In the example they give they predict the future of a simulation. If you have a perfect simulation, with perfect observation, why not just run the simulation forwards? Well, the goal is to apply this to the real world, where you possibly have only an approximate model and observations - which are noisy and imprecise. So, instead, try predicting the future based on noisy observations. Due to exponential divergence, it seems unlikely this would work. Looking through the paper, it looks like they do not analyze the performance under noisy observation - they just analyze their ability to estimate the Lyapunov exponents under noise, which is much easier.

So the real world application (in terms of forward forecasting) seems like it's limited to cases where chaotic divergence between simulation and real world is due to simulation model errors rather than observation error - the latter is still a fundamental limit. Otherwise, this demonstrates that DNNs can be trained to solve diff eqs well, which is fairly well-trodden work.

As a result, claiming that this can enhance ex. weather prediction is highly misleading at this point, as nothing has demonstrated any prediction performance improvements under noisy observation, which is really the fundamental (and practical) limit in chaotic systems. While it still may be possible to use DNNs to do that (by ex. learning how to most accurately estimate the components of state with the largest Lyapunov constants) I don't think this work demonstrates the feasibility of that yet. The language in the paper is more reserved, in that they primarily claim that they can use the DNN to learn a dynamic model even with observation noise and then that model can be used to estimate Lyapunov constants accurately. That's much more reasonable.