Neural Dynamical Operator: Continuous Spatial-Temporal Model with Gradient-Based and Derivative-Free Optimization Methods
This work addresses a bottleneck in data-driven spatial-temporal modeling for engineering applications by enabling efficient use of heterogeneous data, though it is incremental as it builds on neural operator methods.
The authors tackled the problem of modeling complex dynamical systems with varying spatial and temporal data resolutions by proposing a neural dynamical operator framework that is continuous in space and time, achieving resolution-invariance and stable long-term simulations using only short-term data.
Data-driven modeling techniques have been explored in the spatial-temporal modeling of complex dynamical systems for many engineering applications. However, a systematic approach is still lacking to leverage the information from different types of data, e.g., with different spatial and temporal resolutions, and the combined use of short-term trajectories and long-term statistics. In this work, we build on the recent progress of neural operator and present a data-driven modeling framework called neural dynamical operator that is continuous in both space and time. A key feature of the neural dynamical operator is the resolution-invariance with respect to both spatial and temporal discretizations, without demanding abundant training data in different temporal resolutions. To improve the long-term performance of the calibrated model, we further propose a hybrid optimization scheme that leverages both gradient-based and derivative-free optimization methods and efficiently trains on both short-term time series and long-term statistics. We investigate the performance of the neural dynamical operator with three numerical examples, including the viscous Burgers' equation, the Navier-Stokes equations, and the Kuramoto-Sivashinsky equation. The results confirm the resolution-invariance of the proposed modeling framework and also demonstrate stable long-term simulations with only short-term time series data. In addition, we show that the proposed model can better predict long-term statistics via the hybrid optimization scheme with a combined use of short-term and long-term data.