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A physics-enhanced bidirectional multi-order graph fusion network for interpretable bearing remaining useful life prediction

Haoxuan Zhang, Dinghao Yang, Kangning Zhang, Shaoyong Guo, Haisheng Li, Rui Yang, Ruijun Liu

arXiv:2609.02190v1cs.CE

TL;DR

Bearing RUL prediction must handle nonlinear, non-stationary degradation while remaining interpretable for engineering use. PE-BMGN combines bidirectional multi-order graph fusion, historical prototype memory, KAN-based mappings, and physics-enhanced loss; experiments on two bearing datasets report superior performance and interpretable degradation analysis.

  • Problem

    Bearing degradation is nonlinear, non-stationary, variable, and often obscured by noise, while existing KAN-based RUL methods do not fully address model interpretability.

  • Method

    PE-BMGN fuses forward and backward multi-order degradation information, adaptively balances directions, retrieves historical degradation prototypes, and uses KAN mappings with physics-enhanced regularization.

  • Results

    Experiments on XJTU-SY and PHM2012 show superior performance across operating conditions, while interpretability analyses expose contributions from graph propagation and the KAN regressor.

  • Takeaways & Limitations

    The framework provides an interpretable view of nonlinear bearing degradation and supports RUL estimation using both current representations and reusable historical degradation knowledge.

  • Takeaways & Limitations

    Future work will add more physically grounded fault and degradation constraints and explore adaptive memory, uncertainty-aware retrieval, and cross-condition transfer.

Abstract

from arXiv · show

Accurate prediction of bearing remaining useful life (RUL) is a key challenge for intelligent maintenance. Although deep learning-based prediction methods have showed effectiveness, existing methods still have limitations in learning nonlinear bearing degradation processes and model interpretability. Especially in engineering applications, the "black box" nature of deep learning models can easily raise concerns about their reliability. Therefore, we propose a physics-enhanced bidirectional multi-order graph fusion network for interpretable bearing RUL prediction. Our network mines complementary information from both forward and backward degradation sequences. Specifically, our network introduces a multi-order graph propagator to capture the local-global degradation dependencies. A gated cross-fusion mechanism is further designed to dynamically balance the feature contributions from both forward and backward directions. Then, our network stores representative historical degradation prototypes in dynamic memory, so that the final RUL prediction no longer depends solely on the current latent features, but is guided by reusable historical degradation knowledge. To reveal how our model learns the nonlinear degradation process, the feature mapping parts utilize the Kolmogorov-Arnold network, which allows the nonlinear mapping to be visualized using learnable functions. Finally, a physics-enhanced dynamic loss function is developed to help our network learn effective and reliable degradation representations. Extensive experiments on two public datasets show that our method achieves the lowest error while providing more conservative estimates than existing methods. Our code is available at https://github.com/IMGresearcher/PE-BMGN.

1. Introduction

Bearing RUL prediction is important for proactive maintenance, but complex degradation and limited interpretability challenge existing methods. PE-BMGN addresses these issues with bidirectional graph fusion, historical prototypes, KAN mappings, and physics-enhanced training.

  • Bearing degradation is nonlinear, non-stationary, and individually variable because of changing loads, fatigue, lubrication, and environmental disturbances.
  • Existing KAN-based RUL methods do not address interpretability or tailor surrounding modules to nonlinear degradation modeling.
  • KAN is embedded in graph-to-degradation and degradation-to-RUL mappings, making nonlinear functions more explicit and analytically visualizable.
  • PE-BMGN uses forward and backward graph views, multi-order Chebyshev convolution, and gated fusion to model degradation dependencies.
  • A dynamic memory bank retrieves representative historical degradation prototypes to guide prediction alongside current latent features.
  • Experiments on XJTU-SY and PHM2012 report superior performance across operating conditions and interpretable views of degradation learning.

2. Problem Definition

The problem is formulated as mapping each bearing’s complete degradation run from monitoring inputs to a normalized RUL sequence. Labels decrease from 1 at the first monitoring step to 0 at the last.

  • Each complete degradation run of one bearing is treated as a temporal sample for learning input-to-RUL mapping.
  • The input sequence contains monitoring steps, signal channels, and signal samples for each acquisition.
  • The two signal channels correspond to horizontal and vertical vibration signals.
  • RUL labels are normalized to [0, 1] using y_t = (T_b−t)/(T_b−1), with the first step equal to 1 and the last equal to 0.

