How to define inputs out of the geometry and average of output as an input

[AJAXJR24]

Dear @lululxvi I want to solve a system of equation consist of an energy balance PDE and Arrhenisu equation. My inputs are the Spatial and Temporal dimensions and the ambient temperature. My outputs are T which is the function of all dimensions and c which is the function of time. How can I define the ambient temperature (is a function of time) as an input? the temperature in the Arrhenius equation is the average temperature in the whole domain in each time step. How can I define the mean Temperature in the Arrhenius equation? Since the temperature is getting solved in each node. Thanks

[praksharma]

Sorry this is not my background, but why do you need ambient temperature as the input to the network? Can you write the PDEs your are trying to solve?

Also, to answer your question, it is quite challenging to solve problems with PINN where the inputs to the network is dependent on the output. The training takes excessively long time to converge even for simple problems. This is because your inputs are not constant, it is changing in each training iteration.

[AJAXJR24]

Sorry this is not my background, but why do you need ambient temperature as the input to the network? Can you write the PDEs your are trying to solve?

There is really no need for that. I was Just saying it exists. I already defined it seperately.

Also, to answer your question, it is quite challenging to solve problems with PINN where the inputs to the network is dependent on the output. The training takes excessively long time to converge even for simple problems. This is because your inputs are not constant, it is changing in each training iteration.

Is there a way though??

[praksharma]

Yes you just need a train the problem for a longer duration. If you provide us with the code, I can look into and give you insights.

[AJAXJR24]

Yes you just need a train the problem for a longer duration. If you provide us with the code, I can look into and give you insights.

Thanks for your reply My problem has changed. I got a convection-conduction balance boundary condition where the T_ambient is a function of time as below: BTW is that tf.reduce_mean ok to define the average temperature in each time step?

How can I define this? if it could be defined by tf.where what should be the shape of the T_amb tensor (shape of time or shape of the whole domain)? Or is there a better way to define this Temperature?

def pde(x,y):

  x = tf.cast(x, tf.float64)

  y = tf.cast(y, tf.float64)

  T_xx = dde.grad.hessian(y,x,component=0,i=0,j=0)

  T_yy = dde.grad.hessian(y,x,component=0,i=1,j=1)

  T_t = dde.grad.jacobian(y,x,i=0,j=2)

  c_t = dde.grad.jacobian(y,x,i=1,j=2)

  ###########################################
  ###########################################

  return [T_t  - (kx/ky)* T_xx  - T_yy + ((H*W)/(ρ*cp*delta_T))*c_t,
              A * y[:,1:2] * (He**2/alpha_y) * tf.exp(-E/(Rc * (tf.reduce_mean(y[:,0:1],axis=0)*delta_T + T_min)))  + c_t]

BC:?

def func_conv_x(x,y):
  x = tf.cast(x, tf.float64)
  y = tf.cast(y, tf.float64)
  condition_1 = tf.less(x[:,2:3],48.604069896509976)
  condition_2 = tf.greater(x[:,2:3],72.90610484476497)
  condition_3 = tf.math.logical_and(x[:,2:3] >= 48.604069896509976, x[:,2:3] <= 72.90610484476497)
  #################################################################################################
  mm = 0.03647356906232812
  yy = -1.6921837228041898
  #################################################################################################
  T_ext_1 = tf.where(condition_1, tf.ones_like(x[:,2:3])*0.08058017727639, tf.zeros_like(x[:,2:3]))
  T_ext_2 = tf.where(condition_2, tf.ones_like(x[:,2:3])*0.96696212731668,tf.zeros_like(x[:,2:3]))
  T_ext_3 = tf.where(condition_3, mm * x[:,2:3] + yy, tf.zeros_like(x[:,2:3]))
  T_ext = tf.add_n([T_ext_1,T_ext_2,T_ext_3])
  tf.ones_like(x)*T_ext,y[:,0:1]
  print((x[:,0:1].shape),(x[:,1:2].shape),(x[:,2:3].shape))
  return ((h*He)/kx) * (tf.ones_like(x)*T_ext-y[:,0:1])

I used one Neural Network with 3 inputs and 2 outputs but the project I am working on said that there are 2 different Neural Networks each with three inputs and one output. Is my approach correct? If not what is the solution? Thanks