Here is the RHS of the SOI DiffEq model based on two different techniques. The representation in blue is from maximizing the correlation coefficient of a DiffEq solution via Mathematica (see [Comment #17](https://forum.azimuthproject.org/discussion/comment/14461/#Comment_14461) ). The red dotted profile is based on letting loose the Eureqa symbolic regression solver on the LHS constraint and unknown RHS:
The solver then combined algebraic operators and sinusoids to arrive at an analytical expression which matched the residual RHS data. This is an extremely powerful technique.
Amazing that so little guidance is required for Eureqa to find a solution that essentially maps to the hypothesized aggregate input of QBO, TSI, and CW that was fed into the Mathematica DiffEq solution.
The only RHS constraint that was provided was a *suggestion* that a conditional building block centered at 1974 (i.e Time > 1974 or Time < 1974) could be used as a "no penalty" factor in arriving at a solution. The transition between 1974 and 1980 is obvious and shows a clear shift from an approximate 22 year beat envelope to a stronger 11 year beat envelope.
This is also clearly seen in the wavelet spectrum that I described in the white paper. In the figure below, month 1200 corresponds to the year 1980. Above ~1980, a strong lower frequency beat component is seen in the scalogram.
Things are still holding together, and if anything the ENSO solution is as far from being a chaotic mess as I can imagine. Certainly, the answer is more complex than a single sinusoid, but not that unusual considering what is routinely solved in other scientific and engineering disciplines -- e.g. band structures, etc.
The solver then combined algebraic operators and sinusoids to arrive at an analytical expression which matched the residual RHS data. This is an extremely powerful technique.
Amazing that so little guidance is required for Eureqa to find a solution that essentially maps to the hypothesized aggregate input of QBO, TSI, and CW that was fed into the Mathematica DiffEq solution.
The only RHS constraint that was provided was a *suggestion* that a conditional building block centered at 1974 (i.e Time > 1974 or Time < 1974) could be used as a "no penalty" factor in arriving at a solution. The transition between 1974 and 1980 is obvious and shows a clear shift from an approximate 22 year beat envelope to a stronger 11 year beat envelope.
This is also clearly seen in the wavelet spectrum that I described in the white paper. In the figure below, month 1200 corresponds to the year 1980. Above ~1980, a strong lower frequency beat component is seen in the scalogram.
Things are still holding together, and if anything the ENSO solution is as far from being a chaotic mess as I can imagine. Certainly, the answer is more complex than a single sinusoid, but not that unusual considering what is routinely solved in other scientific and engineering disciplines -- e.g. band structures, etc.
Version 2.1.8p2
