Fault Diagnosis of Irregular Sequences by Adjoint Learning in Continuous-Time Model Space

Fault Diagnosis (FD) on sequential data suffers from irregular sampling (with missing values), limited training data, and varying underlying environments. In response, this paper proposes FD by adjoint learning in continuous-time model space. Model-Space Learning employs well-fitted models that capture data's dynamics (i.e., changing information) as more stable and concise representations of the original data. The Continuous-Time Reservoir Computing Network (CT-Res) is first introduced, which embeds Ordinary Differential Equation (ODE) within the reservoir-based hidden layer to govern continuous-time hidden-state evolution, naturally handling irregular sampling without relying on fixed time steps and effectively capturing intrinsic data dynamics. By fitting each sequence via CT-Res and representing it with the fitted model, the original sequences are mapped from the data space into the continuous-time model space. We further develop an adjoint learning strategy by incorporating a discrete-time "adjoint Echo State Network (ESN)" that shares structure and parameters with CT-Res, thus enabling efficient training by bypassing the computationally intensive ODE solver, with joint optimization of fitting accuracy and class discrimination in the model space. Experiments on multiple FD benchmarks highlight the effectiveness and efficiency of our study, particularly with missing values and scarce training data.