A deflection-basin fusion transformer for predicting critical pavement strains from falling weight deflectometer

Target responses

• εac is the horizontal tensile strain at the bottom of the asphalt concrete layer, and εsg is the vertical compressive strain at the top of the subgrade.Both strains are reported in microstrain (με).Source: Nomenclature · p. 2

Structure and moduli

• The layer inputs are AC, base, and subbase thicknesses h1–h3; E1–E4 denote the backcalculated moduli of the four pavement layers.Thicknesses are in millimetres and moduli in megapascals.Source: Nomenclature · p. 2

Measurements and model terms

• The basin notation includes deflection Dr at radial offset r and indices including SCI, BDI, BCI, AREA, and AUPP.The glossary also defines FiLM, FWD, RMSi, wi, λf, and Poisson’s ratio ν.Source: Nomenclature · p. 2

1.1 Motivation

• The two critical design responses are asphalt-bottom tensile strain and subgrade-top compressive strain, both inferred in routine practice rather than measured directly.These responses enter design transfer functions and remaining-life estimates.Source: §1.1 · p. 2
• Conventional modulus backcalculation is ill-posed, sensitive to measurement and thickness errors, and requires iterative office processing and expert screening.The resulting moduli feed a forward analysis, so errors from the inverse step carry into the strain estimates.Source: §1.1 · pp. 2–3
• Field validation assesses agreement with backcalculation-consistent mechanics, not agreement with strain measured by embedded gauges.The conventional inference chain is removed at prediction time, but backcalculation remains in the provenance of the reference labels.Source: §1.1 · p. 3

1.2 State of the art and research gap

• The literature review identifies missing evaluation of simulation-to-field transfer and whole-road holdouts, alongside input representations that discard offset or fail to let structure modulate basin interpretation.Reported point-level splits can allow interpolation along roads partly seen during training.Source: §1.2 · p. 4

1.3 Novelty and contributions of this study

• DBFT is trained jointly on 14,174 layered-elastic solutions and 7,651 field measurements from 96 sections, with three routes withheld entirely.The field corpus comes from the Thai Department of Highways network.Source: §1.3 · p. 4
• The model represents the nine geophone readings as an offset-indexed attention sequence, uses thicknesses to modulate basin tokens, and decodes both strains directly without layer moduli as inputs.Five routine basin indices are auxiliary tokens, and two strain queries decode the targets.Source: §1.3 · p. 4
• Measured and simulated strain labels are harmonised under the same physics, and the combined surrogate-and-field objective addresses domain shift.The study also resamples simulations over field structure space to examine coverage and deploys the trained model for fast per-drop evaluation.Source: §1.3 · pp. 4–5

2.1 The measurement and the design decision it feeds

• Nine geophones at radial offsets of 0–1800 mm record peak surface deflections that form the basin, the structural signal returned by the survey.The near basin is dominated by upper layers and the outer geophones by the subgrade; neither is observed directly.Source: §2.1 · p. 5
• Design uses asphalt-bottom tensile and subgrade-top compressive strains to estimate allowable repetitions, remaining life, overlay thickness, and rehabilitation ranking.Every decision inherits error in the two strains.Source: §2.1 · p. 5

2.2 The conventional two-step inference

• The conventional route iteratively backcalculates layer moduli to fit the basin, then uses those moduli in layered-elastic analysis to compute strains.The inverse mapping is non-injective and sensitive to thickness and deflection errors, seed values, convergence criteria, and operator judgment.Source: §2.2 · p. 5
• Figure 1 contrasts that conventional two-step route with the proposed single step from measured basin to rehabilitation decision.Its measurement panel identifies the impulse load, geophones, basin, and critical strain locations.Source: Figure 1 · p. 6, §2.2 · p. 7

3.1 Machine learning along the FWD evaluation chain

• Prior FWD machine-learning studies mainly replace modulus search with regressors, predict condition descriptors, or model scalar indices over time.The first strand retains the ill-posed inverse step; the others predict descriptors rather than the strains consumed by design transfer functions.Source: §3.1 · pp. 7–8
• 10.6–64.3% mean absolute percentage error on measured data versus 4.68% on fitted data was reported for Phutthananon et al.’s equations.The equations predict both strains from SCI, BDI, and BCI, but use a fixed form calibrated separately for each pavement archetype.Source: §3.1 · p. 8

