# Difference-Based Mutation Operation for Neuroevolution of Augmented Topologies +
## 2 Materials and Methods  
### Neuroevolution  
Searching for better neural architectures stumbles upon the problem of solutions representation. The representation should be designed so that it would allow encoding various types of networks, and an efficient search at the same time. An individual in the developed evolutionary algorithm should contain the encoding of neurons, connections and weights. Some of the early studies on neuroevolution use the scheme, which is often referred to as the Topology and Weight Evolving Artificial Neural Networks (TWEANNs). For example, such methods, described in [11; 12], encode the appearance of each node and connection in the phenotype. In [13] the evolutionary algorithm for neural networks training utilized five mutation operators to add or delete nodes and connections, as well as the back-propagation for weights training.  
Despite the ability of TWEANNs to perform the search for optimal architectures, they are characterized by a number of significant disadvantages, including the competing conventions problem, described in [9]. The competing conventions problem occurs when the encoding scheme allows several ways of expressing the same solution, i.e., the solution is the same, but the encodings are different. The historical markers, or innovation markers mechanism, proposed in the NEAT approach allows solving the problem of competing conventions.  
The original studies on the NEAT algorithm have shown that it could be applied to a variety of problems, including classification, regression, control algorithms, playing games, and so on. In [14] a review of neuro-evolutionary approaches is given, describing the application to various benchmark problems, for example, pole balancing and robot control. Some of the recent studies considered the landscape analysis of the neuroevolutionary approaches [15], other studies investigated applying autocorrelation and measuring ruggedness based on entropy for problem difficulty estimation. Some of the investigations used the hyperheuristic approach to train a NN, which determined the types of operators to be used in the optimization process [16]. In [17] the new loss function is generated using neuroevolution, which allowed improving the training process of classic neural networks.  
Although the NEAT algorithm and its modifications have found a wide area of applications, only a few studies have addressed the problem of weight tuning in the trained network. Several studies have tried to solve the problem of weights tuning, for example, in [18] the cooperative coevolution of synapses was proposed, showing significant improvements to the algorithm capabilities. In [19] the cooperative co-evolutionary differential evolution was applied to train ANN weights, however, this approach is not applicable for the case of network topology optimization. In a recent study, [20] the weight tuning for the evolutionary exploration of augmenting memory models, a variation of neuroevolution, is considered. In particular, weights of the same connections in both parents are identified and generated based on an equation, similar to the one used in differential evolution. The results show that the so-called Lamarckian weight inheritance allows providing additional benefit to the selection of weights improves the way the weight search space is traversed. Considering these studies, it can be concluded that the finding suboptimal weight values for evolved architectures represent a significant problem, so that the development of new general weight search operators for neuroevolution is an important research direction, as it would allow improving the performance of many other modifications. In particular, it seems reasonable to transfer the search operators applied in differential evolution, which is a simple yet efficient numerical optimizer.  
The next section of this paper contains the description of the encoding scheme used in NEAT, and then the newly developed operators are presented.