Calibrating and optimizing operator terms via a neural Galerkin Projection on the Shallow Water Equations

Successful reduced order models (ROMs) can enable a computationally inexpensive framework for quickly modeling partial differential equations (PDEs). With Galerkin Projection ROMs (GP-ROMs), a system of PDEs are turned into a system of ordinary differential equations (ODEs) by representation of the state as an expansion into a reduced-dimension, orthogonal basis, which is then projected back onto the PDEs. However, GP-ROMs have particular difficulty predicting long-time dynamics, typically as selection of a reduced-order basis removes important interaction of higher modes. The GP-ROMs can be calibrated, for example, by basis-weighting methods or by introducing scaling terms in the ODEs, but this can often be difficult and is highly problem specific. Alternatively, neural networks can learn calibration terms directly from the data and optimize them, as in the recent work solving compressible Navier-Stokes GP-ROMs with differentiable programming [1]. In this work, we extend these methods to the shallow water equations (SWE). We show by training and parameterization on SWE simulations, its long-time stability and accuracy dynamics are significantly improved at a low computation cost, thereby empowering rapidly deployable solutions for urgent forecasting problems.

[1] Chakrabarti, Surya, et al. "Full trajectory optimizing operator inference for reduced-order modeling using differentiable programming." arXiv preprint arXiv:2301.10804 (2023).