3.2 Research gaps

• The review identifies gaps in direct strain prediction, offset-aware basin inputs, structure-conditioned interpretation, and validation on measured basins with whole roads held out.Only two of twelve studies reported a direct response, and no study combined measured-basin testing with a whole-road holdout.Source: §3.2 · p. 10
• No retrieved study used geophone offsets as inputs or weighted sensors case by case, and structure did not condition how the basin was read.Prior models used indices as flat inputs or the entire input, rather than alongside raw readings.Source: §3.2 · p. 10
• No study reports both measured-basin testing and whole-road holdout; the review notes a domain reversal and factorial-trained R² ≤0.14 on real routes.Phutthananon et al. tested measured basins but did not hold out whole roads.Source: §3.2 · p. 10
• DBFT predicts both design strains from the basin and thicknesses, tags readings by physical offset, and uses case-dependent sensor weighting.Thickness modulates each basin token, while the five practice indices enter the same representation.Source: §3.2 · pp. 8, 10, 11
• The combined objective trains one model on exact layered-elastic solutions and measured basins using labels placed on identical physics.The review distinguishes this response-prediction use of a shared objective from governing-equation constraints applied to deterioration models.Source: §3.2 · p. 11

4.1 Overview; 4.5 Extended layered-elastic surrogate

• The combined framework uses one physical scale for field and surrogate data, while the extended surrogate adds 11,744 solutions sampled over field backcalculated structure space, bringing the dataset to 14,174 rows.The extended set is a drop-in replacement for the factorial in surrogate-only and combined training; generating it takes minutes.Source: §4.1 · p. 11, §4.5 · p. 15, Table 5 · p. 15

4.2 Layered-elastic surrogate dataset

• The base surrogate contains 2,430 layered-elastic cases; inputs are nine deflections and three thicknesses, excluding unknown layer moduli, with separate training, validation, and test partitions.The targets span 81–433 με for AC tensile strain and 186–706 με for subgrade compressive strain.Source: §4.2 · p. 12, Table 4 · p. 12

4.3 Field dataset: the Thai DOH network

• The field corpus retains 7,651 points from 16 DOH routes and 96 sections; held-out-route sections fall within the network’s lower-to-middle centre-deflection range.Surveys span 17 provinces, and basins are rescaled to 707 kPa; 90% of retained drops were already within ±5% of that target pressure.Source: §4.3 · pp. 12–13

4.4 Harmonisation of field labels with surrogate physics

• The field labels are recomputed at the surrogate’s load and material assumptions; verification shows they preserve ELMOD ranking and scale while sharing identical physics with surrogate targets.The recomputation does not remove the ill-posed backcalculation that produced the field moduli; basin-fit RMS is retained as a label-quality weight.Source: §4.4 · pp. 13, 15

4.6 DBFT architecture

• DBFT encodes deflections with Fourier features of physical geophone offsets, embeds five basin indices alongside them, conditions basin tokens on thicknesses, and uses separate strain queries to read the fused memory.Physical-offset encoding respects non-uniform sensor spacing; thicknesses modulate how the basin is read, rather than acting as an additive offset.Source: §4.6 · p. 17

4.7 Training and the combined loss

• Training uses standardised targets, Huber loss, AdamW, and early stopping; joint training weights field points by basin-fit RMS and combines field and surrogate examples as complementary constraints.The compared modes are surrogate-only, field-only, and combined training at λf values of 0.25, 0.5, and 1.0.Source: §4.7 · pp. 18–19

4.8 Evaluation protocol

• Synthetic performance is evaluated on held-out surrogate data, while field performance uses 1,374 points from three entirely held-out routes rather than a random point split.Random forest, gradient boosting, and a 128–128 multilayer perceptron serve as baselines on the same inputs and splits.Source: §4.8 · p. 19

