Nonlinear differential equations rarely admit closed-form solutions, thus requiring numerical time-stepping algorithms to approximate solutions. Further, many systems characterized by multiscale physics exhibit dynamics over a vast range of timescales, making numerical integration computationally expensive due to numerical stiffness. In this work, we develop a hierarchy of deep neural network time-steppers to approximate the flow map of the dynamical system over a disparate range of time-scales. The resulting model is purely data-driven and leverages features of the multiscale dynamics, enabling numerical integration and forecasting that is both accurate and highly efficient. Moreover, similar ideas can be used to couple neural network-based models with classical numerical time-steppers. Our multiscale hierarchical time-stepping scheme provides important advantages over current time-stepping algorithms, including (i) circumventing numerical stiffness due to disparate time-scales, (ii) improved accuracy in comparison with leading neural-network architectures, (iii) efficiency in long-time simulation/forecasting due to explicit training of slow time-scale dynamics, and (iv) a flexible framework that is parallelizable and may be integrated with standard numerical time-stepping algorithms. The method is demonstrated on a wide range of nonlinear dynamical systems, including the Van der Pol oscillator, the Lorenz system, the Kuramoto-Sivashinsky equation, and fluid flow pass a cylinder; audio and video signals are also explored. On the sequence generation examples, we benchmark our algorithm against state-of-the-art methods, such as LSTM, reservoir computing, and clockwork RNN. Despite the structural simplicity of our method, it outperforms competing methods on numerical integration.
Hierarchical Multiscale Structure-Function Coupling for Brain Connectome Integration
School of Medicine, University of Dundee, UK（邓迪大学医学院）；School of Science and Engineering, University of Dundee, UK（邓迪大学科学与工程学院）；College of Medicine and Biological Information Engineering, Northeastern University, China（东北大学医学院与生物信息工程学院）；College of Medical Informatics, Chongqing Medical University, China（重庆医科大学医学信息学院）；School of Airspace Science and Engineering, Shandong University, China（山东大学航空航天科学与工程学院）；Chinese Institute for Brain Research, China（中国脑科学研究院）；Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK（剑桥大学应用数学与理论物理系）；Department of Clinical Neurosciences, University of Cambridge, UK（剑桥大学临床神经科学系）
MCbiF: Measuring Topological Autocorrelation in Multiscale Clusterings via 2-Parameter Persistent Homology
Department of Mathematics, Imperial College London, UK（英国伦敦帝国理工学院数学系）；Department of Mathematical Modeling and Machine Learning, University of Zurich, Switzerland（瑞士苏黎世大学数学建模与机器学习系）
Image Processing Group – Signal Theory and Communications Department（图像处理组——信号理论与通信系）；Inorganic and Organic Chemistry Department and Institute of Theoretical and Computational Chemistry（无机和有机化学系及理论与计算化学研究所）；Department of Physics and Materials Science（物理与材料科学系）
A Generalized Framework for Multiscale State-Space Modeling with Nested Nonlinear Dynamics: An Application to Bayesian Learning under Switching Regimes
School of Complex Adaptive Systems（复杂适应系统学院）；Arizona State University（亚利桑那州立大学）
Prediction of Multiscale Features Using Deep Learning-based Preconditioner-Solver Architecture for Darcy Equation in High-Contrast Media
School of Mathematics and Physics, Xi'an Jiaotong-Liverpool University（数学与物理学院，西安交通大学利物浦大学）；Department of Mathematical Sciences, University of Liverpool（数学科学系，利物浦大学）；School of Mathematics and Information Science, Yantai University（数学与信息科学学院，烟台大学）
Bayesian multiscale deep generative model for the solution of high-dimensional inverse problems
ShanghaiTech University（上海科技大学）；University of Notre Dame（圣母大学）；Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences（中国科学院上海微系统与信息技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）