3. Methodology

PE-BMGN is a unified pipeline for improving degradation-correlation modeling and prediction interpretability. It processes vibration data through multiscale decomposition, adaptive graph construction, bidirectional propagation, and memory-guided prediction.

  • PE-BMGN aims to improve degradation-correlation modeling and prediction interpretability within one unified framework.
  • The pipeline first generates multiscale degradation representations from raw vibration sequences using learnable discrete wavelet decomposition.
  • It then constructs an adaptive graph, propagates information bidirectionally, and performs memory-augmented RUL prediction.

3.1. Dynamic degradation graph construction

The graph-construction module converts non-stationary vibration signals into multiscale node features and a sparse adaptive topology. Learnable wavelet branches preserve complementary temporal and spectral degradation information before correlation-based graph formation.

  • Dynamic graph construction: Raw vibration sequences are transformed into graph nodes representing degradation states and edges representing correlations between states.
  • Learnable wavelet decomposition: Trainable wavelet filters adapt decomposition to different bearings and degradation stages instead of relying on fixed signal bases.
  • Learnable wavelet decomposition: Two convolutional branches project each level into complementary coefficient spaces for oscillatory degradation information and dominant temporal responses.
  • Learnable wavelet decomposition: Frequency-domain gating recalibrates spectral magnitude while retaining phase, emphasizing degradation-related frequencies and weakening irrelevant fluctuations.
  • Learnable wavelet decomposition: Residual max pooling strengthens dominant responses while preserving original coefficient information, complementing the gated frequency branch.
  • Node representation: Enhanced coefficients from multiple levels are compressed, aggregated, concatenated, and adaptively pooled into node features containing temporal and spectral information.
  • Dynamic graph topology construction: Query-key projections and cosine similarity quantify pairwise degradation correlations, with temperature controlling similarity sharpness.
  • Dynamic graph topology construction: Top-K sparsification removes weak redundant connections, producing a sparse adaptive adjacency matrix for subsequent graph propagation.

3.2. Bidirectional multi-order graph fusion network

The bidirectional graph fusion network models bearing degradation from complementary forward and backward views, combining multi-order dependencies with nonlinear, interpretable mappings and adaptive cross-fusion.

  • Bidirectional graph propagation: Forward and backward graph views encode complementary degradation information across the bearing life trajectory.The forward branch emphasizes degradation accumulation, while the backward branch provides reverse calibration cues.
  • Multi-order graph propagation: Chebyshev propagation captures dependencies from adjacent states to distant, multi-hop degradation states.Zeroth-, first-, and higher-order responses preserve node states and progressively expand the receptive field.
  • Nonlinear graph mapping: KAN mappings are applied after each Chebyshev propagation order to represent complex degradation interactions through structured spline-based functions.This replaces purely linear post-aggregation transformations while maintaining functional interpretability.
  • Gated cross fusion: A learnable gate dynamically balances forward and backward features because their relative usefulness changes across degradation stages.Backward cues help calibrate weak early signatures, whereas forward information may become more important near severe degradation.
  • Fused representation: The fused degradation representation is passed to the memory-augmented prediction network for prototype retrieval and final RUL regression.The fusion module combines directional features before downstream prediction.

3.3. Memory-augmented prediction module

The memory-augmented prediction module supplements the current degradation representation with relevant historical prototypes before applying an interpretable KAN regressor to estimate RUL.

  • Dynamic memory: A dynamic memory bank stores representative historical degradation prototypes for reuse during prediction.Each memory vector represents historical degradation information, and the bank has configurable capacity and dimension.
  • Prototype retrieval: Current degradation states are projected into a query space, and softmax-normalized similarities determine prototype relevance.The query and learnable projection parameters support matching the current state to stored prototypes.
  • Prototype retrieval: Only the top-K_m most relevant prototypes are selected, with their weights renormalized before aggregation.This retrieval strategy excludes unrelated historical patterns and preserves relative relevance among selected prototypes.
  • Memory-enhanced representation: Weighted aggregation gives greater contribution to prototypes that better match the current degradation state.The module therefore uses discriminative historical knowledge rather than simply averaging retrieved prototypes.
  • Memory adaptation: Momentum updates keep memory prototypes stable while adapting them to the evolving feature distribution during training.The update combines previous memory states with the current mini-batch’s average memory query representation.
  • RUL regression: The current query and retrieved prototype are concatenated, then a KAN regressor maps the combined representation to predicted RUL.KAN layers use structured functional mappings instead of conventional fully connected transformations.

