Basis Functions and MLP Dimensions

[lyncdw19]

Hello everyone,

I have recently began investigating the possibility of using Allegro to simulate some materials of interest. I am a bit confused as to how the dimensions of the various MLPs match up and have a few questions that hopefully will be simple to answer.

1. The initial scalar feature embedding takes Z_i, Z_j, and N_{basis} basis functions into the 2-body MLP to generate the x^{ij,L=0} term. How do we define N_{basis} in the yaml settings file to determine how many basis functions to use?

2. I am a bit confused with regards to the various MLP dimensions. For instance, if I have the following dimensions defined:

              two_body_latent_mlp_latent_dimensions: [128, 256, 512, 1024]
   

If I understand this correctly, this will define the MLP to have 4 hidden layers, with 128, 256, 512, and 1024 neurons respectively. But how are the input and output layers defined? Is the 1024 the output dimension and therefore x^{ij,L=0} contains 1024 features? What about the input dimensions?

3. Regarding the initial tensor feature embedding, the equation is given:

             V_{n,l,p}^{ij,L=0} = w_{n,l,p}^{ij,L=0} Y_{l,p}^{ij}
   

where the w_{n,l,p}^{ij,L=0} terms are defined by the MLP_{embed} neural network. Since this neural network has the same argument x^{ij,L=0} for all n,l,p combinations, does this mean that each n,l,p combination has a unique MLP_{embed} with unique weights and biases that are learned for that specific n,l,p choice?

4. It is my understanding that the V^{ij} terms are higher order objects containing l_{max} irreducible representations of SO(3). Why is there a need to have n channels containing multiple "duplicate" Y_l^m terms of the same order? Wouldn't having a single set of Y_l^m terms be sufficient given that they are being multiplied by the learned weights w_{n,l,p}^{ij,L=0} ? What is the intuition regarding the need for multiple channels and how many channels are "sufficient" for the representation?

Thanks in advance.