A Pretrained Model for Bayesian Filtering in Heterogeneous Dynamical Systems
Ke Fan, Wen Wu, Haitao Su, Yilin Wang, Yuan Zhou, Chao Zhang
AI 速览：提出首个面向贝叶斯滤波的预训练模型 FilterFM，用单一检查点跨异构动力系统估计隐状态，并报告零样本滤波与缩放行为。
Filtering has not yet had its foundation-model moment: state estimation is still dominated either by online Bayesian filtering algorithms that require known dy- namics as inputs, or by learning-based neural filters trained from scratch for a single system or parametric family. The core challenge is structural heterogene- ity: dynamical systems differ in their governing dynamics, physical scales, and noise levels, precluding a shared representation space for a reusable filter. To bridge this gap, we introduce FilterFM, the first pretrained model for Bayesian filtering that estimates latent states across heterogeneous dynamical systems with a single checkpoint. FilterFM employs a hierarchical two-level observation se- quence encoder and a prior-to-posterior state-reconstruction module that together handle varying observation and state dimensions across heterogeneous systems, while modeling state uncertainty via quantile prediction. Trained on a mixture of heterogeneous filtering tasks, FilterFM exhibits superior in-domain performance, delivering significant gains in estimation accuracy, runtime efficiency, and noise robustness compared to existing baselines. Beyond in-domain performance, it marks the first model to successfully enable zero-shot filtering on unseen dynamics, bridging the generalization gap in state estimation. Furthermore, we uncover clear scaling behaviors in foundation-model-based Bayesian filtering, which combined with comprehensive component ablation studies, validate the structural optimality of our design and broaden the generalization scope for this research field.
Jia-Nan Wang, Zixun Huang, Kairui Li, Zilin Wang, Lei Wu
We study when and how momentum improves large-batch training in the one-pass regime, using power-law kernel regression as a tractable setting. We first characterize stochastic stability through the \emph{critical learning rate}, defined as the largest learning rate for stable training, and obtain $η_{\mathrm{SGD}}^{\mathrm{crit}}\eqsim 1$, $η_{\mathrm{Polyak}}^{\mathrm{crit}}\eqsim \min\{1,B(1-β)\}$, and $η_{\mathrm{Nesterov}}^{\mathrm{crit}}\eqsim \min\{1,B^q(1-β)\}$, where $B$ is the batch size, $β\in[0,1)$ is the momentum factor, and \(q>1\) is the capacity exponent governing spectral decay. Within the stable region, we derive scaling laws for the full risk dynamics, capturing the progression from an early transient through power-law decay to a noise floor. We then optimize the final-step risk under a fixed data budget, obtaining a three-regime phase diagram that reveals how the benefits of momentum change with batch size. Notably, Polyak enlarges the critical batch size, the largest batch size preserving the optimal small-batch data-scaling exponent, thereby allowing fewer sequential updates without sacrificing data efficiency. In contrast, Nesterov improves the optimal data-scaling exponent in the large-batch regime through look-ahead-induced suppression of noise amplification. Experiments on PLK regression quantitatively support the theoretical scaling predictions, while neural-network experiments provide qualitative evidence for the predicted batch-size dependence and stability behavior.