Unsupervised operator learning approach for dissipative equations via Onsager principle

TL;DR

This paper introduces a novel unsupervised deep operator learning method, DOOL, based on the Onsager principle, which efficiently solves dissipative equations without labeled data and outperforms supervised methods.

Contribution

The paper presents the DOOL framework that leverages the Onsager variational principle for unsupervised training and introduces a spatiotemporal decoupling strategy to improve efficiency.

Findings

DOOL effectively solves dissipative equations with high accuracy.

It outperforms supervised DeepONet and MIONet in experiments.

The method extends to second-order wave models with dissipation.

Abstract

Existing operator learning methods rely on supervised training with high-fidelity simulation data, introducing significant computational cost. In this work, we propose the deep Onsager operator learning (DOOL) method, a novel unsupervised framework for solving dissipative equations. Rooted in the Onsager variational principle (OVP), DOOL trains a deep operator network by directly minimizing the OVP-defined Rayleighian functional, requiring no labeled data, and then proceeds in time explicitly through conservation/change laws for the solution. Another key innovation here lies in the spatiotemporal decoupling strategy: the operator's trunk network processes spatial coordinates exclusively, thereby enhancing training efficiency, while integrated external time stepping enables temporal extrapolation. Numerical experiments on typical dissipative equations validate the effectiveness of the DOOL method, and systematic comparisons with supervised DeepONet and MIONet demonstrate its enhanced performance. Extensions are made to cover the second-order wave models with dissipation that do not directly follow OVP.