# A Review of Neural Network-Based Emulation of Guitar Amplifiers
## 3 Neural Network-Based Methods
### Approaches  
Neural networks are black-box by nature but in recent years they have started to be integrated into both gray- and even white-box modelling methods. Indeed, the recent works of [PERSON] et al. [41], [PERSON] et al. [43], [PERSON] et al. [4], and [PERSON] [42] fall into the category of gray-box approaches. [PERSON] et al. present the State Trajectory Network (STN) in their gray-box approach [41], which is a method of integrating neural networks, namely, a MLP here, into a State-Space model. This method aims to augment the black-box neural network approach by integrating the internal values into the training data for a more accurate simulation. A number of distortion circuits are tested, namely a simplified version of the main distortion stage in the Boss DS-1 pedal was used in the form of a second-order diode clipper. Results show that this method is viable as all the circuits modeled were said to be indistinguishable from the targets in informal listening tests however the network training can be unstable [41; 42]. [PERSON] augments the STN in his Masters thesis [42] to include recurrence using a GRU and compares this to both the original STN and a black-box network using only the GRU trained on input-output recordings. This work confirms the instability in training \begin{table}
\begin{tabular}{c c c} \hline \hline
**Architectures** & **Details** & **Type** \\ \hline \multirow{3}{*}{NARX} & RNN modified for limited & \multirow{3}{*}{Recurrent} \\  & connectivity to avoid gradient & \\  & problems. Used in [30] & \\ \hline \hline Multi-layer LSTM & Used in [31] & Recurrent \\ \hline
1 layer LSTM (150 units) + FC & Used in [35] & Recurrent \\ \hline
2 layers LSTM (150 units) & Used in [35] & Recurrent \\ \hline Sequence-to-sequence LSTM & Used in [35] & Recurrent \\ \hline Parametric LSTM & Used in [35] & Recurrent \\ \hline Convolutional LSTM & Used in [35] & Hybrid \\ \hline
3-layer MLP & Used in [35] & DNN \\ \hline CNN with pooling & Used in [35] & Convolutional \\ \hline Convolutional RNN & Used in [35] & Hybrid \\ \hline CAFcx & DNN latent space [37] & Autoencoder \\ \hline \multirow{2}{*}{FeedForward WaveNet} & Dilated convolutions used & \multirow{2}{*}{Convolutional} \\  & in [23; 25; 26] & \\ \hline CRAFx & Bi-LSTM latent space [39] & Autoencoder \\ \hline Single-layer LSTM \& GRU & Followed by FC layer [33] & Recurrent \\ \hline CWAFx & WaveNet latent space [38] & Autoencoder \\ \hline Shallow TCN & Used in [28] & Convolutional \\ \hline STN & Gray-box MLP used in [41] & DNN \\ \hline State-Space GRU & Gray-box RNN used in [42] & Recurrent \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of the architectures for distortion circuit modelling comprising recurrent, convolutional, and hybrid approaches. for the STN as it was unable to model a Boss SD-1 pedal. Although the State-Space GRU was able to emulate this pedal, it was outperformed by its black-box equivalent. However the state-space model managed to outperform the black-box one when applied to a Moog ladder filter due to its self-oscillatory nature. Some of the gray- and white-box approaches making an appearance in the neural network landscape result from the introduction of the Differentiable Digital Signal Processing (DDSP) library from Magenta [44]. DDSP enables the integration of classic signal processing elements, such as filters or oscillators which are typically non-differentiable and thus cannot be dealt with via gradient-based optimization techniques, into the deep learning pipeline in Tensorflow. [PERSON] et al. [4] explore the idea of differentiable IIR filters using the DDSP library. The authors present the link between IIR filters and RNN and present a Wiener-Hammerstein model using differentiable IIR filters. This model is used to emulate a Boss DS-1 distortion pedal and compared with a simple convolutional layer as a baseline. None of the models were able to fit the target data perfectly using this method.