Extend the sampler from invariant measures to time-dependent laws or SDE solutions

Extend the amortized neural sampler from invariant measures to the full time-dependent law $\mu_t$ of a stochastic differential equation, or to the weak solution of the stochastic differential equation.

Background

The current framework learns a coefficient-dependent transport from a reference measure to the stationary invariant measure. It does not address the transient distribution at a finite time or directly learn the weak solution process of the stochastic differential equation.

The conclusion explicitly identifies learning the full time-dependent law or weak solution as an unresolved extension.

References

Several directions remain open. First, the supervised results rely on access to training samples from target invariant measures, which may require substantial offline simulation effort, and the amortized approach is most useful when many related SDE instances must be solved and mixing is slow. Data preparation, unsupervised training, and training without samples need further investigation. Second, the current framework targets the invariant measure, and a natural extension is to learn the full law $\mu_t$ that depends on time, or the weak solution of the SDE.

— Deep operator learning for efficient sampling from invariant measures of stochastic differential equations  (2609.11376 - Guo et al., 10 Sep 2026) in Section 7, Conclusion