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

2026-09-15

AI Summary

本文提出了一种基于变分潜在马尔可夫算子的方法，通过引入潜在分布和概率转移来解决长时域PDE预测中的误差累积问题。

Abstract

Neural PDE solvers provide efficient surrogates for time-dependent physical systems, but autoregressive prediction over long horizons remains challenging because local errors can induce distribution shift and accumulate under recursive deployment. We develop a variational approach to this problem by introducing latent Markov dynamics in which physical states are represented by latent distributions and evolved through probabilistic transitions. The framework is formulated directly on function spaces and specialized to functional Gaussian models, where structured latent perturbations induce a spectral geometry and variational transition alignment regularizes the learned dynamics. We further analyze how these mechanisms affect autoregressive error propagation, providing a theoretical connection between variational training and long-horizon prediction. We instantiate the framework as the Variational Autoencoding Markov Operator (VAMO), which combines spatially resolved latent fields, structured Gaussian perturbations, and a neural-operator transition. Empirically, we demonstrate the effectiveness of VAMO on several fluid-dynamics benchmarks with prediction horizons extending substantially beyond those represented during training, where it consistently reduces error accumulation and improves rollout stability over several deterministic and noise-injection baselines. Overall, these results highlight variational modeling as a complementary approach to robust long-horizon neural PDE dynamics.

Problem

• neural PDE solvers
• long-horizon prediction
• autoregressive error propagation
• latent Markov dynamics
• variational modeling

Innovation

• variational latent Markov dynamics
• functional Gaussian models
• spectral geometry
• autoregressive error propagation
• VAMO