This topic will gain steam in the coming years. The following paper generates quite a good cross-validation for SOI, shown in the figure below.
- Xiaoqun, C. et al. ENSO prediction based on Long Short-Term Memory (LSTM). IOP Conference Series: Materials Science and Engineering, 799, 012035 (2020).
The x-axis appears to be in months and likely starts in 1979, so it captures the 2016 El Nino (El Nino is negative for SOI). Still have no idea how the neural net arrived at the fit other than it being able to discern the cyclic behavior from the historical waveform between 1979 and 2010. From the article itself, it appears that neither do the authors.
The following are fragments that I am working on.
High-resolution (5-day) MJO model fit
Back extrapolation to historical SOI
The forcing for SOI and high-res MJO is aligned, not degrading at all regions that are outside the training interval (prior to 1980)/
The LTE modulation over the entire span (above) and just over the post 1979 MJO data interval (below)
Power spectrum of forcing shows the expected 13.66 day tropical fortnightly signal, but nearly inseparable from the next strongest 27.55 day anomalistic monthly signal. With the annual impulse mixing these show up as 3.795 year and 3.917 year periods, which thus gives rise to a long beat frequency of 121 years, which is likely the low-frequency peak.
IMPA=399, IMPB=400, SCALING=0.1, SAMPLING=365, REF_TIME=1880
GeoEnergy Math 
Very short-term forecasting skill here
Hu, S., et al (2020). Improved ENSO prediction skill resulting from reduced climate drift in IAP‐DecPreS: A comparison of full‐field and anomaly initializations. Journal of Advances in Modeling Earth Systems
https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2019MS001759
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https://www.nature.com/articles/s41586-019-1559-7
Ham, YG., Kim, JH. & Luo, JJ. Deep learning for multi-year ENSO forecasts. Nature 573, 568–572 (2019).
“During the validation period from 1984 to 2017, the all-season correlation skill of the Nino3.4 index of the CNN model is much higher than those of current state-of-the-art dynamical forecast systems. The CNN model is also better at predicting the detailed zonal distribution of sea surface temperatures, overcoming a weakness of dynamical forecast models. A heat map analysis indicates that the CNN model predicts ENSO events using physically reasonable precursors. The CNN model is thus a powerful tool for both the prediction of ENSO events and for the analysis of their associated complex mechanisms”
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