This work proposes a general framework for capturing noise-driven transitions in spatially extended non-equilibrium systems and explains the emergence of coherent patterns beyond the instability onset. The framework relies on stochastic parameterizations to reduce the original equations' complexity while capturing the key effects of unresolved scales. It works for both Gaussian and Levy-type noise. Our parameterizations offer two key advantages. First, they approximate stochastic invariant manifolds when the latter exist. Second, even when such manifolds break down, our formulas can be adapted by a simple optimization of its constitutive parameters. This allows us to handle scenarios with weak time-scale separation where the system has undergone multiple transitions, resulting in large-amplitude solutions not captured by invariant manifold or other time-scale separation methods. The optimized stochastic parameterizations capture how small-scale noise impacts larger scales through the system's nonlinear interactions. This effect is achieved by the very fabric of our parameterizations incorporating non-Markovian coefficients into the reduced equation. Such coefficients account for the noise's past influence using a finite memory length, selected for optimal performance. The specific "memory" function, which determines how this past influence is weighted, depends on the noise's strength and how it interacts with the system's nonlinearities. Remarkably, training our theory-guided reduced models on a single noise path effectively learns the optimal memory length for out-of-sample predictions, including rare events. This success stems from our "hybrid" approach, which combines analytical understanding with data-driven learning. This combination avoids a key limitation of purely data-driven methods: their struggle to generalize to unseen scenarios, also known as the "extrapolation problem."

翻译：本研究提出一个通用框架，用于捕捉空间扩展非平衡系统中噪声驱动的跃迁现象，并解释超越失稳阈值的相干图案形成机制。该框架基于随机参数化方法，在保留未解析尺度关键效应的同时降低原始方程的复杂度，适用于高斯噪声与莱维型噪声。我们的参数化方案具有两大优势：首先，当随机不变流形存在时，该方案可对其实现近似；其次，即使此类流形失效，通过对其本构参数进行简单优化即可调整公式形式。这使得我们能够处理弱时间尺度分离的场景——当系统经历多重跃迁时，会产生不变流形或其他时间尺度分离方法无法捕捉的大幅值解。优化后的随机参数化方案通过系统非线性相互作用，揭示了小尺度噪声如何影响大尺度动力学。这种效应源于我们参数化方案的核心架构：将非马尔可夫系数嵌入约化方程。这些系数通过有限记忆长度来量化噪声的历史影响，该长度经过优化选择以实现最佳性能。决定历史影响权重的具体"记忆"函数，取决于噪声强度及其与系统非线性特征的相互作用机制。值得注意的是，基于单条噪声路径训练我们理论引导的约化模型，即可有效学习适用于样本外预测（包括罕见事件）的最优记忆长度。这一成功源于"混合"方法——将解析理解与数据驱动学习相结合。该组合规避了纯数据驱动方法的关键局限：难以泛化至未观测场景，即所谓的"外推问题"。