构造即稳定：用于长时程偏微分方程预测的变分潜在马尔可夫算子
Stable by Construction: Variational Latent Markov Operators for Long-Horizon PDE Prediction
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- Duke University（杜克大学）
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中文总结 AI 辅助
本文提出变分潜在马尔可夫算子（VAMO），通过潜在分布与概率转移建模物理状态，减少长时程自回归预测中的误差累积，并在流体动力学基准上验证了其稳定性和有效性。
中文摘要 AI 辅助
神经偏微分方程求解器为时变物理系统提供了高效的替代模型，但在长时程上的自回归预测仍然具有挑战性，因为局部误差可能导致分布偏移并在递归部署下累积。我们通过引入潜在马尔可夫动力学来发展一种变分方法，其中物理状态由潜在分布表示，并通过概率转移进行演化。该框架直接在函数空间上构建，并专门针对函数高斯模型，其中结构化潜在扰动引发谱几何，变分转移对齐正则化学习到的动力学。我们进一步分析了这些机制如何影响自回归误差传播，为变分训练与长时程预测之间提供了理论联系。我们将该框架实例化为变分自编码马尔可夫算子（VAMO），它结合了空间分辨的潜在场、结构化高斯扰动和神经算子转移。在实验上，我们在多个流体动力学基准上展示了VAMO的有效性，其预测时程远超训练时所见，并且相较于多个确定性和噪声注入基线，它持续减少了误差累积并提高了滚动稳定性。总体而言，这些结果凸显了变分建模作为稳健长时程神经偏微分方程动力学的补充方法。
英文摘要
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.