4.9 Interpretability protocol

• Interpretability combines per-target decoder cross-attention over 15 memory tokens with SHAP attribution over 12 field-measurable inputs.SHAP explanations cover 250 sampled held-out field points and 250 extended-surrogate test points.Source: Table 6 · p. 18, §4.9 · pp. 19–20

5.1 Characterisation of the surrogate-to-field domain gap

• Field basins decay more steeply than factorial basins: median D1800/D0 is 0.105 versus 0.309, and only about 45% of field points fall within factorial strain ranges.The factorial also omits 50 mm asphalt layers carried by 26% of field sections.Source: §5.1 · p. 20, Figure 6 · p. 22
• D1800–strain correlation reverses between domains: −0.50 and −0.48 in the factorial versus +0.10 and +0.33 in the field, where variable subgrade stiffness dominates.The factorial fixes subgrade modulus at 40 MPa; in the field, a larger D1800 indicates a softer subgrade and larger strain.Source: §5.1 · pp. 20, 23

5.2 Coverage of the field domain by the extended surrogate

• Field target-space coverage rises from 22.4% to 100%; median nearest-neighbour distance falls from 20.2 to 0.75 standard deviations.The extended surrogate also places 72.3% of field points within one standard deviation of a surrogate case, versus 0% for the factorial.Source: §5.2 · pp. 23–24, Figure 8 · p. 23
• With subgrade modulus varying from 30–900 MPa, extended-set D1800–strain correlations shift to −0.01 and +0.43, closer to the field’s +0.10 and +0.33.This addresses the functional mismatch as well as the coverage gaps.Source: §5.2 · p. 24

5.3 Accuracy on the layered-elastic surrogate

• 0.40% MAPE for AC tensile strain and 0.30% for subgrade compressive strain; DBFT halves the strongest baseline’s RMSE on both factorial targets.On the same 2,430-case set, the closed-form equations reached RMSE = 12.57 με and 31.59 με for the two criteria.Source: §5.3 · p. 25, Table 7 · p. 25
• Removing the thickness branch inflates RMSE three- to fivefold, showing that the deflection basin alone does not determine critical strains.FiLM helps the AC query and index tokens help the subgrade query, each at a slight cost to the other.Source: §5.3 · p. 27, Table 8

5.4 Learned attention

• Attention concentrates the AC query on SCI, AUPP and near-load deflections, while the subgrade query emphasizes D450–D600 and layer thickness.The AC query weights SCI at 0.23 and AUPP at 0.22.Source: §5.4 · p. 27

5.5.1 Failure of the surrogate-only model

• R² falls to −0.045 for AC and 0.141 for subgrade when the factorial-trained model is applied unchanged to held-out field routes.It regresses toward the factorial mean, producing a nearly constant prediction band.Source: Figure 14 · p. 28, §5.5.1 · p. 28

5.5.2 Combined loss on the factorial surrogate

• With the factorial surrogate, combined training raises held-out-route R² to 0.95–0.96 for AC and 0.85–0.86 for subgrade; field-only training loses the layered-elastic mapping.Field-only training has synthetic R² ≈ −2.5, and performance varies by less than 0.02 across λf = 0.25–1.0.Source: Table 9 · p. 28, §5.5.2 · pp. 28–29

5.5.3 Combined loss on the extended surrogate

• 0.78/0.50 to 0.96/0.88: adding field measurements improves held-out-route R² for AC/subgrade strains, while the deployed model retains synthetic R² ≥0.993.DBFT also lowers the best baseline’s field RMSE by 19% for fatigue and 11% for rutting.Source: Table 11 · p. 29, §5.5.3 · pp. 30–31

5.5.4 Sensitivity of field accuracy to label quality

• For AC, held-out-route R² reaches 0.987 at 17.9 με on the 692 points fitted to better than 2%.Error rises from 15.4 to 41.4 με across fit-quality strata; subgrade label quality cannot be separated from structural condition.Source: Table 12 · p. 31, §5.5.4 · p. 31

