some questions; info for reading

[ian204]

Hello Lu, Let me first express my thanks to you for bringing to us this package which simplifies the application of NN to PDEs. My interests are on solving time-dependent NS equations, with various BC and IC. I have read your guide and FAQ list but I need some extra readings for understanding better DeepXDE package. If you, or anyone else in this group, can answer/comment my following questions I would appreciate very much. Thank you. Ian

Q1: Where I can find proper description of the syntax for using "lambda x, or lambda x, y or lambda_" ? This would help to better understand the implementation of soft/hard BC/IC. Q2: How can I use my optimization function to minimize the loss function? Q3: How can I access the gradient of loss function wrt to all variables (weights and biases) or with respect to a subset of them? Q4: How can I access the Hessian of the loss function wrt to a subset of variables (weights and biases)? Q5: Does DeepXDE have the option to move the geometry? Q6: Is there any explicit example for time-dependent NS equations (2D/3D)?

[lululxvi]

• Q1; This is standard Python grammar. You should find lots of tutorials online.
• Q2:I have no experience with this. You need to figure out how to write your own optimizer instead of those in Tensorflow.
• Q3 &Q4: I also don't have experience with this. There should a way in tensorflow. Or actually you can implement the network from scratch (instead of using something like tf.layers), which is very easy to do for fully-connected networks, and then you have the full control, and can directly use tf.gradients to get the gradient. Apply twice to get Hessian.
• Q5: what do you mean by "move the geometry"?
• Q6: I currently don't have NS equation example, but the implementation of NS is quite similar to other problems.

[ian204]

Thank you for the comments, Lu. I have added some further explanation as it seems that I didn't describe properly my questions in my original message.

Ian

For Q2: Definitely, I will write my own optimization function - this is not object of my question. The question is how to call this new function instead of the ones (adam, ...) that tensorflow uses. For Q5: I mean how one could modify the domain. Think of NS equation in a moving domain ... For Q6: Actually, it seems the the code with Beltrami flow is a NS code, just with specific BC/IC

[lululxvi]

Here are some details of the optimizer: https://www.tensorflow.org/api_docs/python/tf/compat/v1/train/Optimizer#apply_gradients You can use compute_gradients to get the gradients, and I think you need to implement your own apply_gradients. So basically you need to write a subclass of tf.compat.v1.train.Optimizer

For moving domain, you can use a sufficiently large domain that covers your true domain. Then instead of asking DeepXDE to sample the training points, you can sample the points X by yourself, and then define PDE(..., anchors=X). Or you can implement a subclass of Geometry

[ian204]

Thank you Lu. It gives me some ideas to work on.