The Sun's most complex mysteries could soon be solved thanks to artificial intelligence. On August 20, IBM and NASA announced the launch of Surya, a foundation model for the sun. Having been trained on large datasets of solar activity, this AI tool aims to deepen humanity's understanding of solar weather and accurately predict solar flares--bursts of electromagnetic radiation emitted by our star that threaten both astronauts in orbit and communications infrastructure on Earth. Surya was trained with nine years of data collected by NASA's Solar Dynamics Observatory (SDO), an instrument that has orbited the sun since 2010, taking high-resolution images every 12 seconds. The SDO captures observations of the sun at various different electromagnetic wavelengths to estimate the temperature of the star's layers.

There's no way to prevent these sorts of effects, but being able to predict when a large solar flare will occur could let people work around them. However, as Louise Harra, an astrophysicist at ETH Zurich, puts it, "when it erupts is always the sticking point." Scientists can easily tell from an image of the sun if there will be a solar flare in the near future, says Harra, who did not work on Surya. But knowing the exact timing and strength of a flare is much harder, she says. That's a problem because a flare's size can make the difference between small regional radio blackouts every few weeks (which can still be disruptive) or a devastating solar superstorm that would cause satellites to fall out of orbit and electrical grids to fail.

This paper revisits classical works of Rauch (1963, et al. 1965) and develops a novel method for maximum likelihood (ML) smoothing estimation from incomplete information/data of stochastic state-space systems. Score function and conditional observed information matrices of incomplete data are introduced and their distributional identities are established. Using these identities, the ML smoother $\widehat{x}_{k\vert n}^s =\argmax_{x_k} \log f(x_k,\widehat{x}_{k+1\vert n}^s, y_{0:n}\vert\theta)$, $k\leq n-1$, is presented. The result shows that the ML smoother gives an estimate of state $x_k$ with more adherence of loglikehood having less standard errors than that of the ML state estimator $\widehat{x}_k=\argmax_{x_k} \log f(x_k,y_{0:k}\vert\theta)$, with $\widehat{x}_{n\vert n}^s=\widehat{x}_n$. Recursive estimation is given in terms of an EM-gradient-particle algorithm which extends the work of \cite{Lange} for ML smoothing estimation. The algorithm has an explicit iteration update which lacks in (\cite{Ramadan}) EM-algorithm for smoothing. A sequential Monte Carlo method is developed for valuation of the score function and observed information matrices. A recursive equation for the covariance matrix of estimation error is developed to calculate the standard errors. In the case of linear systems, the method shows that the Rauch-Tung-Striebel (RTS) smoother is a fully efficient smoothing state-estimator whose covariance matrix coincides with the Cram\'er-Rao lower bound, the inverse of expected information matrix. Furthermore, the RTS smoother coincides with the Kalman filter having less covariance matrix. Numerical studies are performed, confirming the accuracy of the main results.