Behler-Parrinello-Neural-Network

The neural network topology implemented by Fortnet was proposed by J. Behler and M. Parrinello in 2007 and has worked its way into computational materials science as one of the most famous neural network architectures to represent potential-energy surfaces [1].

By representing the systems total energy (or targets in general) as a sum of atomic contributions, in combinations with suitable symmetry mappings (c.f. ACSF), it overcomes the limitations of conventional fixed structure topologies. The BPNN super-nn contains several sub-nn’s, each of which is assigned to a certain type of atom (i.e. element or species). Accordingly, the subnets are fed (forward) with the features of the atoms of their specific type and added up in the case of global targets (i.e. system properties like the total energy).

To clarify this admittedly abstract explanation of the Behler-Parrinello-Neural-Network architecture, the figure below shows the topology described for the special case of two atom species \(A\) and \(B\) with a single output node:

In general, the activation status \(a_i^l\) of a single neuron \(i\) in layer \(l\) can be calculated as follows:

\[a_i^l = f_l\left(\sum_{k=1}^{N_{l-1}}w_k^{l-1}a_k^{l-1} + b_k^l\right)\]

There the index \(k\) runs over all neurons of the adjacent layer \(l-1\). Once this fundamental equation is known, the input \(a_i^1\) (i.e. ACSF) can be propagated through the entire subnet, with the arguments of the activation functions being logged for later backpropagation. For global system properties the quantity of interest finally follows as a summation of atomic contributions of the species:

\[E = \sum_{i=1}^{N_\mathrm{type}}\sum_{j=1}^{N^\mathrm{at}_i} E_j^i\]

In the case of atomic properties (i.e. forces) the atomic contributions can obviously be used directly, since a summation would be superfluous.