition
   ode.add_endog_condition("y", 
                               "y_t(SV)", {"SV": torch.zeros((1, 1))},
                               Comparator.EQ,
                               "2", {},
                               label="ic2") # add a second initial condition
   

3. Train and evaluate

   ode.train_model("./models/second_order_ode3", "second_order_ode3.pt", True)
   ode.eval_model(True)
   

4. To load a trained model

   ode1.load_model(torch.load("./models/ode1/ode1.pt"))
   

5. Plot the solutions

   fig, ax = plt.subplots(1, 3, figsize=(18, 6))
   x = np.linspace(0, 0.25)
   ax[0].plot(x, 50/81+5/9*x+31/81*np.exp(9*x)-2*np.exp(x), label="y_true")
   ax[1].plot(x, 5/9+31/9*np.exp(9*x)-2*np.exp(x), label="y_t_true")
   ax[2].plot(x, 31*np.exp(9*x)-2*np.exp(x), label="y_tt_true")
   ode.endog_vars["y"].plot("y", {"t": [0, 0.25]}, ax=ax[0])
   ode.endog_vars["y"].plot("y_t", {"t": [0, 0.25]}, ax=ax[1])
   ode.endog_vars["y"].plot("y_tt", {"t": [0, 0.25]}, ax=ax[2])
   plt.subplots_adjust()
   plt.show()