Multiwavelet-based Operator Learning for Differential Equations

Abstract

The solution of a partial differential equation can be obtained by computing the inverse operator map between the input and the solution space. Towards this end, we introduce a \textit{multiwavelet-based neural operator learning scheme} that compresses the associated operator's kernel using fine-grained wavelets. By explicitly embedding the inverse multiwavelet filters, we learn the projection of the kernel onto fixed multiwavelet polynomial bases. The projected kernel is trained at multiple scales derived from using repeated computation of multiwavelet transform. This allows learning the complex dependencies at various scales and results in a resolution-independent scheme. Compare to the prior works, we exploit the fundamental properties of the operator's kernel which enable numerically efficient representation. We perform experiments on the Korteweg-de Vries (KdV) equation, Burgers' equation, Darcy Flow, and Navier-Stokes equation. Compared with the existing neural operator approaches, our model shows significantly higher accuracy and achieves state-of-the-art in a range of datasets. For the time-varying equations, the proposed method exhibits a ($2X-10X$) improvement ($0.0018$ ($0.0033$) relative $L2$ error for Burgers' (KdV) equation). By learning the mappings between function spaces, the proposed method has the ability to find the solution of a high-resolution input after learning from lower-resolution data.

arXiv comment31 pages, 9 figures

References in corpus (7)

• Fourier Neural Operator for Parametric Partial Differential Equations
• Neural Operator: Graph Kernel Network for Partial Differential Equations
• A physics-informed operator regression framework for extracting data-driven continuum models
• Meta-learning Pseudo-differential Operators with Deep Neural Networks
• Learning to Optimize Multigrid PDE Solvers
• Neural Vortex Method: from Finite Lagrangian Particles to Infinite Dimensional Eulerian Dynamics
• Design of a Simple Orthogonal Multiwavelet Filter by Matrix Spectral Factorization

Cited by in corpus (11)

• Choose a Transformer: Fourier or Galerkin
• Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems
• Applications of Scientific Machine Learning for the Analysis of Functionally Graded Porous Beams
• Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers
• Learning Physical Models that Can Respect Conservation Laws
• Multi-scale Time-stepping of Partial Differential Equations with Transformers
• Neural Dynamical Operator: Continuous Spatial-Temporal Model with Gradient-Based and Derivative-Free Optimization Methods
• Local neural operator for solving transient partial differential equations on varied domains
• Deciphering and integrating invariants for neural operator learning with various physical mechanisms
• Learnable-Differentiable Finite Volume Solver for Accelerated Simulation of Flows
• Evaluation of Deep Neural Operator Models toward Ocean Forecasting