Hierarchical deep learning of multiscale differential equation time-steppers

Area of Science:

• Computational Physics
• Applied Mathematics
• Machine Learning

Background:

• Nonlinear differential equations often lack closed-form solutions, necessitating numerical methods.
• Multiscale physics systems present computational challenges due to vast timescale dynamics.
• Existing numerical integration methods can be computationally expensive for long-term forecasting.

Purpose of the Study:

• To develop a hierarchy of deep neural network (DNN) time-steppers for approximating dynamical system flow maps.
• To enable accurate and efficient numerical integration and forecasting using a data-driven approach.
• To address the computational expense of simulating systems with multiscale physics.

Main Methods:

• A purely data-driven, hierarchical deep neural network time-stepping scheme was developed.
• The method approximates the dynamical system flow map across a range of timescales.
• The framework is designed to be parallelizable and integrable with classical numerical methods.

Main Results:

• The hierarchical DNN time-steppers demonstrated accurate and efficient numerical integration.
• The approach successfully captured dynamics across a range of timescales.
• The method showed improved accuracy compared to leading neural network architectures and state-of-the-art sequence generation models.
• Demonstrated effectiveness on diverse nonlinear systems (e.g., Van der Pol oscillator, Lorenz system) and signal processing tasks.

Conclusions:

• The proposed hierarchical time-stepping scheme offers significant advantages for numerical integration of nonlinear dynamical systems.
• The data-driven DNN approach provides an efficient and accurate alternative for multiscale physics simulations and long-time forecasting.
• The flexible framework can be coupled with traditional numerical methods, enhancing predictive capabilities in various scientific domains.