Learning Dynamical Coupled Operator for High-Dimensional Black-Box Partial Differential Equations

Abstract

The deep operator networks (DON), a class of neural operators that learn mappings between function spaces, have recently emerged as surrogate models for parametric partial differential equations (PDEs). However, their full potential for accurately approximating general black-box PDEs remains underexplored due to challenges in training stability and performance, primarily arising from difficulties in learning mappings between low-dimensional inputs and high-dimensional outputs. Furthermore, inadequate encoding of input functions and query positions limits the generalization ability of DONs. To address these challenges, we propose the Dynamical Coupled Operator (DCO), which incorporates temporal dynamics to learn coupled functions, reducing information loss and improving training robustness. Additionally, we introduce an adaptive spectral input function encoder based on empirical mode decomposition to enhance input function representation, as well as a hybrid location encoder to improve query location encoding. We provide theoretical guarantees on the universal expressiveness of DCO, ensuring its applicability to a wide range of PDE problems. Extensive experiments on real-world, high-dimensional PDE datasets demonstrate that DCO significantly outperforms DONs.