Description:
- Physical modeling, including aerospace and complex fluid flow
- Biological modeling, including bacterial dynamics and neuron potential
- Epidemiology
Abstract
USC researchers introduce a novel method to solve partial differential equations (PDEs) using a multiwavelet-based neural operator learning scheme. By compressing the operator's kernel using fine-grained wavelets and learning the projection of the kernel onto fixed multiwavelet polynomial bases, our model achieves state-of-the-art accuracy compared to existing neural operator approaches. The method’s efficacy has been demonstrated on various PDEs, including Korteweg-de Vries, Burgers', Darcy Flow, and Navier-Stokes equations, with (2X-10X) improvement in relative L2 error.
Benefits
- Allows for learning complex dependencies at multiple scales and results in a resolution-independent scheme
- Improves accuracy and parallels state-of-the-art in a range of datasets
Market Application
There is a growing need for partial differential equations (PDEs) in various fields, such as aerospace and complex fluids. Deep neural networks (NNs) have been applied to learning PDEs from trajectories of variables. However, the challenge lies in generalizing NNs to other resolutions, coefficients, and structures, and obtaining sufficient data. The "Neural Operators" approach has shown promise in learning operator maps without prior knowledge of the underlying PDEs, but data scarcity remains an issue. Exploring fundamental properties of the operators could lead to more efficient data representation and better solutions.
Publications
Gupta, Gaurav, Xiongye Xiao, and Paul Bogdan. "Multiwavelet-based operator learning for differential equations." Proceedings of the 35th International Conference on Neural Information Processing Systems. 2021.
Stage of Development
- Method developed
- Available for licensing