Multi-Term Fourier Graph Neural Network with Sample Relationship Learning for Enhanced Remaining Useful Life Prediction

I. INTRODUCTION

• ST-GNNs can require costly graph-structure learning, separate spatial from temporal modeling, and rely on fixed lookback windows.The introduction also notes that conventional RUL methods often use only the final test window, leaving earlier test-sequence information unused.Source: §I · p. 2
• MTFGN-SRL treats each sample as a complete graph and uses a Fourier Graph Neural Network to model spatio-temporal information in the frequency domain.The method uses the Discrete Fourier Transform and aims to avoid separate spatial and temporal modeling.Source: §I · p. 2
• Multi-Term FGN generates samples using varying lookback windows to capture both short- and long-term dependencies.It can optionally produce multiple predictions for ensemble averaging.Source: §I · p. 2
• Sample Relationship Learning uses a graph of samples and a heterogeneous message-passing network to model inter-sample relationships.The introduction illustrates that temporally adjacent samples can be strongly related, more distant samples weakly related, and samples from separate time series unrelated.Source: §I · p. 2, Figure 1 · p. 2
• MTFGN-SRL achieves competitive performance against state-of-the-art traditional sequence models and ST-GNN methods on the benchmark.The introduction identifies it as a widely used benchmark but does not name it in this passage.Source: §I · p. 2

A. Deep learning models for RUL prediction

• Deep learning models for RUL include recurrent, convolutional, and Transformer approaches, while ST-GNNs represent each time step as a graph to capture variable relationships.Graph construction in prior work has used domain knowledge, sensor correlations, or graph-structure learning.Source: A. Deep learning models for RUL… · p. 3

B. Multi-scale learning in time series prediction

• Unlike multi-scale methods that keep input dimensions uniform, multi-term learning creates inputs with different lookback lengths, some emphasizing short-term information and others long-term dependencies.Prior approaches described include multi-scale kernels, pyramids, patches, and varying CNN filters.Source: B. Multi-scale learning in time series… · p. 3

C. Sample relationship learning in time series prediction

• Prior time-series graph work targets variable, within-series, or whole-series relationships; MTFGN-SRL instead models relationships between samples themselves.It treats each time-series sample as a complete graph and builds a separate sample-relationship graph.Source: C. Sample relationship learning in… · p. 3

A. Preliminaries and motivations

• A sliding window of size T forms each sample from T successive observations and pairs it with an output label.Moving the window across the dataset generates multiple samples.Source: A. Preliminaries and motivations · p. 4
• For a dataset of M sequences with lengths Li, the number of samples generated with a window of length T is expressed as N = Σ(Li − T + 1).The supplied expression gives the per-sequence contribution; the sum runs across sequences.Source: A. Preliminaries and motivations · p. 4
• The described ST-GNN formulation uses graph processing for spatial dependencies and a temporal network for temporal dependencies.The passage contrasts this with traditional sequence models that encapsulate temporal information.Source: A. Preliminaries and motivations · p. 4

B. Multi-Term FGN (MTFGN)

• MTFGN adds multi-term learning to FGN to address traditional models’ limited ability to learn long-term dependencies.The approach uses multiple lookback windows.Source: B. Multi-Term FGN (MTFGN) · p. 4

1) FGN for Time Series Prediction:

• FGN treats each time-series sample as one complete graph rather than as a sequence of graphs.This formulation is used to learn unified spatio-temporal dependencies.Source: 1) FGN for Time Series Prediction · p. 4
• FGN maps node features into embeddings, applies a Discrete Fourier Transform, combines Fourier Graph Operators in frequency space, then uses an Inverse Discrete Fourier Transform and fully connected layers to produce labels.The paper states that FGN can encode potential spatio-temporal interdependencies in sensor signals while reducing noise, without requiring explicit graph-structure learning.Source: 1) FGN for Time Series Prediction · pp. 4–5

2) Multi-term Training Process:

