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Toward Reliable Railway-Bogie Response Prediction Using Multifidelity TDNN and Physics-Informed Residual Learning

Gyeolhee Lee, Moosun Kim, Taewook Kwon, Jaehun Kim, Changsung Jeon, Dongjin Lee

arXiv:2609.12018v2cs.LGphysics.app-ph

TL;DR

Limited physical testing means railway simulations must predict responses across operating conditions beyond those directly measured, while agreement at representative conditions does not ensure accuracy elsewhere. The paper combines simulation-informed multifidelity learning with physics-constrained residual correction for bogie-response histories. At the held-out 385 km/h condition, the corrected response achieves a mean R2 of 0.8197, with broader generalization still unestablished.

  • Problem

    Physical tests cannot cover all railway operating conditions, and agreement with measurements at representative conditions does not guarantee accuracy elsewhere.

    Source: §1 · p. 3

  • Method

    The method uses MBD histories as a low-fidelity trend and roller-rig measurements to train a physics-constrained correction for multichannel bogie responses.

    Source: §1 · p. 4

  • Results

    At the held-out fifth speed condition, all four corrected responses achieve R2 > 0.79, with a mean R2 of 0.8197, mean NRMSE of 4.6055 %, and mean NMAE of 1.9297 %.

    Source: §4.4 · p. 23

  • Takeaways & Limitations

    The held-out 385 km/h condition provides initial evidence of extrapolative response prediction beyond the development-speed range.

    Source: §4.4 · p. 23, §5 · p. 25

    A single evaluation condition is insufficient to establish generalization across broader speed, excitation, loading, suspension, or boundary-condition ranges.

    Source: §5 · p. 25

Abstract

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Railway engineers need simulation models that predict vehicle responses across operating scenarios that cannot be tested exhaustively. Agreement with representative measurements provides essential evidence, but calibration at a limited set of conditions does not guarantee accuracy elsewhere. We present a multifidelity railway-bogie response-correction method that treats multibody simulation histories as low-fidelity information and roller-rig measurements as high-fidelity evidence. This method combines an experiment-anchored fidelity assignment with physics-informed discrepancy learning for multichannel bogie-response histories. A time-delay neural network (TDNN) represents the condition-dependent simulation trend, and development-fitted amplitude alignment defines the low-fidelity baseline. A residual-correction network then models the reproducible response component not explained by this baseline and adds it to the baseline. An effective dynamic-balance equation constrains the learned discrepancy by representing differences in inertia, damping, stiffness, and external forcing between the simulated and physical systems. The training objective combines this constraint with residual matching, temporal smoothness, and a combined channel-2 acceleration loss selected using displacement-acceleration consistency evidence. For the evaluated reconstruction case, the corrected response gives a mean coefficient of determination of 0.8197, a mean normalized root-mean-square error (NRMSE) of 4.6055 %, and a mean normalized mean absolute error (NMAE) of 1.9297 %. These results provide initial evidence of accurate response prediction at the held-out 385 km/h condition.

Nomenclature

The nomenclature defines symbols for bogie responses and operating conditions, model components, correction losses, and evaluation metrics.

