Tensorflow to learn an input dependent and an input independent variable

The problem is that you are creating the nu variable with a hard-coded batch size of 1. That is why it is only working with a sample size of 1 and no more. It is hard to say what you want to do, but you can try something like this:

import tensorflow as tf

class NuLayer(tf.keras.layers.Layer):

  def __init__(self, batch_dim, initial='he_uniform'):
    super(NuLayer, self).__init__()
    self.batch_dim = batch_dim

  def build(self, input_shape):
    self.nu = tf.Variable(initial_value = tf.ones((self.batch_dim,1)), trainable = True)

  def call(self, inputs):
    return self.nu

inp_1 = tf.keras.layers.Input(shape=(2,))  #setting the size of the input layer
initial = 'he_uniform'
x = tf.keras.layers.Dense(20,kernel_initializer= initial, activation = 'tanh', bias_initializer=initial)(inp_1)
x = tf.keras.layers.Dense(20,kernel_initializer= initial, activation = 'tanh', bias_initializer=initial)(x)
x = tf.keras.layers.Dense(20,kernel_initializer= initial, activation = 'tanh', bias_initializer=initial)(x)
x = tf.keras.layers.Dense(20,kernel_initializer= initial, activation = 'tanh',bias_initializer=initial)(x)
x = tf.keras.layers.Dense(20,kernel_initializer= initial, activation = 'tanh',bias_initializer=initial)(x)
x = tf.keras.layers.Dense(20,kernel_initializer= initial, activation = 'tanh',bias_initializer=initial)(x)

x = tf.keras.layers.Dense(1,kernel_initializer= initial,  activation = 'tanh',bias_initializer=initial)(x)

nu = NuLayer(batch_dim=4)
nu = nu(inp_1)
out = tf.keras.layers.Concatenate(axis=1)([x, nu])
model = tf.keras.Model(inputs=inp_1, outputs=out)

def residualValOfPDE(xt):
    
    x = xt[:, 0:1] # x coordinate
    t = xt[:, 1:2] # t coordinate
    with tf.GradientTape(persistent=True) as tape:
        
        tape.watch(x) 
        tape.watch(t)
        u, nu  = model(tf.stack([x[:, 0], t[:, 0]], axis=1) )[0]        
        u_x = tape.gradient(u, x)   
        
    u_t  = tape.gradient(u, t)        
    u_xx = tape.gradient(u_x, x)

    return u_t + u*u_x - nu*u_xx

xt_f = tf.random.normal((10000, 2))

print( residualValOfPDE(xt_f[1:3,:])) 

tf.Tensor(
[[0.5751909]
 [0.       ]], shape=(2, 1), dtype=float32)

If you want to examine a different batch size, then change the batch size in the Input layer:

nu = NuLayer(batch_dim=4)
print( residualValOfPDE(xt_f[1:5,:])) 

[[-0.51205623]
 [ 0.        ]
 [ 0.        ]
 [ 0.        ]], shape=(4, 1), dtype=float32)