حساب التفاضل والتكامل والتحسين

import numpy as np
import matplotlib.pyplot as plt

class="tok-comment"># ── Numerical derivatives (educational) ──────────────────────────────────────
def numerical_grad(f, x, h=class="tok-num">1e-5):
    class="tok-str">"""Central difference approximation: (f(x+h) - f(x-h)) / 2h"""
    grad = np.zeros_like(x, dtype=float)
    for i in range(len(x)):
        x_plus  = x.copy(); x_plus[i]  += h
        x_minus = x.copy(); x_minus[i] -= h
        grad[i] = (f(x_plus) - f(x_minus)) / (class="tok-num">2 * h)
    return grad

class="tok-comment"># ── class="tok-num">1. Gradient Descent on simple quadratic ───────────────────────────────────
def loss(theta):
    return (theta[class="tok-num">0] - class="tok-num">3)**class="tok-num">2 + (theta[class="tok-num">1] + class="tok-num">1)**class="tok-num">2  class="tok-comment"># minimum at (class="tok-num">3,-class="tok-num">1)

def grad_loss(theta):
    return np.array([class="tok-num">2*(theta[class="tok-num">0]-class="tok-num">3), class="tok-num">2*(theta[class="tok-num">1]+class="tok-num">1)])

theta = np.array([class="tok-num">0., class="tok-num">0.])
lr = class="tok-num">0.1
history = [theta.copy()]

for step in range(class="tok-num">50):
    g = grad_loss(theta)
    theta -= lr * g
    history.append(theta.copy())
    if np.linalg.norm(g) < class="tok-num">1e-6:
        print(fclass="tok-str">"Converged at step {step}")
        break

print(fclass="tok-str">"Final θ: {theta.round(class="tok-num">4)}")  class="tok-comment"># ≈ [class="tok-num">3, -class="tok-num">1]

class="tok-comment"># ── class="tok-num">2. Adam optimizer ────────────────────────────────────────────────────────
def adam(grad_fn, theta_init, lr=class="tok-num">0.01, n_steps=class="tok-num">100, b1=class="tok-num">0.9, b2=class="tok-num">0.999, eps=class="tok-num">1e-8):
    theta = theta_init.copy().astype(float)
    m, v = np.zeros_like(theta), np.zeros_like(theta)
    history = [theta.copy()]
    for t in range(class="tok-num">1, n_steps+class="tok-num">1):
        g = grad_fn(theta)
        m = b1*m + (class="tok-num">1-b1)*g
        v = b2*v + (class="tok-num">1-b2)*g**class="tok-num">2
        m_hat = m / (class="tok-num">1 - b1**t)
        v_hat = v / (class="tok-num">1 - b2**t)
        theta -= lr * m_hat / (np.sqrt(v_hat) + eps)
        history.append(theta.copy())
    return theta, history

theta_adam, hist_adam = adam(grad_loss, np.array([class="tok-num">0., class="tok-num">0.]), lr=class="tok-num">0.1)
print(fclass="tok-str">"Adam θ: {theta_adam.round(class="tok-num">4)}")

class="tok-comment"># ── class="tok-num">3. Chain rule in action (manual backprop) ─────────────────────────────────
class="tok-comment"># f(x) = (2x + class="tok-num">1)^class="tok-num">2. df/dx = class="tok-num">2 * (2x+class="tok-num">1) * class="tok-num">2 = class="tok-num">4*(2x+class="tok-num">1)
x = class="tok-num">3.0
class="tok-comment"># Forward pass
u = class="tok-num">2*x + class="tok-num">1    class="tok-comment"># u = class="tok-num">7
f = u**class="tok-num">2       class="tok-comment"># f = class="tok-num">49

class="tok-comment"># Backward pass (chain rule)
df_du = class="tok-num">2*u    class="tok-comment"># = class="tok-num">14
du_dx = class="tok-num">2      class="tok-comment"># constant
df_dx = df_du * du_dx   class="tok-comment"># = class="tok-num">28
print(fclass="tok-str">"df/dx at x=class="tok-num">3: {df_dx}")  class="tok-comment"># analytical: class="tok-num">4*(class="tok-num">2*class="tok-num">3+class="tok-num">1) = class="tok-num">28 ✓

class="tok-comment"># ── class="tok-num">4. Learning rate sensitivity ─────────────────────────────────────────────
fig, axes = plt.subplots(class="tok-num">1, class="tok-num">3, figsize=(class="tok-num">12,class="tok-num">3))
for ax, lr_val in zip(axes, [class="tok-num">0.01, class="tok-num">0.1, class="tok-num">0.9]):
    theta = np.array([class="tok-num">0.])
    losses = []
    for _ in range(class="tok-num">100):
        g = class="tok-num">2*(theta[class="tok-num">0] - class="tok-num">5)
        theta[class="tok-num">0] -= lr_val * g
        losses.append((theta[class="tok-num">0]-class="tok-num">5)**class="tok-num">2)
    ax.semilogy(losses)
    ax.set_title(fclass="tok-str">"lr = {lr_val}")
    ax.set_xlabel(class="tok-str">"Steps")
    ax.set_ylabel(class="tok-str">"Loss")
plt.tight_layout()
plt.show()  class="tok-comment"># lr=class="tok-num">0.01: slow, lr=class="tok-num">0.1: perfect, lr=class="tok-num">0.9: oscillates