Discovery of Governing Equations with Recursive Deep Neural Networks
This work addresses a bottleneck in mathematical modeling for researchers dealing with sparse experimental data, though it is incremental as it builds on existing neural network methods.
The paper tackles the problem of discovering governing equations from limited or inefficiently sampled time-series data by introducing a recursive deep neural network (RDNN). The approach improves approximation accuracy with increasing recursive stages and shows superior performance in recovering models from data with large time lags, as confirmed by numerical comparisons on dynamical systems.
Model discovery based on existing data has been one of the major focuses of mathematical modelers for decades. Despite tremendous achievements of model identification from adequate data, how to unravel the models from limited data is less resolved. In this paper, we focus on the model discovery problem when the data is not efficiently sampled in time. This is common due to limited experimental accessibility and labor/resource constraints. Specifically, we introduce a recursive deep neural network (RDNN) for data-driven model discovery. This recursive approach can retrieve the governing equation in a simple and efficient manner, and it can significantly improve the approximation accuracy by increasing the recursive stages. In particular, our proposed approach shows superior power when the existing data are sampled with a large time lag, from which the traditional approach might not be able to recover the model well. Several widely used examples of dynamical systems are used to benchmark this newly proposed recursive approach. Numerical comparisons confirm the effectiveness of this recursive neural network for model discovery.