Nomenclature

  • FW, FB, RW, and RB denote the front and rear axleboxes and bogie frames, while Y and Y/Q denote wheel–rail lateral force and derailment quotient.Q is the corresponding compressive vertical wheel load.Source: Nomenclature · p. 1
  • The operating-condition vector x comprises speed, excitation amplitude, and frequency, while qs and qe denote simulation and experimental displacement-response vectors.ξ is the experiment-only PINN input vector containing speed and harmonic-phase features.Source: Nomenclature · p. 2
  • The ML baseline notation distinguishes the input-history window, unaligned simulation representation, amplitude-aligned baseline, observed residual, latent discrepancy, and measurement error.The learned correction is added to the baseline to form the discrepancy-corrected response.Source: Nomenclature · p. 2
  • The effective dynamic-residual notation uses reduced-system mass, damping, stiffness, and force terms, with discrepancy matrices for differences between nominal and physical systems.Fext and F0 denote forces transmitted through the external rod-bush markers and the internal preload force.Source: Nomenclature · p. 2
  • The correction objective notation covers residual matching, temporal smoothness, dynamic-residual losses, and the combined channel-2 acceleration loss.The nomenclature also defines the scale matrices and channel-2 acceleration quantities used by that loss.Source: Nomenclature · p. 2
  • R2, RMSE, MAE, NRMSE, and NMAE name the response-evaluation metrics, while the TDNN-assisted residual model has its own residual-matching loss.The nomenclature also specifies oracle temporal alignment and the corresponding oracle-aligned corrected response.Source: Nomenclature · pp. 2–3

1. Introduction

Railway simulations can cover operating conditions that physical tests cannot exhaustively sample, but calibration against representative measurements does not establish accuracy elsewhere. This work proposes an experiment-informed, physics-constrained correction of multichannel bogie responses.

1. Introduction

  • Simulation can assess responses across many operating and design conditions, but physical testing cannot cover them all and calibration at representative conditions does not ensure accuracy elsewhere.The introduction notes that simulations do not replace physical procedures required for safety, certification, or vehicle acceptance.Source: §1 · p. 3
  • Prior railway studies use MBD models and measurements to validate responses across conditions, while calibration on one dataset does not establish predictive accuracy elsewhere.The cited examples include model-acceptance procedures, roller-rig comparisons, and parameter identification.Source: §1 · p. 3
  • Railway ML studies report limits involving training-domain coverage, high-frequency impact prediction, available operating conditions, and practical generalization.These limits motivate using simulation for a condition-dependent trend and measurements to learn reproducible simulation–test residuals.Source: §1 · p. 3
  • Prior railway multifidelity applications combine numerical models of differing fidelity, but do not address experiment-informed correction of multichannel bogie-response histories.Examples include railway-bridge reliability, bridge sensor prediction, and high-speed-train crash response.Source: §1 · p. 4
  • The proposed method combines low-fidelity MBD histories and high-fidelity roller-rig measurements, using a TDNN baseline and an effective dynamic-balance constraint for the learned correction.Its objective also combines residual matching, temporal smoothness, and a channel-2 acceleration loss selected using displacement–acceleration consistency evidence.Source: §1 · p. 4

2. Roller-Rig Test Configuration and Modeling Foundations

The section defines the roller-rig evidence and multibody representations, then frames simulation histories as a low-fidelity trend corrected against measured responses. It also introduces the finite-history TDNN used to represent simulation response histories.

2.3. Physics-Based Vehicle Model and Simulation Implementation

  • The roller-rig and full-vehicle multibody configurations share bogie-level components but have distinct boundaries and evidentiary roles.The rig model represents a semi-full-car installation, while the full-vehicle model contains one car body, two bogies, and four wheelsets.Source: §2.1 · p. 4, §2.3 · pp. 7–8

2.1. Test Vehicle, Operating Conditions, and Roller-Rig Boundary

  • The 300–360 km/h conditions are intended for development, with 385 km/h held for evaluation only if all target-informed operations exclude it.The physical rig is rated to 420 km/h, and the present tests extend to 385 km/h.Source: §2.1 · pp. 4–5

2.2. Instrumentation and Response Data

  • Four displacement channels are indexed by front/rear axlebox and bogie-frame locations, but physical-axis, sign, and reference correspondence require verification before pairing.Records are segmented to a steady forced-response interval, placed on a common time base, then amplitude-aligned and normalized.Source: §2.2 · pp. 6–7

2.4. Simulation-Model Consistency Assessment

  • Three full-vehicle consistency assessments examine preload, hunting-stability transition, and selected wheel–rail safety-related quantities without establishing standards compliance or vehicle acceptance.The assessments provide limited cross-code context.Source: §2.4 · p. 10