5.6 SHAP attribution of the deployed model

• Thicker asphalt and subbase are associated with lower predicted subgrade strain, while positive asphalt-thickness attribution for AC strain reflects this network’s thin surfacings over stiff cement-bound bases.The reported correlations are r = −0.57 for asphalt thickness and r = −0.84 for subbase thickness against subgrade strain.Source: §5.6 · p. 33
• AC attribution is near-load and curvature-driven, while subgrade attribution includes D900 and D1800, beyond the D900 limit of routine indices.The closed-form equations attain R² = 0.905 (RMSE 31.59 με) for subgrade, versus DBFT RMSE ≤1.24 με on both targets.Source: §5.6 · pp. 31, 33
• Field and surrogate importance profiles are nearly identical, consistent with one learned input–output mapping across both domains.The attribution analysis also aligns AC importance with D0 and D300, the sensors defining SCI.Source: §5.6 · pp. 33–34

6.1 Why the combined loss works

• The surrogate anchors the model to exact layered-elastic theory, while the field term extends it to real basins, structures and backcalculation-consistent responses.Field-only training has synthetic R² ≈ −2.5; once the surrogate covers field structures, the best field weighting is λf = 1.0.Source: §6.1 · p. 34

6.2 Evaluation honesty and spatial structure

• Entirely held-out routes yield R² of 0.88–0.96; route-level subgrade R² is 0.80–0.84, and random point splits would overstate field performance.The discussion suggests fine-tuning on backcalculated sections of a new route.Source: §6.2 · p. 34

6.3 Practical implications

• A fatigue screening model with R² ≈0.78 can be obtained from simulation alone when its parameter ranges match the local network’s structures.The discussion recommends adding measured data as it accumulates.Source: §6.3 · pp. 34–35

6.4 What the field validation establishes

• Field validation establishes agreement with the conventional pipeline’s backcalculated strain on unseen roads, not accuracy against true strain.No instrumented sections were available; layered-elastic idealisation bias and modulus non-uniqueness remain outside the test.Source: §6.4 · p. 35

6.5 Limitations

• Linear load normalisation neglects nonlinear subgrade response; clustered drop pressures support, but do not eliminate, this approximation.Temperature is not an explicit input, and uncertainty quantification and cross-device transfer remain open extensions.Source: §6.5 · p. 35

7.1 Evaluation at the moment of the drop

• 1.6 ms returns both strains on one CPU thread, and interpretation no longer constrains survey progress, enabling on-site resampling and full-resolution archive review.Batch evaluation takes 0.15 ms per drop; an attribution adds two to three seconds.Source: §7.1 · pp. 35–36

7.2 Implementation

• Four access modes use the same model weights and return the same strains and attribution, with physically invalid points refused rather than silently evaluated.The model is available through a command-line program, a service, an interactive application, and a hosted browser version.Source: §7.2 · p. 36

7.3 Towards an unattended pavement investigation system

• An unattended investigation chain still needs layer thicknesses from records or coring and confidence estimates to refer extrapolating sections for engineering judgement.The paper identifies ground-penetrating radar or an equivalent vehicle-mounted system as a way to supply thicknesses.Source: §7.3 · p. 36

