Unsupervised operator learning approach for dissipative equations via Onsager principle
This addresses the problem of high computational costs in operator learning for researchers in computational physics and machine learning, though it is incremental as it builds on existing operator learning frameworks.
The authors tackled the computational cost of supervised operator learning for dissipative equations by proposing an unsupervised method called deep Onsager operator learning (DOOL), which eliminates the need for labeled data and shows enhanced performance in numerical experiments compared to supervised methods.
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.