ROMNet: Learning Partial Differential Equation Dynamics from Data Using Reduced Order Model Neural Networks

Data-driven modeling of dynamical systems is an active area of research. However, current techniques very often require extensive prior knowledge of the governing equations, or are limited to linear or first-order equations. In this work, we propose a neural net-based approach for learning the dynamics of systems described by Partial Differential Equations (PDEs), without requiring any prior knowledge of the system. Specifically, we propose a novel deep learning framework, called Reduced Order Model Network (ROMNet), which consists of three modules responsible for (i) learning a lower dimensional representation of the data, (ii) learning the dynamics and advancing the solution in the reduced latent space, and (iii) mapping the advanced solution from the latent space to the original space. We demonstrate the effectiveness of ROMNet for learning PDE dynamics on complex simulated and real-world data, showing that our model accurately learns unknown linear and nonlinear PDEs (in 2D and 3D). We compare our approach to conventional numerical schemes and find that ROMNet advances the dynamics considerably faster and more efficiently in addition to having comparable accuracy. Our results showcase the implications of deep learning models (such as ROMNet) in learning complex PDEs and the potential to significantly enhance current numerical methods for large systems, as well as to improve the analysis of systems with limited prior knowledge.