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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 expensive. In this work, we develop a hierarchy of deep neural network time-steppers to approximate the dynamical system flow map over a range of time-scales. The model is purely data-driven, enabling accurate and efficient numerical integration and forecasting. Similar ideas can be used to couple neural network-based models with classical numerical time-steppers. Our hierarchical time-stepping scheme provides advantages over current time-stepping algorithms, including (i) capturing a range of timescales, (ii) improved accuracy in comparison with leading neural network architectures, (iii) efficiency in long-time 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 numerous 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. This article is part of the theme issue 'Data-driven prediction in dynamical systems'. [ABSTRACT FROM AUTHOR]

Multiscale is a hallmark feature of complex systems, presenting challenges for traditional numerical methods due to their reliance on local Taylor series constraints. Further, multiscale techniques often face limitations in identifying appropriate heuristic closures used to handle unresolved scales or processes within the system being modeled. In this study, we develop an efficient time-stepping strategy for multiscale problems. The proposed method's novelty lies in synergizing the hierarchical deep learning formalism with an adaptive step-size estimation process to approximate dynamical system flow maps across timescales efficiently. The model is purely data-driven and provides improved forecasting accuracy compared to fixed-step neural network time steppers. To demonstrate the effectiveness of the proposed scheme, we provide numerical simulations for several canonical nonlinear systems and source codes for their implementation. We believe the method has the potential to benefit multiscale analysis of complex systems and encourage further investigation in this area. • Identifies potential drawbacks in the current multiscale methods. • Introduces efficient time-stepping algorithm for multiscale problems using hierarchical deep learning. • Demonstrates the application of adaptive time-stepping on canonical ODEs and PDEs with noisy measurements. [ABSTRACT FROM AUTHOR]

Multiscale is a hallmark feature of complex systems, presenting challenges for traditional numerical methods due to their reliance on local Taylor series constraints. Further, multiscale techniques often face limitations in identifying appropriate heuristic closures used to handle unresolved scales or processes within the system being modeled. In this study, we develop an efficient time-stepping strategy for multiscale problems. The proposed method's novelty lies in synergizing the hierarchical deep learning formalism with an adaptive step-size estimation process to approximate dynamical system flow maps across timescales efficiently. The model is purely data-driven and provides improved forecasting accuracy compared to fixed-step neural network time steppers. To demonstrate the effectiveness of the proposed scheme, we provide numerical simulations for several canonical nonlinear systems and source codes for their implementation. We believe the method has the potential to benefit multiscale analysis of complex systems and encourage further investigation in this area. • Identifies potential drawbacks in the current multiscale methods. • Introduces efficient time-stepping algorithm for multiscale problems using hierarchical deep learning. • Demonstrates the application of adaptive time-stepping on canonical ODEs and PDEs with noisy measurements. [ABSTRACT FROM AUTHOR]