3.4. Physics-enhanced dynamic loss function

The physics-enhanced dynamic loss jointly optimizes prediction accuracy, physical consistency of signal decomposition, and compactness and stability of KAN mappings.

  • Prediction objective: Prediction loss minimizes the discrepancy between predicted and ground-truth normalized RUL sequences.
  • Physical consistency: Orthogonality and energy constraints preserve physically plausible behavior in the learnable wavelet decomposition filters.Orthogonality reduces redundant branch responses, while energy regularization limits excessive amplification or attenuation.
  • KAN regularization: KAN activation and entropy regularization constrain nonlinear mappings in the graph and prediction modules.These terms suppress excessive spline activations and overly dispersed functional responses.
  • Joint optimization: The total objective combines prediction, physical constraint, and KAN regularization losses.Joint optimization is intended to reduce prediction errors while preserving physically meaningful decomposition and interpretable mappings.

4. Experimentation and analysis

Experiments evaluate PE-BMGN on the XJTU-SY and PHM2012 bearing datasets using RMSE, MAE, and EAS, alongside comparative and interpretability analyses. On XJTU-SY, PE-BMGN achieves the lowest overall RMSE, while learned functions provide evidence of stable or complex degradation representations across bearings.

  • Comparative experiments: PE-BMGN achieves the lowest XJTU-SY Total_Avg RMSE of 0.069, outperforming DyWave-BiAGCN at 0.096 and SSPGKAN at 0.224.The result indicates fewer severe RUL estimation errors across bearing trajectories.
  • Comparative experiments: PE-BMGN obtains the best XJTU-SY C1_Avg and C2_Avg RMSE values of 0.025 and 0.068, respectively, although its C3_Avg RMSE of 0.114 exceeds DyWave-BiAGCN’s 0.106.The strongest condition-level gains occur under Conditions I and II.
  • Interpretability analysis: For Br1_3, RMSE, MAE, and EAS are 0.019, 0.014, and 0.0012, while forward and backward learned functions remain continuous and stable.Most mappings are monotonic or weakly nonlinear without obvious abrupt oscillations or boundary jumps.
  • Interpretability analysis: Br1_3 representations become progressively richer across graph transformations: shallow mappings preserve smooth trends, whereas deeper mappings introduce active but continuous nonlinear responses.The functional hierarchy suggests gradual refinement rather than reliance on unstable local fluctuations.
  • Interpretability analysis: The Br1_3 KAN regressor remains smooth and continuous, mapping fused degradation features to RUL through a stable nonlinear transformation.The interpretation links stable graph propagation with smooth final regression.
  • Interpretability analysis: Br2_1 has larger errors—RMSE 0.171, MAE 0.142, and EAS 0.0124—with deeper graph functions showing local peaks, endpoint sensitivity, and stronger high-order responses.The final regressor remains relatively smooth but cannot fully compensate for unstable graph-level responses.

4.5. Case study 2: PHM2012 bearing dataset

On the PHM2012 dataset, PE-BMGN delivers the strongest reported quantitative performance and generally follows ground-truth degradation trends. Its interpretability analysis shows complementary directional functions and smoother final RUL mappings.

  • Comparative experiments: PE-BMGN achieves the lowest PHM2012 Total_Avg RMSE of 0.095, outperforming all baselines.The advantage is especially evident under Conditions II and III.
  • Comparative experiments: Under PHM2012 Condition II, PE-BMGN reaches C2_Avg RMSE 0.078 versus the best baseline value of 0.129 from STACP-GCN.For Br3_3, it also obtains the lowest RMSE of 0.052 under Condition III.
  • Comparative experiments: PE-BMGN records PHM2012 C1_Avg, C2_Avg, and Br3_3 MAE values of 0.094, 0.060, and 0.042, respectively.The method achieves the lowest MAE values under all three operating conditions.
  • Comparative experiments: PE-BMGN obtains Total_Avg EAS 0.0067, below STACP-GCN’s 0.0093 and DyWave-BiAGCN’s 0.0094.EAS evaluates engineering risk from biased RUL prediction.
  • Comparative experiments: Figure 11 shows predicted RUL curves generally following the ground-truth decreasing tendency, with closer agreement under Conditions II and III.This agrees with the lower C2_Avg and Br3_3 errors.
  • Interpretability analysis: In Condition II, forward low-order functions preserve degradation trends, while backward mappings combine increasing and decreasing responses to provide reverse structural cues.The backward branch supplies complementary information for weak or ambiguous degradation stages.
  • Interpretability analysis: The final KAN regressor produces smoother functions than the graph branches, compressing complex graph-level responses into lower-complexity RUL mappings.This functional transformation is linked to strong quantitative performance under Condition II.

