Implicit Neural Representations Meets Interpretable Parameterized Mesh-agnostic Stability-Preserving Reduced-Order Modeling

Learning interpretable reduced-order models of nonlinear PDE dynamics in fluid dynamics has been a long-standing problem in the data-driven modeling of dynamical systems. Early works can be traced back to Operator Inference, Sparse Identification of Nonlinear Dynamics (SINDy), and SINDy-Autoencoder, among others. However, these approaches still suffer scalability issues in scalable nonlinear dimensionality reduction. On the other hand, novel dimensionality reduction frameworks that leverage implicit neural representation, such as Neural Implicit Flow, show great promise for scalable 3D PDE data, even on dynamic meshes. Here, we propose a novel framework combining the idea of implicit neural representation with learning interpretable nonlinear dynamics from data. We compare our framework against state-of-the-art operator learning techniques (e.g., FNO) and a recent related work called DINo that leverages a vanilla feedforward neural network to learn the nonlinear latent dynamics. Furthermore, we extend our interpretable reduced-order learning framework to a parametric setting. Our testing cases range from 2D wave propagation with varying wave speeds and forced 2D Navier-Stokes equations with varying viscosity to incompressible flow over a 2D cylinder with varying Reynolds numbers.