I am trying to get started with pinns so i build this simple script to train an mlp using only the ode.
The ode im trying to solve is y = dy/dx. This ode is very simple but i am using it as a toy example to get to know pinns. I am using numpy in this script and not other ml frameworks. The problem i have is that the network is not learning even though i tried tweaking the learning rate, the size of the network, the weights initialisation and the contribution of the ode loss and the boundary condition loss.

import numpy as np
import matplotlib.pyplot as plt

def tanh(x):
    return np.tanh(x)

def tanh_derivative(x):
    return 1 - np.tanh(x)**2

def initialize_adam(parameters):
    v = {}
    s = {}
    
    for key in parameters.keys():
        v[key] = np.zeros_like(parameters[key])
        s[key] = np.zeros_like(parameters[key])
        
    return v, s

def initialize(n1, n2, n3, n4):
    
    W1 = np.random.normal(0,0.1,size = (n2, n1))
    B1 = np.random.uniform(-0.5, 0.5, size=(n2, 1))
    W2 = np.random.normal(0,0.1,size = (n3, n2))
    B2 = np.random.uniform(-0.5, 0.5, size=(n3, 1))
    W3 = np.random.normal(0,0.1,size = (n4, n3))
    B3 = np.random.uniform(-0.5, 0.5, size=(n4, 1))

    parameters = {'W1':W1,'W2':W2,'W3':W3,'B1':B1,'B2':B2,'B3':B3}
    return parameters


def forward_propagation(A0, parameters:dict):
    W1 = parameters['W1']
    W2 = parameters['W2']
    W3 = parameters['W3']
    B1 = parameters['B1']
    B2 = parameters['B2']
    B3 = parameters['B3']

    Z1 = np.dot(W1, A0) + B1
    A1 = tanh(Z1)
    Z2 = np.dot(W2, A1) + B2
    A2 = tanh(Z2)
    Z3 = np.dot(W3, A2) + B3
    Y_hat = Z3 #Y_hat

    forward = {'Z1':Z1,'A1':A1,'Z2':Z2,'A2':A2,'Z3':Z3,'Y_hat':Y_hat}
    return forward

def back_propagation(A0,Y,forward:dict,parameters:dict,dydx):
    W1 = parameters['W1']
    W2 = parameters['W2']
    W3 = parameters['W3']

    A1 = forward['A1']
    A2 = forward['A2']
    Y_hat = forward['Y_hat']

    Z1 = forward['Z1'] 
    Z2 = forward['Z2']
    Z3 = forward['Z3']


    network_initial_condtion = forward_propagation(A0[0][0]*np.ones_like(A0), parameters) #netowrk ouput at x = A0[0][0]
    Y0 = network_initial_condtion ['Y_hat']

    
    initial_condition_error = Y0 - Y[0]
    ode_error = Y_hat - dydx


    l1 = 1
    l2 = 0.01
    
    m = A0.shape[1]

    dldy = (1/m) * (l1*ode_error + l2*initial_condition_error)

    dldz3 = dldy
    dldw3 = np.dot(dldz3, np.transpose(A2))
    dldb3 = np.sum(dldz3,axis=1, keepdims=True)

    dlda2 = np.dot(W3.T, dldz3)
    dldz2 = dlda2 * tanh_derivative(Z2)
    dldw2 = np.dot(dldz2, np.transpose(A1))
    dldb2 = np.sum(dldz2,axis=1, keepdims=True)

    dlda1 = np.dot(W2.T, dldz2)
    dldz1 = dlda1 * tanh_derivative(Z1)
    dldw1 = np.dot(dldz1, np.transpose(A0))
    dldb1 = np.sum(dldz1,axis=1, keepdims=True)


    backward = {'W1':dldw1,'B1':dldb1,'W2':dldw2,'B2':dldb2,'W3':dldw3,'B3':dldb3}

    return backward


def gradient(forward:dict,parameters:dict):
    W1 = parameters['W1']
    W2 = parameters['W2']
    W3 = parameters['W3']

    Y_hat = forward['Y_hat']
    Z1 = forward['Z1'] 
    Z2 = forward['Z2']
   
    # graidents in relation to network outpout Y_hat
    dydy = np.ones_like(Y_hat )
    dydz3 = dydy

    dyda2 = np.dot(W3.T, dydz3)
    dydz2 = dyda2 * tanh_derivative(Z2)
   

    dyda1 = np.dot(W2.T, dydz2)
    dydz1 = dyda1 * tanh_derivative(Z1)

    dydx = np.dot(W1.T, dydz1)

    return dydx


def update_adam(parameters, grads, v, s, t, learning_rate=None, beta1=0.9, beta2=0.999, epsilon=1e-8):
    v_corrected = {}
    s_corrected = {}
    
    for key in parameters.keys():
        v[key] = beta1 * v[key] + (1 - beta1) * grads[key]
        s[key] = beta2 * s[key] + (1 - beta2) * (grads[key] ** 2)

        v_corrected[key] = v[key] / (1 - beta1 ** t)
        s_corrected[key] = s[key] / (1 - beta2 ** t)

        parameters[key] -= learning_rate * (v_corrected[key] / (np.sqrt(s_corrected[key]) + epsilon))

    return parameters, v, s

def mean_squared_error(Y_hat, Y):
    mse = np.mean((Y_hat - Y) ** 2)
    return mse



def train(A0, Y, epochs, a, n1, n2, n3, n4):

    parameters = initialize(n1, n2, n3, n4)
    v,s = initialize_adam(parameters)

    for i in range(1, epochs + 1):

        forward = forward_propagation(A0, parameters)
        dydx = gradient(forward,parameters)
        gradients = back_propagation(A0,Y,forward,parameters, dydx)
        parameters,v,s = update_adam(parameters, gradients, v, s, i, learning_rate=a)

        Y_hat = forward['Y_hat'].flatten()

        if i % 100 == 0 or i==1:
            mse = mean_squared_error(Y_hat, Y)
            print(f'Epoch: {i}/{epochs} MSE: {mse}')

    return Y_hat

n1 = 1   #input
n2 = 18  #hidden 1
n3 = 18  #hidden 2
n4 = 1   #output

X = np.linspace(0, 3, 300)

Y = np.exp(X) #ode solution

m = len(X)
A0 = X.reshape((n1, m))

epochs = 100000
a = 0.001  #learning rate


Y_hat = train(A0, Y, epochs, a, n1, n2, n3, n4)


plt.figure(figsize=(12, 8))
plt.plot(X, Y, label='Original', color='b')
plt.plot(X, Y_hat, label='Predicted', color='black', linewidth=2)
plt.title('PINN')
plt.xlabel('X')
plt.ylabel('Y')
plt.legend()
plt.grid()
plt.show()