用于揭示非平衡系统中转变的非马尔可夫约化模型
Non-Markovian Reduced Models to Unravel Transitions in Non-equilibrium Systems
AI总结：
本研究提出结合随机参数化与非马尔可夫系数的通用框架，适配高斯和Levy型噪声，通过理论引导与数据驱动的混合训练，可捕捉非平衡系统噪声驱动转变，解决纯数据驱动方法的外推难题。
摘要：
本研究提出了一个通用框架，用于捕捉空间扩展非平衡系统中由噪声驱动的转变，并解释不稳定性起始点之上相干模式的涌现。该框架依托随机参数化方法，在捕捉未解析尺度关键效应的同时，降低原始方程的复杂度，适用于高斯型和Levy型两类噪声。我们的参数化方法具备两大核心优势：其一，当随机不变流形存在时，可对其进行近似；其二，即便此类流形失效，仅需对其构成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时间尺度分离较弱的场景——这类场景中系统已经历多次转变，产生大振幅解，而不变流形或其他时间尺度分离方法无法捕捉这类解。优化后的随机参数化方法可捕捉小尺度噪声如何通过系统的非线性相互作用影响更大尺度，这一效应通过参数化方法的核心结构实现：将非马尔可夫系数引入约化方程。这类系数利用有限记忆长度来解释噪声的历史影响，记忆长度经选取以实现最优性能。决定历史影响权重方式的特定“记忆”函数，取决于噪声强度及其与系统非线性的相互作用方式。值得注意的是，在单条噪声路径上训练我们的理论引导型约化模型，就能有效学习到最优记忆长度，用于包括稀有事件在内的样本外预测。这一成果源于我们的“混合”方法，该方法将解析理解与数据驱动学习相结合，规避了纯数据驱动方法的一个关键局限：难以泛化到未见过的场景，也就是所谓的“外推问题”。
英文摘要：
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."
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
