Discontinuous and Oscillating Function

The full solution can be found at function_approximation.ipynb.

Problem Setup

\[ y= \begin{cases} 5 + \sum_{k=1}^4 \sin(kx), x<0\\ \cos(10x), x\geq 0 \end{cases} \]

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
   Here, we use the default training configuration, and default setup for learnable endogenous variable.

   discont_approx2 = PDEModel("discontinuous_approximator", {"num_epochs": 50000}) # define PDE model with 50,000 epochs
   discont_approx2.set_state(["x"], {"x": [-3., 3.]}) # set the state variable "x" with its range
   discont_approx2.add_endog("y", { 
       "hidden_units": [40, 40], # specify the architecture with two hidden layers of 40 units each
       "activation_type": ActivationType.SiLU, # set the activation function to SiLU
   })
   discont_approx2.add_endog_equation("y=5+sin(x)+sin(2*x)+sin(3*x)+sin(4*x)") 
   discont_approx2.add_endog_equation("y=cos(10*x)") 
   

3. Train and evaluate

   discont_approx2.train_model("./models/discont_approx2", "discont_approx2.pt", True)
   discont_approx2.eval_model(True)
   

4. To load a trained model

   discont_approx2.load_model(torch.load("./models/discont_approx2/discont_approx2_best.pt"))
   

5. Plot the solutions

   fig, ax = plt.subplots(1, 3, figsize=(18, 6))
   x = np.linspace(-3, 3)
   x_neg = np.linspace(-3, 0)
   x_pos = np.linspace(0, 3)
   ax[0].plot(x_neg, 5+np.sin(x_neg)+np.sin(2*x_neg)+np.sin(3*x_neg)+np.sin(4*x_neg), label="5+sin(x)+sin(2x)+sin(3x)+sin(4x)", color="red")
   ax[0].plot(x_pos, np.cos(10*x_pos), label="cos(10x)", color="red")
   ax[1].plot(x_neg, np.cos(x_neg)+2*np.cos(2*x_neg)+3*np.cos(3*x_neg)+4*np.cos(4*x_neg), label="cos(x)+2cos(2x)+3cos(3x)+4cos(4x)", color="red")
   ax[1].plot(x_pos, -10*np.sin(10*x_pos), label="-10sin(10x)", color="red")
   ax[2].plot(x_neg, -np.sin(x_neg)-4*np.sin(2*x_neg)-9*np.sin(3*x_neg)-16*np.sin(4*x_neg), label="-sin(x)-4sin(2x)-9sin(3x)-16sin(4x)", color="red")
   ax[2].plot(x_pos, -100*np.cos(10*x_pos), label="-100cos(10x)", color="red")
   discont_approx2.endog_vars["y"].plot("y", {"x": [-3, 3]}, ax=ax[0])
   discont_approx2.endog_vars["y"].plot("y_x", {"x": [-3, 3]}, ax=ax[1])
   discont_approx2.endog_vars["y"].plot("y_xx", {"x": [-3, 3]}, ax=ax[2])
   plt.subplots_adjust()
   plt.show()