Benchmarking Long Roll-outs of Auto-regressive Neural Operators for the Compressible Navier-Stokes Equations with Conserved Quantity Correction

Deep learning has been proposed as an efficient alternative for the numerical approximation of PDE solutions, offering fast, iterative simulation of PDEs through the approximation of solution operators. However, deep learning solutions have struggle to perform well over long prediction durations due to the accumulation of auto-regressive error, which is compounded by the inability of models to conserve physical quantities. In this work, we present conserved quantity correction, a model-agnostic technique for incorporation physical conservation criteria within deep learning models. Our results demonstrate consistent improvement in the long-term stability of auto-regressive neural operator models, regardless of the model architecture. Furthermore, we analyze the performance of neural operators from the spectral domain, highlighting significant limitations of present architectures. These results highlight the need for future work to consider architectures that place specific emphasis on high frequency components, which are integral to the understanding and modeling of turbulent flows.