Self-composing neural operators for high-frequency and multiscale PDE surrogates

Abstract

Addressing the computational challenges of high-frequency and multiscale partial differential equations (PDEs), this work introduces a self-composing neural operator (SC-NO) framework. Inspired by classical fixed-point iterative solvers (e.g., multigrid, domain decomposition), the proposed architecture constructs a deep operator by repeatedly applying a single, parameter-efficient backbone block. This design mimics the update step of a numerical solver, allowing the model to progressively resolve complex solution features without increasing the parameter count. For practical training, we develop an adaptive ``Train-and-Unroll'' strategy that grows the composition depth during training, acting as a curriculum from shallow to deep self-composed models. We demonstrate the efficacy of this framework on the Helmholtz equation for ultrasound computed tomography (USCT), a problem characterized by high-frequency wave propagation in highly heterogeneous media. By instantiating the backbone with a multigrid-inspired architecture, the SC-NO effectively mitigates the spectral bias often observed in standard operator learning baselines. Numerical experiments show that our method reduces the prediction error significantly compared to Fourier Neural Operators (FNO) and their variants in the 300--500 kHz regime. Furthermore, we provide theoretical analysis linking the self-composition depth to approximation accuracy.