U-NO: U-shaped Neural Operators
This work addresses memory and depth limitations in neural operators for PDE learning, offering improvements in prediction accuracy for computational physics applications.
The paper tackled the problem of high memory usage and shallow architectures in neural operators for learning solution operators of partial differential equations, proposing U-NO, a U-shaped memory-enhanced architecture that achieved an average of 26% and 44% prediction improvement on Darcy's flow and turbulent Navier-Stokes equations, respectively, over state-of-the-art methods.
Neural operators generalize classical neural networks to maps between infinite-dimensional spaces, e.g., function spaces. Prior works on neural operators proposed a series of novel methods to learn such maps and demonstrated unprecedented success in learning solution operators of partial differential equations. Due to their close proximity to fully connected architectures, these models mainly suffer from high memory usage and are generally limited to shallow deep learning models. In this paper, we propose U-shaped Neural Operator (U-NO), a U-shaped memory enhanced architecture that allows for deeper neural operators. U-NOs exploit the problem structures in function predictions and demonstrate fast training, data efficiency, and robustness with respect to hyperparameters choices. We study the performance of U-NO on PDE benchmarks, namely, Darcy's flow law and the Navier-Stokes equations. We show that U-NO results in an average of 26% and 44% prediction improvement on Darcy's flow and turbulent Navier-Stokes equations, respectively, over the state of the art. On Navier-Stokes 3D spatiotemporal operator learning task, we show U-NO provides 37% improvement over the state of art methods.