2.5. Time-Delay Neural Network

  • The TDNN maps a finite window of current and preceding simulation responses plus speed to a multichannel response representation, not an independent forecast.Unseen-condition forecasting would require predictive inputs that exclude the target response history and separate validation.Source: §2.5 · p. 11

2.6. Multifidelity Model-Discrepancy Learning

  • Development measurements set amplitude alignment for a high-fidelity-informed low-fidelity baseline, and residual learning models the remaining discrepancy against experimental responses.The experimental reference is not noise-free ground truth, and the fidelity assignment applies to selected response quantities and tested conditions.Source: §2.6 · pp. 11–12
  • Limited experimental responses anchor predictions for unseen conditions, while simulation responses supply condition-dependent physical trends.The modeling problem is to correct simulation–experiment discrepancy while preserving simulation trends.Source: §2.3 · p. 10, §2.6 · p. 12

3. Proposed Multifidelity Simulation–Test Response Correction Method

The method uses simulation histories to establish a condition-dependent baseline, then uses development measurements and a physics-informed network to correct the residual response. Its workflow separates baseline fitting, amplitude alignment, discrepancy correction, and evaluation with fitted quantities frozen.

3.1. Proposed Method Overview and Response Decomposition

  • The two-stage method learns a simulation-informed response trend, aligns it with measurements, and adds a physics-informed correction for the reproducible discrepancy.The effective dynamic balance represents differences in inertia, dissipation, restoring behavior, and external forcing between simulated and physical systems.Source: §3.1 · p. 12

3.2. Development Training of the TDNN Baseline

  • The TDNN is fitted only to simulation responses from four development conditions, and its fitted parameters are frozen before amplitude alignment.The TDNN represents simulation histories before the development-fitted amplitude transformations are applied.Source: §3.2 · pp. 12–13

3.3. Development-Fitted Amplitude Alignment

  • A single nonnegative amplitude factor estimated from development records aligns the frozen machine-learning response and is then held fixed.The factor’s definition requires a nonzero denominator.Source: §3.3 · p. 13

3.4. Physics-Informed Residual Correction

  • The correction maps time, the aligned baseline, and PINN inputs to a reduced-coordinate correction, which is projected into measured channels using a verified sensor-to-coordinate map.The identity mapping is valid only when the measured outputs form a mechanically consistent reduced-coordinate vector.Source: §3.4 · pp. 13–14
  • The effective-dynamics residual regularizes the correction using system-level mass, damping, stiffness, and force differences; these matrices are not identified in the analysis.It is a corrective-source regularizer, not an exact reparameterization of the complete nonlinear wheel–roller model.Source: §3.4 · p. 14
  • Training combines residual matching, temporal smoothness, effective-dynamics balance, and three weighted channel-2 acceleration penalties, whose individual contributions are not isolated.The acceleration penalties compare predicted and measured acceleration, displacement-derived and measured acceleration, and the two predicted forms; the common passband is 3.5–4.5 Hz.Source: §3.4 · p. 15

3.5. Overall Training and Inference Workflow

  • At evaluation, development-fitted transformations and networks remain fixed; the solver-independent interface requires compatible response histories.Figure 8 distinguishes development operations from evaluation with target-informed quantities frozen.Source: §3.5 · pp. 15–16, Figure 8 · p. 16

4. Results and Discussion

The results assess simulation-model consistency, compare learning-based response reconstructions, and examine the proposed correction at a held-out speed. The corrected response provides initial evidence of extrapolative prediction at 385 km/h, while the study identifies limits on broader generalization and oracle-aligned evaluation.

