Implementing initial condition as hard constraint using values obtained from interpolation between arbitrary values

[Rahul-Patel25]

Hello,
Thanks for this useful library.

I am working on the heat equation example.

To incorporate the initial condition as a soft constraint I can use

ic = dde.icbc.IC(
    geomtime,
    lambda x: np.sin(np.pi * x[:, 0:1]),
    lambda _, on_initial: on_initial,
)

or to implement the same initial condition as hard constraint I can transform the output as

def output_transform(x, y):
    return x[:, 0:1] * (1 - x[:, 0:1]) * y * x[:, 1:2] +  tf.sin(pi * x[:, 0])

and will get the same results.

Now, rather than specifying the sinusoid function in the ICs, I want to specify arbitrary points (let’s say 7) of X and specify the corresponding solution (y) at that point for the initial state (t=0). For intermediate values of x, the values of y should be obtained via interpolation (the end goal is to pass these arbitrary solution values (y) at t=0 as an input to the Neural network to implement arbitrary initial condition).

For soft constraints, I was able to implement them as follows.

def initial_funcval(x):
    size_vector = x[:, 0:1].shape[0]
    outval = np.zeros([size_vector])
    xx = np.linspace(0, 1, 7)
    yy = [0, 0.5,0.866,1, 0.866,0.5,0]

    for i in range(size_batch):
                f = interpolate.interp1d(xx, yy, kind='linear')
        outval[i] = f(x[i, 0:1])

    return outval.reshape(size_vector, 1)

ic = dde.icbc.IC(
    geomtime,
    lambda x: initial_funcval(x),
    lambda _, on_initial: on_initial,
)

interpolate.interp1d is an 1D interpolation function of SciPy.

How can I implement the same as hard constraints by transforming the output layer? Could you please guide me through it.

[lululxvi]

You cannot use SciPy. You need to implement it in TensoFlow/pyTorch/etc.