Using neural networks with jumps in stock returns
In standard models that try to characterise jumps, such as the Jump Diffusion model from Mertons (a short introduction), the model consists of two main parts:


Brownian motion; a random walk to account for the random path, perhaps with some drift - when the values head upwards or downwards in a consisten manner. And
An additive Possion process, which with some probability add a jump in a time-step, with a given probability.


Drawing from this approach, you could also consider using two models that work independently (or separately) to model the overall market and trend, along with a model that introduces jumps at certain timesteps and itnervals.

You could try using different input data for the second model, such as signals taken from text, such as news feeds or newpapers that discuss current market dynamics/politics, possible decisions from SEC and the like. The first (stable) part could be modelled by your current neural network.



This is just a high level idea, and I haven't actually seen any research that already tried it, so unfortunately cannot provide any links to literature.