Data-assisted reduced-order modeling of extreme events in complex dynamical systems

The prediction of extreme events, from avalanches and droughts to tsunamis and epidemics, depends on the formulation and analysis of relevant, complex dynamical systems. Such dynamical systems are characterized by high intrinsic dimensionality with extreme events having the form of rare transitions that are several standard deviations away from the mean. Such systems are not amenable to classical order-reduction methods through projection of the governing equations due to the large intrinsic dimensionality of the underlying attractor as well as the complexity of the transient events. Alternatively, data-driven techniques aim to quantify the dynamics of specific, critical modes by utilizing data-streams and by expanding the dimensionality of the reduced-order model using delayed coordinates. In turn, these methods have major limitations in regions of the phase space with sparse data, which is the case for extreme events. In this work, we develop a novel hybrid framework that complements an imperfect reduced order model, with data-streams that are integrated though a recurrent neural network (RNN) architecture. The reduced order model has the form of projected equations into a low-dimensional subspace that still contains important dynamical information about the system and it is expanded by a long short-term memory (LSTM) regularization. The LSTM-RNN is trained by analyzing the mismatch between the imperfect model and the data-streams, projected to the reduced-order space. The data-driven model assists the imperfect model in regions where data is available, while for locations where data is sparse the imperfect model still provides a baseline for the prediction of the system state. We assess the developed framework on two challenging prototype systems exhibiting extreme events. We show that the blended approach has improved performance compared with methods that use either data streams or the imperfect model alone. Notably the improvement is more significant in regions associated with extreme events, where data is sparse.

从雪崩、干旱到海啸、流行病等极端事件的预测，有赖于对相关复杂动力系统的建模与分析。此类动力系统的内在维度极高，极端事件表现为偏离均值数个标准差的稀有跃迁。由于底层吸引子的内在维度极高，且瞬态事件复杂度较高，此类系统无法通过对控制方程进行投影来应用经典降阶方法。与之相对，数据驱动技术旨在利用数据流，通过延迟坐标扩展降阶模型的维度，以量化特定关键模态的动力学特性。然而这类方法在相空间中数据稀疏的区域存在显著局限，而极端事件的发生区域恰好属于此类场景。本研究提出了一种新颖的混合框架：该框架将不完善的降阶模型与通过循环神经网络（Recurrent Neural Network, RNN）架构集成的数据流相结合。降阶模型采用投影至低维子空间的投影方程形式，该子空间仍保留系统的关键动力学信息，并通过长短期记忆（Long Short-Term Memory, LSTM）正则化进行扩展。LSTM循环神经网络通过分析不完善模型与投影至降阶空间的数据流之间的偏差完成训练。数据驱动模型可在数据充足的区域辅助不完善模型进行预测，而在数据稀疏的位置，不完善模型仍可作为系统状态预测的基准。我们在两个具备极端事件特性的挑战性原型系统上对所提框架进行了评估。实验结果表明，相较于仅使用数据流或仅使用不完善模型的方法，该混合方法的预测性能更优。尤为关键的是，在数据稀疏的极端事件相关区域，性能提升更为显著。