4.6. Ablation experiments

Ablation experiments show that the complete PE-BMGN performs best, with bidirectional multi-order graph fusion contributing most strongly and memory retrieval providing additional gains. Hyperparameter results indicate dataset-dependent moderate graph density and compact memory capacity are preferable.

  • Model ablation experiments: The complete PE-BMGN achieves the lowest errors across both datasets and all three metrics in the model ablation experiments.Model1 is the complete framework, while Models2–4 remove memory, graph fusion, or both modules.
  • Model ablation experiments: Removing memory increases XJTU-SY RMSE, MAE, and EAS from 0.069, 0.055, and 0.0049 to 0.094, 0.075, and 0.0067.On PHM2012, the corresponding metrics increase from 0.095, 0.074, and 0.0067 to 0.098, 0.081, and 0.0074.
  • Model ablation experiments: Removing bidirectional multi-order graph fusion raises RMSE from 0.069 to 0.163 on XJTU-SY and from 0.095 to 0.228 on PHM2012.MAE and EAS also increase substantially, indicating that graph fusion is the most critical ablated component.
  • Model ablation experiments: Removing both modules yields RMSE values of 0.096 and 0.125 on XJTU-SY and PHM2012, respectively, better than Model3 but worse than the complete model.The results suggest memory gains depend on sufficiently discriminative bidirectional graph representations.
  • Hyperparameter ablation experiments: On XJTU-SY, increasing retained neighbors from 80 to 160 lowers RMSE and MAE, but further increases to 180–240 worsen errors.The best performance in this group occurs at k=160, balancing dependency coverage and graph selectivity.
  • Hyperparameter ablation experiments: On PHM2012, performance is best around k=110 and worsens at k=130 and k=150, showing that optimal graph density depends on the dataset.The authors relate the smaller optimum to shorter samples and fewer training bearings.
  • Hyperparameter ablation experiments: A memory bank around 128 entries is most stable on XJTU-SY, whereas PHM2012 achieves lowest RMSE and MAE with 64 entries.Both insufficient and excessive prototype capacity degrade retrieval effectiveness.

5. Conclusion

PE-BMGN combines bidirectional graph fusion, historical degradation memory, KAN mappings, and physics-enhanced loss to predict bearing RUL and analyze nonlinear degradation. Experiments support its degradation modeling and interpretability, while future work targets stronger physical constraints and deployment robustness.

  • Conclusion: PE-BMGN extracts complementary forward and backward degradation information, adaptively balances their contributions, and retrieves historical degradation prototypes for prediction.The framework also uses KAN mappings to analyze nonlinear degradation-to-RUL relationships and a dynamic loss to constrain physical consistency and functional stability.
  • Conclusion: Experiments on XJTU-SY and PHM2012 show that PE-BMGN captures global degradation trends for most bearings and provides interpretable functional responses.Smooth, coherent functions are associated with accurate predictions, while peaks and boundary-sensitive changes indicate atypical degradation trajectories.
  • Conclusion: Future work will add more physically grounded constraints related to bearing fault mechanisms and degradation dynamics.The authors also plan adaptive memory updating, uncertainty-aware retrieval, and cross-condition transfer for improved robustness and deployability.

CRediT authorship contribution statement

The authors contributed across conceptualization, data curation, formal analysis, methodology, software, visualization, validation, investigation, funding acquisition, and writing or editing.

  • CRediT authorship contribution statement: The contribution statement assigns roles spanning conceptualization, data curation, methodology, analysis, software, visualization, validation, investigation, funding, and manuscript preparation.Specific author assignments are listed in the contribution statement.
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