beginner/examples_tensor/polynomial_numpy

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Warm-up: numpy

Created On: Dec 03, 2020 | Last Updated: Sep 29, 2025 | Last Verified: Nov 05, 2024

A third order polynomial, trained to predict \(y=\sin(x)\) from \(-\pi\) to \(\pi\) by minimizing squared Euclidean distance.

This implementation uses numpy to manually compute the forward pass, loss, and backward pass.

A numpy array is a generic n-dimensional array; it does not know anything about deep learning or gradients or computational graphs, and is just a way to perform generic numeric computations.

99 1214.9896820257532
199 836.3601848140029
299 577.2327233776957
399 399.68616132676254
499 277.8970823124964
599 194.25998851126863
699 136.7583549995971
799 97.180962579619
899 69.91040750200776
999 51.09933003602482
1099 38.10965063063173
1199 29.130410530971567
1299 22.917033973005314
1399 18.61322937050296
1499 15.629197796444608
1599 13.55825096837986
1699 12.119659935713948
1799 11.119437882291038
1899 10.423398716074576
1999 9.938628011086182
Result: y = -0.031841392378030964 + 0.8424362758215231 x + 0.005493170028266546 x^2 + -0.09129568891423226 x^3

import numpy as np
import math

# Create random input and output data
x = np.linspace(-math.pi, math.pi, 2000)
y = np.sin(x)

# Randomly initialize weights
a = np.random.randn()
b = np.random.randn()
c = np.random.randn()
d = np.random.randn()

learning_rate = 1e-6
for t in range(2000):
    # Forward pass: compute predicted y
    # y = a + b x + c x^2 + d x^3
    y_pred = a + b * x + c * x ** 2 + d * x ** 3

    # Compute and print loss
    loss = np.square(y_pred - y).sum()
    if t % 100 == 99:
        print(t, loss)

    # Backprop to compute gradients of a, b, c, d with respect to loss
    grad_y_pred = 2.0 * (y_pred - y)
    grad_a = grad_y_pred.sum()
    grad_b = (grad_y_pred * x).sum()
    grad_c = (grad_y_pred * x ** 2).sum()
    grad_d = (grad_y_pred * x ** 3).sum()

    # Update weights
    a -= learning_rate * grad_a
    b -= learning_rate * grad_b
    c -= learning_rate * grad_c
    d -= learning_rate * grad_d

print(f'Result: y = {a} + {b} x + {c} x^2 + {d} x^3')