PDE Solving Demo?

[cwoolfo1]

I've noticed that in the PDE solving demo the ground truth solution for the PDE is used to evaluate the boundary condition loss and is thus used in the training process.

            # boundary loss
            bc_true = sol_fun(x_b)
            bc_pred = model(x_b)
            bc_loss = torch.mean((bc_pred-bc_true)**2)

Doesn't this defeat the entire purpose of using KAN models to solve PDEs? Have you tried training a KAN without giving the model previous knowledge of the ground truth solution?

[cwoolfo1]

By simply commenting out the bc loss contribution to the loss calculation and backward step I was able to get similar activation functions in the trained KAN.

[cwoolfo1]

Training on just interior points has yielded a similar result for this simple demo. Have you tried implementing other methods for training on the boundary that do not rely on knowing the solution to the PDE?

[KindXiaoming]

Hi, to solve a PDE, we need both the PDE and boundary condition. Very likely I didn't fully get your question, but maybe you'll find this helpful: https://en.wikipedia.org/wiki/Physics-informed_neural_networks where a boundary loss is provided (assuming we know about boundary values).

[chaous]

I've noticed that in the PDE solving demo the ground truth solution for the PDE is used to evaluate the boundary condition loss and is thus used in the training process.

            # boundary loss
            bc_true = sol_fun(x_b)
            bc_pred = model(x_b)
            bc_loss = torch.mean((bc_pred-bc_true)**2)

Doesn't this defeat the entire purpose of using KAN models to solve PDEs? Have you tried training a KAN without giving the model previous knowledge of the ground truth solution?

I was able to solve PDE (navier-stokes) without true solution only with differential equations and boundary conditions https://github.com/KindXiaoming/pykan/blob/master/tutorials/physics_informed_kan.ipynb