## 5 Conclusions  
FSs are well-known machine learning models, but have difficulty in dealing with high-dimensional data. This paper proposed the DNFS to enable FSs to effectively solve high-dimensional regression problems on the basis of ensuring accuracy and interpretability. Inspired by deep learning, the SC-ANFIS was proposed and adopted as a submodule to construct the structure of the DNFS in a bottom-up way. This paper also performed an in-depth study on the deep structure and the combination of submodule inputs for the DNFS to improve the performance of the DNFS.  
The experimental results on five real-world regression datasets indicated that:  
1. The DNFS had higher prediction accuracy. Compared with the well-known models: MBGD-RDA, RBF, GRNN, and LSTM, the average values of the DNFS on MAE, RMSE, and SMAPE decreased by 38.9%, 33.7%, and 40.0%, respectively;
2. The DNFS had less complexity and better interpretability. The number of parameters of the DNFS was far less than the other algorithms, especially when the dimensionality of the input was very high. It also can be concluded that the structure of the DNFS is simpler and more interpretable;
3. The DNFS can solve high-dimensional regression problems in a more reasonable time than the previous FSs, while ensuring excellent performance;
4. The DNFS with five submodule inputs and two shared inputs had the best comprehensive performance. Additionally, dispersing the features that had a high correla  
Figure 11: Performance of the DNFS on the five datasets for the training and testing. (**a**) The test MAE comparison. (**b**) The test RMSE comparison. (**c**) The test SMAPE comparison.  
tion with the output into each submodule, better improvement of the DNFS could be achieved.  
Additionally, this paper proposes future research directions: On the one hand, how to reduce the time consumption under the same structure will be further considered. Removing the submodules with poor performance or more fuzzy rules is the current preliminary idea, which is likely to reduce the computational cost and improve the accuracy. On the other hand, how to determine the optimal input combination for each submodule is also another future direction.  
Conceptualization, D.C. and J.C.; methodology, J.C.; validation, formal analysis, J.C. and Y.H.; investigation, resources, data curation, writing--original draft preparation, D.C. and J.C.; writing--review and editing, J.C., Y.H. and Y.L.; supervision, D.C. and Y.L.; project administration, funding acquisition, D.C. All authors read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China under Grant 61976055, the special fund for education and scientific research of Fujian Provincial Department of Finance under Grant GY-Z21001 and the open project of State Key Laboratory of Management and Control for Complex Systems under Grant 20210116.  
Not applicable.  
The data are contained within the article.  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} ANFIS & Adaptive neuro-fuzzy inference system \\ SC & Subtractive clustering \\ SC-ANFIS & Subtractive clustering-based ANFIS \\ DNFS & Deep neural fuzzy system \\ FSs & Fuzzy systems \\ EAs & Evolutionary algorithms \\ GD & Gradient descent \\ LSE & Least squares estimation \\ PCA & Principal component analysis \\ MFs & Membership functions \\ MAE & Mean absolute error \\ RMSE & Root mean squared error \\ SMAPE & Symmetric mean absolute percentage error \\ AIC & Akaike information criterion \\ RBF & Radial basis function \\ GRNN & Generalized regression neural network \\ LSTM & Long-short term memory \\ \end{tabular}