Waveformer for modelling dynamical systems

TL;DR

Waveformer is a novel neural operator that combines wavelet transforms and transformers to accurately model long-term dynamics of complex PDEs, outperforming existing methods especially in extrapolation tasks.

Contribution

It introduces wavelet-based multi-scale analysis combined with transformer architecture for improved long-term prediction of dynamical systems.

Findings

Outperforms existing neural operators by up to an order of magnitude.

Effectively captures multi-scale spatial behavior and long-term dynamics.

Demonstrates high accuracy in solving complex PDEs like Navier-Stokes.

Abstract

Neural operators have gained recognition as potent tools for learning solutions of a family of partial differential equations. The state-of-the-art neural operators excel at approximating the functional relationship between input functions and the solution space, potentially reducing computational costs and enabling real-time applications. However, they often fall short when tackling time-dependent problems, particularly in delivering accurate long-term predictions. In this work, we propose "waveformer", a novel operator learning approach for learning solutions of dynamical systems. The proposed waveformer exploits wavelet transform to capture the spatial multi-scale behavior of the solution field and transformers for capturing the long horizon dynamics. We present four numerical examples involving Burgers's equation, KS-equation, Allen Cahn equation, and Navier Stokes equation to illustrate the efficacy of the proposed approach. Results obtained indicate the capability of the proposed waveformer in learning the solution operator and show that the proposed Waveformer can learn the solution operator with high accuracy, outperforming existing state-of-the-art operator learning algorithms by up to an order, with its advantage particularly visible in the extrapolation region