Parameterized Neural Ordinary Differential Equations: Applications to Computational Physics Problems

Such examples include predicting input/output responses, design, and optimization [55]. These ODEs and their solutions often depend on a set of input parameters, and such ODEs are denoted as parameterized ODEs. Examples of such input parameters within the context of fluid dynamics include Reynolds number and Mach number. In many important scenarios, high-fidelity solutions of parameterized ODEs are required to be computed i) for many different input parameter instances (i.e., many-query scenario) or ii) in real time on a new input parameter instance. A single run of a high-fidelity simulation, however, often requires fine spatiotemporal resolutions. Consequently, performing real-time or multiple runs of a high-fidelity simulation can be computationally prohibitive. To mitigate this computational burden, many model-order reduction approaches have been proposed to replace costly high-fidelity simulations. The common goal of these approaches is to build a reduced-dynamical model with lower complexity than that of the high-fidelity model, and to use the reduced model to compute approximate solutions for any new input parameter instance. In general, model-order reduction approaches consist of two components: i) a low-dimensional latent-dynamics model, where the computational complexity is very low, and ii) a (non)linear mapping that constructs high-dimensional approximate states (i.e., solutions) from the low-dimensional states obtained from the latent-dynamics model.