Hierarchical Deep Learning of Multiscale Differential Equation Time-Steppers (2008.09768v1)

Abstract: 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.

• The paper presents a hierarchical deep neural network architecture that advances multiscale differential equation time-stepping by efficiently approximating flow maps.
• The method improves long-term simulation fidelity and computational efficiency by bypassing numerical stiffness typical in conventional time-stepping schemes.
• The approach integrates with classical numerical methods, offering a promising hybrid framework for complex system simulations in various scientific domains.

Hierarchical Deep Learning of Multiscale Differential Equation Time-Steppers

The paper introduces a novel computational framework that leverages deep learning to address challenges in solving nonlinear differential equations, especially those characterized by multiscale dynamics. Specifically, it proposes a hierarchical deep neural network (DNN) architecture to develop time-stepper solutions that can efficiently approximate the flow map of dynamical systems across various time scales. The goal is to overcome issues of numerical stiffness, which contribute to the computational cost of integrating systems with disparate time scales.

Key Contributions

1. Hierarchical Time-Stepping Scheme: The paper proposes a multiscale time-stepping approach utilizing a hierarchy of DNN models. These models are designed to efficiently capture dynamics across different temporal resolutions, thereby improving computational speed and accuracy over conventional time-stepping methods.
2. Improved Accuracy and Efficiency: By avoiding numerical stiffness and leveraging multiscale features, the hierarchical time-steppers achieve better accuracy than conventional DNN architectures such as LSTM, reservoir computing, and clockwork RNN. The method enables long-term simulations and forecasting with higher fidelity.
3. Integration with Classical Methods: The methodology demonstrates potential for integration with traditional numerical algorithms, suggesting a hybrid computational framework that can exploit the benefits of both deep learning and classical numerical schemes.

Methodology

The proposed approach involves constructing a series of deep residual networks trained to learn flow maps for distinct time scales. This formulation addresses the limitations of local Taylor series expansions generally employed in traditional time-stepping methods like Runge-Kutta. Neural networks, unrestricted by these local constraints, model the flow map directly, allowing for larger step sizes and improved long-term prediction fidelity.

Models across time scales are initially trained individually and then hierarchically coupled. This coupling ensures that each model focuses accurately on its optimal temporal range, simplifying the training process and avoiding issues such as exploding/vanishing gradients typically encountered in recurrent neural network training.

Experimental Evaluation

The paper evaluates the effectiveness of the proposed multiscale hierarchical time-stepping scheme across several nonlinear dynamical systems including the Van der Pol oscillator, the Lorenz system, and the Kuramoto–Sivashinsky equation. Results consistently show that the proposed method outperforms single-scale neural network time-steppers and achieves superior accuracy in comparison to existing state-of-the-art sequence generation and forecasting models.

The work also presents an intriguing hybrid time-stepping approach where neural networks execute large-time steps more efficiently, allowing classical numerical methods to handle small-time steps. This integration suggests potential performance improvements in overall computational simulations.

Implications and Future Directions

The hierarchical deep learning approach for multiscale time-stepping offers significant practical and theoretical implications. On the practical side, it can be applied to complex system simulations, such as fluid dynamics and weather modeling, where multiscale interactions are prevalent. Theoretically, the methodology suggests new avenues for combining data-driven modeling with mathematical physics, potentially leading to advances in adaptive algorithms that adjust time-step sizes based on learned dynamics.

Future research may explore the integration of adaptive time-stepping strategies to further refine the framework. Additionally, applying this multiscale approach to high-dimensional systems could offer deeper insights into the optimization and generalization capacities of deep networks in scientific computing contexts.

In summary, the paper advances the field of scientific computing by demonstrating how deep learning technologies can be harnessed to solve complex, multiscale differential equations with a high degree of accuracy and efficiency, paving the way for their broader application in real-world scenarios.