4.1. Simulation Model Consistency

  • Static assembly checks met screening thresholds, and the nonlinear sweep identified a 1124 km/h critical speed; these support internal consistency, not response-history accuracy.The maximum residual constraint acceleration was 0.0189 m/s2 against a 0.1 m/s2 threshold, and selected loads agreed with static-equilibrium values within 5%.Source: §4.1 · p. 18, Figure 9 · p. 18, Table 5 · p. 18

4.2. Experiment-Only PINN and Simulation-Informed Comparison

  • The experiment-only PINN predicts measured displacements directly without simulation input and uses channelwise SDOF regularization instead of simulation–experiment discrepancy parameters.Its additional acceleration-related penalties were omitted after they did not improve held-out displacement accuracy.Source: §4.2 · p. 19

4.3. TDNN-Assisted Residual Result

  • The TDNN-assisted residual case amplitude-aligns a frozen TDNN baseline, then trains a separate network to reproduce the observed residual and adds that correction to the baseline.The residual network receives time, speed, and aligned baseline response as inputs; unlike the proposed formulation, it uses no acceleration target, reduced dynamic-balance loss, or smoothness loss.Source: §4.3 · p. 20
  • Against the experiment-only PINN, the TDNN-assisted residual model reports mean R2 0.6515 versus 0.5027, mean NRMSE 6.4180 % versus 7.6414 %, and mean NMAE 3.1637 % versus 4.0873 %.These are numerical differences for that comparison, not a controlled ranking across models.Source: §4.3 · p. 20

4.4. Discrepancy-Corrected Response at the Fifth Speed Condition

  • The discrepancy-correction analysis formalizes an effective-dynamics residual and a combined channel-2 acceleration loss for reconstructing the fifth-speed responses.The correction objective also includes displacement-residual matching, smoothness, and constituent acceleration penalties.Source: §4.4 · p. 22
  • At the fifth speed, all four corrected responses achieve R2 > 0.79; mean R2 is 0.8197, mean NRMSE 4.6055 %, and mean NMAE 1.9297 %.These results provide initial evidence of extrapolative prediction beyond the development-speed range; one evaluation condition does not establish broader generalization.Source: Table 7 · p. 23, §4.4 · p. 23

4.5. Post-Hoc Oracle Temporal Alignment

  • At 385 km/h, oracle alignment raises R2 from 0.8197 to 0.9536, but uses the measured evaluation response and is a conditional best-case reference, not a deployable metric.The increase suggests substantial remaining mismatch is related to temporal misregistration rather than waveform-shape error alone.Source: §4.5 · p. 23, Figure 12 · p. 24

4.6. Displacement–Acceleration Channel Diagnostic

  • Channel 2 shows the strongest repeated displacement–acceleration association, motivating its exclusive use in the proposed acceleration loss; the other three channels remain weak after lag adjustment.The diagnostic does not prove exact kinematic equivalence or establish the isolated contribution of that loss to the corrected response.Source: §4.6 · p. 24

5. Conclusions and Outlook

The proposed multifidelity discrepancy method combines simulation histories, physical measurements, and an effective dynamic-balance constraint; the evaluated reconstruction gives initial evidence of prediction at held-out 385 km/h.

5. Conclusions and Outlook

  • Mean R2 of 0.8197, mean NRMSE of 4.6055 %, and mean NMAE of 1.9297 % are reported for the evaluated reconstruction case.The held-out 385 km/h condition provides initial evidence of extrapolative response prediction beyond the development-speed range.Source: §5 · pp. 24–25
  • The method combines low-fidelity MBD histories, high-fidelity physical measurements, and an effective dynamic-balance constraint to learn response unexplained by simulation.Its objective combines residual matching, temporal smoothness, an effective dynamic residual, and a combined channel-2 acceleration loss.Source: §5 · p. 24
  • The TDNN-assisted residual model and discrepancy-correction formulation give higher aggregate agreement than the experiment-only PINN, but the staged comparison does not isolate each component’s contribution.A response-consistency diagnostic supports selective use of the combined acceleration loss; controlled ablation is still needed to quantify its isolated contribution.Source: §5 · pp. 24–25
  • A single evaluation condition does not establish generalization across broader speed, excitation, loading, suspension, or boundary-condition ranges.The reduced dynamic-model discrepancy terms do not identify physical component errors from four sensor histories.Source: §5 · p. 25

Appendix A. Selected Physics-Based Model Parameters

Appendix A reports selected full-vehicle model parameters and explains a rod-bush preload comparison; some detailed suspension and connection parameters are withheld.

