Data Driven Governing Equations Approximation Using Deep Neural Networks (1811.05537v1)

Abstract: We present a numerical framework for approximating unknown governing equations using observation data and deep neural networks (DNN). In particular, we propose to use residual network (ResNet) as the basic building block for equation approximation. We demonstrate that the ResNet block can be considered as a one-step method that is exact in temporal integration. We then present two multi-step methods, recurrent ResNet (RT-ResNet) method and recursive ReNet (RS-ResNet) method. The RT-ResNet is a multi-step method on uniform time steps, whereas the RS-ResNet is an adaptive multi-step method using variable time steps. All three methods presented here are based on integral form of the underlying dynamical system. As a result, they do not require time derivative data for equation recovery and can cope with relatively coarsely distributed trajectory data. Several numerical examples are presented to demonstrate the performance of the methods.

• The paper introduces a novel DNN framework that uses ResNet blocks to recover governing equations without direct derivative data.
• It employs recurrent and recursive ResNet techniques to capture sequential dynamics and adapt variable time steps in complex systems.
• Numerical experiments demonstrate significant error reduction in equation approximation, underscoring the method's potential in diverse scientific fields.

Summary of "Data Driven Governing Equations Approximation Using Deep Neural Networks"

In the paper titled "Data Driven Governing Equations Approximation Using Deep Neural Networks," the authors Tong Qin, Kailiang Wu, and Dongbin Xiu introduce a computational framework designed to approximate unknown system governing equations through the utilization of deep neural networks (DNNs). This work reflects significant advancement in the field of data-driven modeling, aiming to provide a robust solution for systems where governing equations are not explicitly known.

Objective and Methodology

The primary objective of this research is to employ deep learning, specifically the residual network (ResNet) architecture, as a fundamental component in approximating governing equations from observational data. The authors exploit the ResNet architecture by drawing an analogy between its structure and numerical integration methods, elucidating that each ResNet block can be interpreted as a one-step exact temporal integration procedure.

The framework expands upon traditional ResNet blocks by introducing two innovative multi-step methods:

1. Recurrent ResNet (RT-ResNet): This method involves a uniform time-stepping approach. By leveraging recurrence, the RT-ResNet allows for the capturing of sequential dependencies over time, thereby enhancing the approximation accuracy for time-evolving systems.
2. Recursive ResNet (RS-ResNet): As an adaptive multi-step technique, the RS-ResNet employs variable time steps to accommodate changes in the underlying system dynamics, offering greater flexibility and precision in dealing with non-uniform temporal datasets.

Both approaches are formulated on the integral form of dynamical systems, circumventing the need for direct time-derivative data, and enabling the model to operate effectively even with sparsely distributed trajectory data.

Numerical Results and Performance

The authors present several numerical examples to demonstrate the efficacy of the proposed methods. These examples illustrate the capability of the framework to recover governing equations accurately even with limited or coarsely-sampled data, underscoring the potential of this approach in practical applications. Quantitative results from these experiments show a significant reduction in approximation error compared to more conventional methods, validating the theoretical models proposed.

Implications and Future Directions

This research provides a promising direction for the application of DNNs in uncovering fundamental physical equations from empirical data, particularly in instances where conventional methods fall short due to the lack of derivative information or coarse data resolution. The blend of ResNet's structural efficiency with adaptive time-stepping approaches like RS-ResNet presents a versatile toolkit for researchers and practitioners in fields like climate modeling, biological systems, and engineering, where system behaviors are often complex and not well-understood.

Future advancements could include the extension of the proposed framework to handle multi-dimensional systems and the incorporation of more sophisticated machine learning models that can capture interactions within high-dimensional datasets. Further exploration of hybrid models integrating physics-based constraints with data-driven methodologies might also yield improved interpretability and accuracy.

In conclusion, this paper lays a solid groundwork for future exploration in the integration of deep learning techniques with the discovery and approximation of governing equations, providing valuable insights and tools for further advancements in numerical modeling and data analysis.