• A fixed lookback window must fit the shortest sequence, which can force a small window and limit long-term dependency capture.The multi-term strategy instead uses a set of lookback windows of increasing size.Source: 2) Multi-term Training Process · p. 5
• Training creates window-specific sample groups, discards training sequences too short for a designated window, and left-pads smaller samples to the largest window size.One FGN is trained on the resulting samples in a supervised learning paradigm.Source: 2) Multi-term Training Process · p. 5

3) Adaptive Length-grouped Inference:

• At inference, each test sequence uses multiple lookback windows when its length permits, producing predictions for the same current RUL target.Shorter sequences may use fewer windows and therefore yield fewer predictions than longer sequences.Source: 3) Adaptive Length-grouped Inference · p. 5
• For sample relationship learning, inference uses embeddings from the trained FGN’s penultimate linear layer instead of its direct RUL predictions.The applicable windows are determined by sequence length.Source: 3) Adaptive Length-grouped Inference · p. 5

C. Sample relationship learning

• Adjacent sliding-window samples can overlap, and samples from different lookback sizes may share a label when they end at the same time point.The described relationships concern samples from the same series; samples from different series are expected to have weaker or more distant relationships.Source: C. Sample relationship learning · p. 5

1) Sample Relationship Graph Design:

• The Sample Relationship Graph organizes samples from a single time series, using their MTFGN embeddings as uniform-sized node features.Its heterogeneous structure represents distinct relationship types among samples.Source: 1) Sample Relationship Graph Design · p. 6
• Horizontal edges connect sequentially adjacent samples at the same lookback size, while vertical edges connect different window sizes ending at the same time point.The graph thereby represents temporal continuity and cross-scale relationships; its edge types carry distinct relational semantics.Source: 1) Sample Relationship Graph Design · p. 6
• The graph links samples from the same series while isolating samples from different series, supporting information propagation and aggregation among related samples.Figure 3 depicts samples generated at different lookback sizes and sequentially adjacent samples along a timestamp axis.Source: Figure 3 · p. 6, 1) Sample Relationship Graph Design · p. 7

2) Heterogeneous Message Passing Network:

• Horizontal-based convolution aggregates information from a node’s horizontal neighbors to update its representation.These edges connect sequentially adjacent samples generated with the same lookback size.Source: 2) Heterogeneous Message Passing Network · p. 7
• Vertical-based convolution aggregates information from samples using different lookback windows but ending at the same time point.This operation combines perspectives associated with short- and long-term dependencies.Source: 2) Heterogeneous Message Passing Network · p. 7
• After the two convolutions, element-wise max aggregation combines their node representations to retain features from both edge types.The paper describes this as merging temporal continuity and cross-scale relationships.Source: 2) Heterogeneous Message Passing Network · p. 7
• Graph-to-Sequence Pooling averages node representations along the vertical axis to generate a representation for each time step.Figure 4 illustrates this pooling step for generating predictions at each time step.Source: Figure 4 · p. 7

3) Graph-to-Sequence Pooling:

• Average pooling aggregates HMPN node embeddings for nodes at each time step into a pooled representation.The number of nodes associated with that time step determines the averaging denominator.Source: 3) Graph-to-Sequence Pooling · p. 7
• The pooled representations form a time-aligned sequence that integrates multi-term node information for prediction.The authors state that averaging also reduces noise and variability in these representations.Source: 3) Graph-to-Sequence Pooling · p. 7
• An MLP maps pooled representations to predictions, while an optional LSTM processes their sequence to capture temporal dependencies.A fully connected layer applied to the LSTM output derives the predictions.Source: 3) Graph-to-Sequence Pooling · pp. 7–8

• The proposed framework is evaluated on a benchmark dataset.Source: §IV · p. 8

A. Dataset description

• CMAPSS contains four turbofan-engine subsets, with differences in operating conditions, fault modes, and sequence lengths.Training sequences run to failure, whereas test sequences stop before complete failure.Source: A. Dataset description · p. 8, Table I · p. 8
• A fixed lookback window cannot exceed the shortest test-engine sequence, potentially limiting long-term dependency learning on longer sequences.The constraint arises because predictions must be available for all test engines.Source: A. Dataset description · p. 8

