Pseudo-differential integral autoencoder network for inverse PDE operators

TL;DR

This paper introduces pd-IAEnet, a neural network inspired by pseudo-differential operators, that efficiently and accurately solves inverse PDE problems with robustness to noise and discretization variations.

Contribution

The paper presents a novel pseudo-differential neural network, pd-IAEnet, that improves speed, accuracy, and robustness in inverse PDE problems compared to existing methods.

Findings

Outperforms conventional models in accuracy and speed.

Maintains robustness under measurement noise.

Exhibits discretization invariance across different meshes.

Abstract

Partial differential equations (PDEs) play a foundational role in modeling physical phenomena. This study addresses the challenging task of determining variable coefficients within PDEs from measurement data. We introduce a novel neural network, "pseudo-differential IAEnet" (pd-IAEnet), which draws inspiration from pseudo-differential operators. pd-IAEnet achieves significantly enhanced computational speed and accuracy with fewer parameters compared to conventional models. Extensive benchmark evaluations are conducted across a range of inverse problems, including Electrical Impedance Tomography (EIT), optical tomography, and seismic imaging, consistently demonstrating pd-IAEnet's superior accuracy. Notably, pd-IAEnet exhibits robustness in the presence of measurement noise, a critical characteristic for real-world applications. An exceptional feature is its discretization invariance, enabling effective training on data from diverse discretization schemes while maintaining accuracy on different meshes. In summary, pd-IAEnet offers a potent and efficient solution for addressing inverse PDE problems, contributing to improved computational efficiency, robustness, and adaptability to a wide array of data sources.