用数据驱动方法构建可泛化的随机降维模型，提升复杂系统模拟效率与不确定性量化能力。
Data-driven stochastic reduced-order modeling of parametrized dynamical systems
- 基于变分推断与重参数化技巧，联合学习隐空间动力学与概率编码器。
- 训练成本与数据量和系统刚性无关，对未见参数组合泛化能力强。
- 适合需要高效率、高鲁棒性的工程仿真与决策场景，支持物理先验融合。
在不同条件下的复杂动态系统建模计算成本高昂，常导致高保真仿真不可行。尽管降维模型（ROM）提供潜在解决方案，但现有方法通常难以处理随机动力学且无法量化预测不确定性，限制其在稳健决策中的应用。为此，我们提出一种数据驱动框架，用于学习跨参数空间和外部激励的连续时间随机降维模型。该方法基于渐近随机变分推断，利用马尔可夫高斯过程的重参数化技巧，避免训练中使用昂贵的前向求解器。这使得我们能以与数据集大小和系统刚性无关的计算成本，联合学习概率自编码器与描述隐状态动力学的随机微分方程。此外，若存在物理先验知识，本方法也支持其融入。通过三个挑战性测试问题验证，模型展现出对未见参数组合和激励的良好泛化能力，并相比现有方法实现显著效率提升。
原文摘要 · Abstract (English)
Modeling complex dynamical systems under varying conditions is computationally intensive, often rendering high-fidelity simulations intractable. Although reduced-order models (ROMs) offer a promising solution, current methods often struggle with stochastic dynamics and fail to quantify prediction uncertainty, limiting their utility in robust decision-making contexts. To address these challenges, we introduce a data-driven framework for learning continuous-time stochastic ROMs that generalize across parameter spaces and forcing conditions. Our approach, based on amortized stochastic variational inference, leverages a reparametrization trick for Markov Gaussian processes to eliminate the need for computationally expensive forward solvers during training. This enables us to jointly learn a probabilistic autoencoder and stochastic differential equations governing the latent dynamics, at a computational cost that is independent of the dataset size and system stiffness. Additionally, our approach offers the flexibility of incorporating physics-informed priors if available. Numerical studies are presented for three challenging test problems, where we demonstrate excellent generalization to unseen parameter combinations and forcings, and significant efficiency gains compared to existing approaches.
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