B. Implementation settings

• Preprocessing normalizes 14 selected features, caps labels at 125, and generates samples adaptively using multiple lookback windows.Window sizes start below the shortest sequence length and increase to include longer samples; sequence lengths and window partitions are illustrated in Fig. 5.Source: B. Implementation settings · p. 8, Table II · p. 8, Figure 5 · p. 9
• For FD002 and FD004, the authors cluster and normalize samples using label-correlated features before FGN learning, and use LSTM mapping for these longer sequences.The implementation uses three FGO layers and a single graph convolution layer in SRL.Source: B. Implementation settings · p. 8
• Evaluation uses RMSE and an asymmetric Score that penalizes RUL overestimation more heavily; lower values indicate better performance.Source: B. Implementation settings · p. 9

C. Comparisons with state-of-the-art

• The approach learns in the frequency domain with samples represented as a graph, uses multiple lookback terms, and models relationships across samples.The paper contrasts these choices with time-domain sequence-of-graphs methods and their fixed lookback windows.Source: C. Comparisons with state-of-the-art · p. 9

D. Ablation study

• FGN generally yields lower prediction errors than CNN, while adding Multi-Term improves both architectures, especially on long-sequence subsets FD002 and FD004.The ablation findings describe FGN as responsive to Multi-Term improvements and report reduced errors in the long-sequence scenarios.Source: D. Ablation study · p. 9

C. Comparisons with state-of-the-art

• MTFGN-SRL achieves the lowest average RMSE and Score across CMAPSS subsets, reducing both by 11.6% and 21.6%, respectively, versus LOGO.The comparison covers sequence models and ST-GNNs; Table III reports the RMSE and Score results.Source: C. Comparisons with state-of-the-art · p. 9, Table III · p. 10

D. Ablation study

• Adding SRL to MTFGN reduces FD003 RMSE by 11.08%; MTCNN-SRL also performs competitively on FD002 and FD004.The authors report that the proposed modules can help even a simple CNN surpass many sophisticated ST-GNN architectures.Source: D. Ablation study · p. 10
• FGN uses 4.99 seconds per epoch and 270k trainable weights, versus CNN’s 1.35 seconds and 511k; Multi-Term uses 5.48 seconds and 24k parameters.SRL adds 2.25 seconds per epoch for graph processing; the authors attribute the overall performance to architectural design and feature learning rather than increased model complexity.Source: D. Ablation study · p. 10
• Figures 6 and 7 compare results across the CMAPSS dataset: Figure 6 shows test-data utilization percentages, while Figure 7 shows the numbers of training and test samples.Source: Figure 6 · p. 10, Figure 7 · p. 11

E. Data utilization analysis

• SRL achieves 100% test-data utilization across all CMAPSS subsets, while multi-term windows generate more training and test samples at different scales.The framework generates a richer set of samples particularly on FD002 and FD004.Source: E. Data utilization analysis · p. 11

F. Robust prediction analysis

• MTFGN-SRL produces structured embeddings that delineate temporal degradation and predictions that align more stably with ground truth than FGN or MTFGN.MTFGN embeddings show clearer temporal trends than FGN; the figure compares embeddings and predicted RUL against ground truth.Source: F. Robust prediction analysis · p. 11, Figure 8 · p. 12

V. CONCLUSION

• MTFGN-SRL achieved superior RUL prediction performance to state-of-the-art methods on the CMAPSS benchmark.The framework combines frequency-domain modeling, multi-term sampling, and sample relationship learning.Source: §V · p. 11
• MTFGN-SRL uses an FGN for frequency-domain spatio-temporal modeling, varied lookback windows for short- and long-term dependencies, and a heterogeneous GNN to learn relationships among samples.The Sample Relationship Graph captures relationships between samples generated from sliding time windows.Source: §V · p. 11