Physics-Anchored Dual Network for Conditional SOFC Health-Indicator Trajectory Forecasting Under Variable-Load Operation

Reliable health-indicator forecasting under variable-load operation is important for SOFC durability management, maintenance scheduling, and system control. This study investigates offline conditional forecasting using an equivalent ohmic-resistance health indicator. The indicator is extracted from experimental data through reduced-order model inversion and subsequently processed using causal smoothing. In the test horizon, the future operating-condition profile (t,U,T) is supplied as the conditioning input, whereas future health-indicator labels are hidden during inference and used only for post-test evaluation. The measured current I(t) is used only for label extraction and is not supplied as a future covariate. We propose a Physics-Anchored Dual-Network (PADN) framework that couples a surrogate network and a dynamic network through an anchored dynamic relation built around a calibrated empirical degradation skeleton. The prediction target serves as a macroscopic health indicator rather than as a direct microscopic degradation measurement, so the task is not formulated as RUL prediction or unknown-load forecasting. PADN is evaluated on a public variable-load aging dataset containing two SOFC single cells of about 1700 h and is compared with EXP-only, Direct-MLP, CNN-LSTM, and Transformer baselines using chronological split points at 50%, 63%, and 78% of the recorded timeline; data before each split point are used for model development, including parameter updates and validation, whereas data after it are held out for testing. Across the six cell–split-point combinations, PADN achieves the lowest MAPE in five combinations and the lowest average errors at each split point among the compared methods. In the most data-limited scenario, with the chronological split point at 50% of the recorded timeline, PADN maintains MAPE values of 2.645% and 2.934% for Cell-1 and Cell-2, respectively. Ablation results show that the empirical degradation skeleton, physics-consistency loss, and residual regularization each contribute to improved extrapolation stability under the present two-cell conditional forecasting benchmark. Overall, PADN provides a practical gray-box approach for conditional forecasting of extracted and smoothed SOFC health-indicator trajectories under limited-data variable-load settings, although the current evidence remains limited to two cells from one experimental platform.