This paper introduces a new way to learn and simulate systems whose future depends on their past. The authors target generalized Langevin equations, a class of models used when a few observed variables are influenced by unobserved degrees of freedom. Instead of trying to identify a memory kernel or reconstruct the hidden parts of the system, the method stores a compact memory state and uses a diffusion-based sampler to generate the next-step probability of the observed variables.

The key technical idea is a bank of exponential filters that is updated recursively. Each filter summarizes the recent history at a specific time scale. Because the filters are recursive, the memory state can be updated cheaply and its storage cost does not grow with how far back the history goes, as long as the number of filters (the bank size) stays fixed. The next-step distribution is conditioned on the current observation together with this fixed-size memory state. The authors also propose rules to pick the filter rates and bank size using predictive criteria, and they include an optional linear projection to further reduce the conditioning dimension.

To turn the conditional information into samples, the paper uses a kernel-based score estimator. This produces conditional samples without training a separate score neural network. Those samples are then used to train a neural flow-map model that can run autoregressive simulations forward in time. The full pipeline is evaluated in three numerical examples: a scalar multiscale generalized Langevin test to check long-memory retention with small conditioning dimension; a coupled vector test to study predictive compression and memory-induced coupling; and a stochastic scrape-off-layer model from plasma physics to probe non-Gaussian conditional laws and intermittent bursts. In the plasma example the method reproduces conditional asymmetry and burst statistics reported by the model.