Initial Conditions

[KostasVog20]

Dear @lululxvi and the rest of the community,

I am currently working with DeepXDE to solve a two-dimensional inverse problem by identifying five model parameters of a mathematical system composed of three partial differential equations (PDEs). I have a critical question regarding the initial conditions.

I understand that if I know the mathematical expression required to define the initial conditions, I can implement something like the following:

def T_i_func(x):
return (0.57 * np.exp( - ((((x[:, 0:1] - 0.4) ** 2.0) / (0.1 ** 2.0)) +
(((x[:, 1:2] - 0.4) ** 2.0) / (0.1 ** 2.0)))) + 0.19)

ic_T = dde.icbc.IC(
spacetime, T_i_func, lambda _, on_initial: on_initial, component=0
)

In this case, x[:, 0:1] represents the longitudinal coordinate (x-direction), and x[:, 1:2] represents the lateral coordinate (y-direction).

However, my question pertains to a situation where I do not have a formula for defining the initial condition but have known values at specific spatial points in matrix form. How can I use this matrix (or vector) in place of the formula? Specifically, I am uncertain about the values of x[:, 0:1] and x[:, 1:2]. Should these correspond to the points in the training dataset, or are they distinct from them?

I would greatly appreciate any guidance you could provide on this matter.

Thank you in advance for your assistance.

Best regards,
K.V.