# Multi-Output Regression with Generative Adversarial Networks (MOR-GANs)
## 6 Conclusions and Future Work  
In this paper, we demonstrate that Generative Adversarial Networks (GANs) can perform well for a number of regression tasks, sometimes outperforming a model based on state-of-the-art Gaussian Process Regression (GPR). The particular model used is a Wasserstein GAN (WGAN), which can be easier to train than a standard GAN. For simple regression and multiple regression tasks, both GAN and GPR perform well, although for the dataset which has variable uncertainty (modelled as heteroscedastic noise), the GPR fails to learn any variation in uncertainty, whereas the GAN captures this variation well. Also, for the more challenging problem of multi-modal distributions, the GPR struggles to learn the distribution whereas the GAN is able to reproduce the distribution very well. Furthermore, for multi-output regression, the WGAN also demonstrated good performance, showing that the GAN is able to capture the covariance information between all the output variables (which includes the independent and dependent variables of the regression problem).  
Although the GPR can be modified for improved performance on specific types of data (such as heteroscedastic noise and multi-output regression), we wanted to highlight, here, that the WGAN needs no modification for these problems: one single WGAN model can perform well for all the datasets with which we tested the models.  
Novelties of the work include using a GAN for regression; being able to apply this model to multi-modal data and multi-output regression (MOR-GAN) tasks with no fundamental modifications; the presentation of a prediction algorithm to be used with the trained GAN in order to predict a response for a given independent variable; using the critic to provide a confidence level of the predictions made by the generator, which could ultimately be used to help determine where more data is needed.  
In the future, the methods developed here could be applied to imaging, for example, where, when there is missing data from an image or video, we could attempt to re-construct the missing parts. Being able to reconstruct this image with specified uncertainties would be useful. In modelling, the approach could be applied in high-dimensional space (with applications across computational physics e.g. Computational Fluid Dynamics) to perform data assimilation and analyse remaining uncertainties in the modelling, see [PERSON] et al. [35]. Using the confidence level provided by the discriminator in such applications could determine where better models are needed or where coarser models (that are faster) can be used. Performing a sensitivity analysis of the discriminator could also indicate where the model is most error prone and thus where it needs to be improved.  
Conceptualisation, C.C.P., A.E.P. and K.F.C.; methodology, C.C.P., T.R.F.P. and C.E.H.; software, T.R.F.P., E.B., Q.L. and L.H.; data curation, A.E.P.; writing--original draft preparation, T.R.F.P., E.B. and C.E.H.; writing--review and editing, C.C.P., C.E.H. and K.F.C.; funding acquisition, C.C.P. and K.F.C. All authors read and agreed to the published version of the manuscript.  
The authors would like to acknowledge the following EPSRC grants: INHALE, Health assessment across biological length scales (EP/T003189/1); RELIANT, Risk EvaLutIon fAst iN-telligent Tool for COVID19 (EP/V036777/1); MUFFINS, MUltiphase Flow-induced Fluid-flexible structure InteracioN in Subsea applications (EP/P033180/1); the PREMIERE programme grant (EP/T000414/1) and MAGIC, Managing Air for Green Inner Cities (EP/N010221/1).  
Informed Consent Statement: Not applicable.  
Data Availability Statement: Not applicable.  
Acknowledgments: We would like to thank the reviewers for their comments which have improved the article.  
Conflicts of Interest: The authors declare no conflict of interest.