• The 0.22 M-parameter DBFT predicts asphalt-bottom tensile strain and subgrade-top compressive strain from the nine-geophone basin and layer thicknesses, without inference-time backcalculation.Training used 14,174 exact layered-elastic solutions and 7,651 field measurements; three survey routes were withheld entirely.Source: §8 · pp. 36–37
• R² ≥0.9998 at RMSE ≤1.24 με makes DBFT a lossless substitute for the exact two-step forward computation and improves by an order of magnitude on published index equations.The reported error is below FWD measurement reproducibility, and an ablation found the thickness-fusion branch indispensable.Source: §8 · p. 37
• R² = 0.964 (εac) and 0.879 (εsg) on held-out routes, while retaining R² ≥0.993 on exact theory, with the combined objective.The factorial-trained model alone scored R² ≤0.14 on measured basins; conventional pooled-data baselines reached at best 0.945 and 0.847.Source: §8 · p. 37
• A field-matched surrogate reaches R² = 0.78 for fatigue without measured data in the loss; field measurements raise performance from 0.78 to 0.96.The authors identify coverage of the field structure space as the dominant component of the simulation-to-field gap.Source: §8 · p. 37
• Attention and SHAP recover recognised pavement mechanics, including the use of D1200–D1800, a range no routine basin index reaches.The AC criterion draws on D0 and D300; the subgrade criterion uses the whole basin with the physically correct outer-sensor sign.Source: §8 · p. 37
• 1.6 ms returns both strains on one CPU thread; batch evaluation takes 0.15 ms per drop, and attribution adds a few seconds when requested.The authors report that survey interpretation no longer constrains progress and that archives can be re-read at full resolution in minutes.Source: §8 · p. 37
• Field validation establishes agreement with ELMOD backcalculation-consistent mechanics, not gauge-measured strain; instrumented-section validation remains the definitive test.Temperature is represented only implicitly through its effect on the measured basin, and thicknesses currently come from records or coring.Source: §8 · p. 38

• Sompote Youwai covered conceptualization, methods, software, analysis, investigation, data curation, validation, resources, administration, visualization, and writing; Chana Phutthananon contributed methodology and visualization, and Warat Kongkitkul supervision and writing review.Source: CRediT authorship contribution statement · p. 38

• The public web application evaluates basins interactively and returns local SHAP attribution without installation.Source: Data and code availability · p. 38
• The application source, deployed weights and scalers, command-line interface, and REST API are available from the project repository.Source: Data and code availability · p. 38
• The DBFT repository includes the model, training pipeline, surrogate generator, configuration weights, datasets, field extract, figures, tables, and pseudocode; raw DOH reports are not redistributed.Source: Data and code availability · p. 38

Appendix A. DBFT formulation

• Fourier features encode each deflection’s physical geophone offset, making the model defined for sensor arrays absent from training.The offset features use frequencies log-spaced over 1–2,000 mm.Source: Appendix A · p. 41
• Five basin indices—SCI, BDI, BCI, AREA, and AUPP—are embedded alongside the raw deflection readings as individually addressable tokens.Each index receives a learned type embedding.Source: Appendix A · p. 41
• A structure code derived from layer thicknesses supplies scale and shift parameters that modulate each basin token.The structure therefore changes how the basin is read rather than adding a constant to the result.Source: Appendix A · pp. 41–42
• Three transformer encoder layers process the modulated tokens, and a projected structure code is appended to the memory attended by two target-specific decoder queries.The queries correspond to the AC and subgrade strains.Source: Appendix A · p. 42
• Second-layer cross-attention weights, averaged over four heads, provide each target’s attribution across the 15 memory tokens directly from the computation.The reported attribution requires no surrogate explainer.Source: Appendix A · p. 42
• The forward pass uses nine deflections and three layer thicknesses, with no layer moduli, and returns both strains in microstrain.Inputs are standardised before computation; outputs are inverse-transformed after Eq. (A.14).Source: Appendix A · pp. 41, 43

Appendix B. Workflow pseudocode

• The forward pass maps nine standardised deflections and three standardised thicknesses to AC and subgrade strain predictions.It embeds basin readings and indices, modulates them using a thickness-derived structure code, then encodes, fuses, and decodes the tokens.Source: Appendix B · p. 43
• Routes 22, 2334, and 2378 are held out entirely for field testing, while the surrogate set uses a random 70/15/15 train-validation-test split.Field validation samples 12% of the remaining field points.Source: Appendix B · p. 43
• Training pools surrogate rows at weight 1 with field rows weighted by λ_f times a basin-fit-residual weight, then optimises weighted Huber loss across targets.The row weight is w_i = 1 / (1 + RMS_i / 5).Source: Appendix B · pp. 43–44
• The combined validation loss selects the best model with 120-epoch patience, which is evaluated using R², RMSE, and MAE on held-out routes and the synthetic test set.The pseudocode specifies a maximum of 600 training epochs.Source: Appendix B · p. 44