Hard-constraint Neumann boundary condition

[c-straub]

Hi there,

I'm having difficulties imposing a hard-constraint Neumann boundary condition in DeepXDE.

Concretely, say I want to solve some PDE on the space-time domain $(x,t)\in[-1,1]\times[0,1]$ with Neumann boundary condition $\partial_x u(1,t)=0$. One way to hard-constraint this boundary condition is by transforming the output of a neural network $u_{\mathrm{NN}}=u_{\mathrm{NN}}(x,t)$ as follows:

$$u_{\mathrm{transformed}}(x,t) := x\cdot (u_{\mathrm{NN}}(x,t)-u_{\mathrm{NN}}(1,t)-\partial_xu_{\mathrm{NN}}(1,t)).$$

Is there a way to implement this transformation in DeepXDE using an output_transform function (in particular, the evaluation of the NN and its $x$-derivative at $(1,t)$ ) or should I use callbacks for this task?

Any help would be very much appreciated.

Best,
Christopher

[lululxvi]

This is not easy to implement.