Stable by Construction: Variational Latent Markov Operators for Long-Horizon PDE Prediction

This paper introduces a variational approach to neural PDE solvers by developing latent Markov dynamics where physical states are represented as latent distributions and evolved through probabilistic transitions. The framework directly operates on function spaces to handle the temporal evolution of time-dependent physical systems. By constraining predictions within a stable manifold, the method mitigates distribution shift and error accumulation issues inherent in autoregressive deployment over long horizons.