Waveformer for modelling dynamical systems
N Navaneeth Souvik Chakraborty
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