cs.LGApr 9, 2026

A Machine Learning Framework for Turbofan Health Estimation via Inverse Problem Formulation

Authors: Milad Leyli-Abadi, Lucas Thil, Sebastien Razakarivony, Guillaume Doquet, Jesse Read

Organizations: Institut de recherche technologique SystemX, Palaiseau, France · École Polytechnique, LIX, Palaiseau, France · Safran Tech, Magny-les-Hameaux, France

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Abstract

Estimating the health state of turbofan engines is a challenging ill-posed inverse problem, hindered by sparse sensing and complex nonlinear thermodynamics. Research in this area remains fragmented, with comparisons limited by the use of unrealistic datasets and insufficient exploration of the exploitation of temporal information. This work investigates how to recover component-level health indicators from operational sensor data under realistic degradation and maintenance patterns. To support this study, we introduce a new dataset that incorporates industry-oriented complexities such as maintenance events and usage changes. Using this dataset, we establish an initial benchmark that compares steady-state and nonstationary data-driven models, and Bayesian filters, classic families of methods used to solve this problem. In addition to this benchmark, we introduce self-supervised learning (SSL) approaches that learn latent representations without access to true health labels, a scenario reflective of real-world operational constraints. By comparing the downstream estimation performance of these unsupervised representations against the direct prediction baselines, we establish a practical lower bound on the difficulty of solving this inverse problem. Our results reveal that traditional filters remain strong baselines, while SSL methods reveal the intrinsic complexity of health estimation and highlight the need for more advanced and interpretable inference strategies. For reproducibility, both the generated dataset and the implementation used in this work are made accessible.

Figures & tables

Figure 1: Turbofan engine; with the courtesy of the team of OpenDeckSMR.
Figure 2: Generated degradation trajectories for 10 health indicators and 1 engine
Figure 3: Simulated sensor values (measurements) for the trajectory in Figure 2 and corresponding to Cruise operational condition.
Figure 4: Sensor variables (measurements) distribution grouped by 4 flight phases.
Figure 5: Two model categories considering stationary and non-stationary hypotheses. Based on steady-state hypothesis, the model finds a mapping between inputs and outputs. Based on non-stationary hypothesis, the model takes into account the temporal dynamics over time for prediction of outputs.
Figure 6: SSL Approaches. a. Autoencoder: trained with reconstruction loss on yty_{t} to learn latent ztz_{t} . b. JEPA: predicts masked patches in latent space to learn ztz_{t} . c. State Decoding: predicts xtx_{t} from frozen ztz_{t} to evaluate representations.
ModelMetricHI1HI2HI3HI4HI5HI6HI7HI8HI9HI10Avg
Steady-state
GBSMAPE0.24 ±\pm 0.210.19 ±\pm 0.200.30 ±\pm 0.260.05 ±\pm 0.130.09 ±\pm 0.140.18 ±\pm 0.210.19 ±\pm 0.190.16 ±\pm 0.180.24 ±\pm 0.220.33 ±\pm 0.280.20 ±\pm 0.09
RMSE2.0 ±\pm 0.101.0 ±\pm 0.092.0 ±\pm 0.080.3 ±\pm 0.021.0 ±\pm 0.032.0 ±\pm 0.101.0 ±\pm 0.071.0 ±\pm 0.052.0 ±\pm 0.033.0 ±\pm 0.101.53 ±\pm 0.79
P.Corr.0.93 ±\pm 0.000.96 ±\pm 0.000.88 ±\pm 0.011.00 ±\pm 0.000.99 ±\pm 0.000.97 ±\pm 0.000.96 ±\pm 0.000.97 ±\pm 0.000.93 ±\pm 0.000.87 ±\pm 0.010.95 ±\pm 0.04
MLPSMAPE0.20 ±\pm 0.250.15 ±\pm 0.230.28 ±\pm 0.250.05 ±\pm 0.150.07 ±\pm 0.160.12 ±\pm 0.210.17 ±\pm 0.220.12 ±\pm 0.210.22 ±\pm 0.240.30 ±\pm 0.300.17 ±\pm 0.24
RMSE1.24 ±\pm 0.050.86 ±\pm 0.032.07 ±\pm 0.080.19 ±\pm 0.010.62 ±\pm 0.031.24 ±\pm 0.051.11 ±\pm 0.060.53 ±\pm 0.041.59 ±\pm 0.042.19 ±\pm 0.111.16 ±\pm 0.62
Table 1: Prediction performance for 10 health indicators across model families. Results are reported as mean ±\pm standard deviation over 5-fold cross-validation using SMAPE, RMSE ( ×103\times 10^{3} ), and Pearson correlation. It should be noted that the mean and std reported in avg column is computed directly on the test sets and is not the average of the 10 columns.
Figure 7: True health indicators versus predictions made by different models for four HI trajectories with varying levels of prediction difficulty.
Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

SymbolDescription
ggTransition Model
hhObservation Model
v,wv,wGaussian Noise
fproc,fenc,fdecf_{\text{proc}},f_{\text{enc}},f_{\text{dec}}Process, Encoder, Decoder Models
xtx_{t}HI State Label at timestep tt
yty_{t}Sensor Measurement
Appendix
Table 2: Proposed Notation (to refactor in text).
State LabelModuleQuantityMinMax
deg_CmpFan_s_mapEff_inFanEfficiency-0.050.00
deg_CmpFan_s_mapWc_inFanMass Flow-0.050.03
deg_CmpBst_s_mapEff_inBooster CompressorEfficiency-0.050.00
deg_CmpBst_s_mapWc_inBooster CompressorMass Flow-0.050.03
deg_CmpH_s_mapEff_inHigh-Pressure CompressorEfficiency-0.050.00
deg_CmpH_s_mapWc_inHigh-Pressure CompressorMass Flow-0.050.03
Appendix
Table 3: Health Indicators (HIs): degradation factors per module with their physical bounds. Each indicator represents a subsystem state used to generate observations.
Figure 8: health indicators distribution
Figure 9: Health indicators (degradation trajectories) mean and standard deviation over all the sequences
Figure 10: Auto-correlation (ACF) and partial autocorrelation (PACF). The x-axis represents the lag and y-axis represents the correlation value (-1,1)
Figure 11: t-SNE plot of few trajectories from the VJEPA architecture. The four clusters correspond to trajectories grouped by the amount of maintenance performed.
Figure 12: (Proposal) Multitask Experiment Setup Overview . a) Learning: A model learns a representation from sensor observation which is evaluated on a series of downstream tasks. b) Decoding: The latent zz is used to train a regression head to estimate the HI labels, and another block ϕz\phi_{z} is trained for next state prediction zt+1=ϕz(zt)z_{t+1}=\phi_{z}(z_{t}) . c) Forecasting: the frozen blocks are used for multi-state forecasting over τ\tau timesteps from nn randomly sampled initial states ztz_{t} . We compute the R2R^{2} score of the estimated HIs through time. d) Geometric Interpretation: Through clustering methods, we investigate if the learned representation reflects the underlying geometry in terms of cluster separation of signature degradation modes and maintenance history. e) Prescription: we evaluate the model capabilities in RUL estimation as well as the horizon of maintenance actions.
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