## 7 Conclusions  
In this work, we explored different deep learning architectures for black-box modeling of audio effects. Using raw audio and a given audio effects modeling task, we explored the capabilities of end-to-end DNNs to process the audio accordingly. We tested the models when modeling nonlinear effects with short-term and long-term memory such as a tube _preamp_ and a transistor-based _limiter_ and nonlinear time-varying processors such as the rotating _horn_ and _woofer_ of a _Leslie speaker_ cabinet.  
Through objective perceptual-based metrics and subjective listening tests we found that across all modeling tasks, the architectures that incorporate Bi-LSTMs or, to a lesser extent, latent-space dilated convolutions to explicitly learn long temporal dependencies, outperform the rest of the models. With these architectures we obtain results that are virtually indistinguishable from the analog reference processors. Also, state-of-the-art DNN architectures for modeling nonlinear effects with short-term memory perform similarly when matching the _preamp_ task and considerably approximate the _limiter_ task, but fail when modeling the time-varying _Leslie speaker_ tasks.  
The nonlinear amplifier, rotating speakers and wooden cabinet from the _Leslie speaker_ were successfully modeled. Nevertheless, the crossover filter was bypassed in the modeling tasks since the dry and wet audio were filtered accordingly. This was due to the limited frequency bandwidth of the bass and guitar samples, thus, this modeling task could be further explored with a more appropriate dataset such as Hammond organ recordings.  
As future work, a cost function based on both time and frequency can be used to further improve the modeling capabilities of the models. In addition, since the highest ranked architectures use past and subsequent context input frames, more research is needed on how to adapt these architectures to overcome this latency. Thus, real-time applications would benefit significantly from the exploration of end-to-end DNNs that include long-term memory without resorting to large input frame sizes and the need for past and future context frames. Also, an end-to-end WaveNet architecture with a receptive field as large as the context input frames from _CRAFx_ and _CWAFx_ could also be explored for the time-varying modeling tasks.  
Modeling of artificial reverberators such as a plate or spring can also be explored. Moreover, as shown in [9], the introduction of controls as a conditioning input to the networks can be investigated, since the models are currently learning a static representation of the audio effect. Finally, applications beyond virtual analog can be investigated, for example, in the field of automatic mixing the models could be trained to learn a generalization from mixing practices.  
Conceptualization, M.A.M.R., E.B. and J.D.R.: Data curation, M.A.M.R.: Formal analysis, M.A.M.R.: Investigation, M.A.M.R.: Methodology, M.A.M.R.: Supervision, E.B. and J.D.R.: Validation, M.A.M.R.: Visualization, M.A.M.R.: Writing--original draft, M.A.M.R.: Writing--review & editing, E.B. and J.D.R. All authors have read and agreed to the published version of the manuscript.  
[PERSON] is supported by RAEng Research Fellowship RF/128. [PERSON] is funded by the EPSRC Programme Grant EP/L019981/1, 2014-2019.  
The Titan Xp GPU used for this research was donated by the NVIDIA Corporation. The Queen Mary Ethics of Research Committee approved the listening test with reference number QMREC2165. The _[PERSON] speaker_ samples were recorded with the help of [PERSON].  
Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study: in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.