Appendix A. Selected Physics-Based Model Parameters

  • Table A.1 reports rigid-body properties of the full-vehicle model.Detailed suspension and connection parameters, attachment coordinates, and nonlinear damper characteristics are not reported because of confidentiality and security restrictions.Source: Table A.1 · p. 25, Appendix A · p. 25
  • The coordinate triplets distinguish car-body reference coordinates from bogie-frame and wheelset local-bogie reference coordinates.Center-of-mass triplets list longitudinal, lateral, and vertical components; mass-moment triplets list moments about those axes.Source: Appendix A · p. 25
  • At zero relative deformation, the rod-bush external force is generally a stiffness-weighted fraction of internal preload, so preload comparison uses external marker force.The rod-bush model has series and parallel branches separated by an internal node.Source: Appendix A · p. 25

Appendix B. Displacement–Acceleration Channel-Consistency Evidence

The appendix assesses whether paired displacement and acceleration measurements are consistent under the selected processing, while cautioning that this does not independently validate either sensor.

Appendix B. Displacement–Acceleration Channel-Consistency Evidence

  • The diagnostic compares each measured acceleration with the second derivative of its nominally paired measured displacement.The assessment is made under the selected processing.Source: Appendix B · p. 26
  • The comparison concerns consistency between paired measurements, not independent sensor validation.Source: Appendix B · p. 26
  • The consistency assessment is specific to the selected processing.Source: Appendix B · p. 26

Appendix B.1. Direct time-domain comparison

The appendix presents a channelwise displacement-derived versus measured-acceleration comparison and separately reports direct agreement.

Appendix B.1. Direct time-domain comparison

  • Figure B.1 compares the second time derivative of measured displacement with sign-corrected measured acceleration channel by channel at 300 km/h.The dashed line denotes one-to-one agreement.Source: Figure B.1 · p. 26
  • Table B.1 presents direct agreement between displacement-derived and measured acceleration.Source: Table B.1 · p. 27
  • Coefficient of determination and Pearson correlation are listed separately because they quantify different aspects of agreement; negative R2 is less accurate than the sample-mean baseline under Eq. (14).Source: Appendix B.1 · p. 27

Appendix B.2. Sensitivity to time lag

The appendix examines time-lag sensitivity at 300 km/h using shifted-sequence correlations and an acceleration–displacement-derivative comparison.

Appendix B.2. Sensitivity to time lag

  • Figure B.2 compares measured acceleration with the second derivative of measured displacement at 300 km/h after channelwise lag search.Panel titles report the signed lag and correlation.Source: Figure B.2 · p. 27
  • Table B.2 reports maximum absolute correlation after shifting each sequence by at most 2000 samples.The bound cannot be expressed in seconds without the sample interval.Source: Table B.2 · p. 27, Appendix B.2 · p. 27
  • At 300 km/h, negative signed correlations for channels 1, 3, and 4 indicate that the table’s positive values are interpreted as magnitudes.The figure panel titles show signed lag and correlation after the channelwise lag search.Source: Appendix B.2 · p. 27, Figure B.2 · p. 27

Appendix B.3. Frequency-domain comparison

The appendix compares normalized power-spectral densities at 300 km/h, reporting spectral overlap by channel.

Appendix B.3. Frequency-domain comparison

  • At 300 km/h, spectral-overlap values are 0.2640 (channel 1), 0.8230 (channel 2), 0.3270 (channel 3), and 0.1980 (channel 4), not magnitude-squared coherence.Figure B.3 presents the normalized power-spectral-density comparison.Source: Figure B.3 · p. 28
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