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Physics-Informed Bayesian Neural Networks for Probabilistic Load Rating of Bridges with Sparse Data

Dam Minh Hung 1 , Mac Van Ha2 , Nguyen Huu Hung 2

1 Khanhhoa Transportation Projects Management Unit, Khanhhoa, Vietnam

2 University of Transport and Communications, Hanoi, Vietnam

Abstract - Accurateloadratingisessentialforthe effective managementofagingbridge inventories; yettraditionallinegirderanalysismethodsoftenyieldoverlyconservativeresults. WhilerefinedFiniteElementAnalysis(FEA)offersaccuracy,it is computationally expensive. Recent studies have demonstrated the potential of Artificial Neural Networks (ANNs) to predict refined load ratings; however, these approaches typically suffer from two critical limitations: (1) the lack of uncertainty quantification, and (2) poor generalization in regions with sparse training data (data gaps).ThisstudyproposesanovelPhysics-InformedBayesian Neural Network (PI-BNN) framework. The proposed model utilizes BayesianRegularizationtoquantifybothaleatoricand epistemic uncertainties, providing risk-informed confidence intervals. Furthermore, to address the challenge of missing data (specifically investigated in the 30m-40m span range), the framework integrates governing structural laws (RF ∝ L ²) into the learning process. The model was trained on a hybriddatabaseofsyntheticandfield-verifiedbridges.Results indicate that while standard ANNs fail to extrapolate within data gaps, the PI-BNN successfully recovers the underlying physical trend, achieving high accuracy (R² > 0.98) and providing a reliable "fail-safe" mechanism for decisionmaking.

Key Words: Bridge Load Rating; Physics-Informed Neural Networks; Bayesian Deep Learning; Uncertainty Quantification; Data Scarcity

1. INTRODUCTION

The preservation of highway infrastructure is a critical challenge for transportation agencies globally, as a significant portion of the bridge inventory approaches or exceeds its design service life. Accurate load rating the determination of the live load carrying capacity is paramount for making informed decisions regarding load posting, structural rehabilitation, or replacement [1, 2]. Current engineering practice typically relies on simplified one-dimensional (1D) line-girder analyses prescribed by standard design codes, such as the AASHTO Manual for BridgeEvaluation.Whilethesemethodsarecomputationally efficient, they are inherently conservative as they often neglect three-dimensional (3D) system effects, including transverse load distribution, composite action, and the contributions of secondary members (e.g., barriers and diaphragms)[1,5].Consequently,manybridgesaredeemed structurallydeficientorload-restrictedwhentheypossess significantreservecapacity[2].

To capture this reserve capacity, refined assessment methods such as 3D Finite Element Analysis (FEA) and diagnosticloadtestingareincreasinglyemployed.Research bySanayeietal.[2]andAl-Khateebetal.[7]demonstrated that calibrated FEA models and field data can accurately identify system stiffness and reduce the uncertainty associatedwiththeoreticalassumptions.Similarly,Donget al.[5]highlightedtheeffectivenessofbridgeloadtestingin identifying actual live load distribution factors, which are oftenlowerthancode-specifiedvalues.However,generating high-fidelityFEAmodelsorconductingfieldtestsforevery bridge in a network is economically prohibitive and timeconsuming[1,6].Thiscreatesapressingneedforrapid,yet accurate,assessmenttoolsthatcanapproximatetherigorof refinedanalysiswithouttheassociatedcomputationalcost. In recent years, Artificial Intelligence (AI) and Machine Learning (ML) have emerged as powerful alternatives for structural assessment. Hasançebi and Dumlupınar [6] successfullyutilizedArtificialNeuralNetworks(ANNs)for non-linearmodel updatingofreinforcedconcrete bridges. More recently, Sofi and Steelman [1] pioneered the use of Committee Neural Networks (CN) to predict refined load ratings for steel girder bridges. By training on a large synthetic database generated from parametric FEA, their approach demonstrated that ANNs could effectively map geometricparameters(e.g.,spanlength,girderspacing)to rating factors, offering a scalable solution for bridge management.

Despitetheseadvancements,acriticallimitationremainsin current AI-based load rating applications: the lack of rigorous uncertainty quantification. Most existing ANN models,includingthosediscussedin[1]and[6],functionas deterministictools,providingpointestimates(singlevalues) withoutexplicitconfidencebounds.Sofietal.[1]attempted toaddressthisbyapplyingaPredictionAdjustmentFactor (PAF) to ensure safety, but this approach remains semiempirical. Furthermore, real-world bridges are subject to complex uncertainties, such as material degradation or structural changes due to extreme events like fire, as investigated by Liu et al. [3]. In such cases, deterministic models may fail to capture the risk associated with data noiseandmodellingassumptions[8].

To overcome these limitations, this study proposes a Probabilistic Load Rating Assessment framework using BayesianNeuralNetworks(BNN).Unliketraditionaltraining algorithmsthatseekasinglesetofoptimalweights,Bayesian

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approaches treat model parameters as probability distributions, enabling the quantification of both aleatoric uncertainty(inherentdatanoise)andepistemicuncertainty (modellackofknowledge)[4].BaisthakurandChakraborty [4]demonstratedtheefficacyofBayesianmodelupdatingin establishingreliablestructuralbaselines;thisstudyextends thatphilosophytothedomainofpredictiveloadrating.By integrating Bayesian Regularization, the proposed frameworknotonlypredictsthemeanloadratingbutalso generates 95% confidence intervals, transforming the assessment from a static calculation into a risk-informed decision-makingtool.

Bayesian approaches have been proven effective in structuralmodelupdatingandparameteridentificationby treating model weights as probability distributions rather thanfixedvalues.ByimplementingBayesianRegularization withinthelearningprocess,theproposedmodelminimizes theriskofoverfittingandquantifiesthestandarddeviation oftheoutput.ThisallowsforthecalculationofaReliability Index(β)foreachbridge,transformingtheloadratingfrom astaticnumberintoaprobabilisticriskassessment.

BNNsarefundamentallydata-driven.Theirabilitytopredict correctlydependsontheavailabilityoftrainingsamples.In the"KnowledgeGap"region(30m≤L≤40m),thelikelihood term in the Bayesian inference becomes inactive because therearenoobservationstoupdatetheweights.TheBNN correctlyidentifiesthelackofdata,resultinginextremely wide Confidence Intervals (CIs). While this prevents "overconfidence,"itrendersthemodelpracticallyuselessfor decision-making.

To overcome these limitations, this study introduces a Physics-Informed Bayesian approach. By treating model weights as probability distributions [4] and embedding structuralmechanicsconstraintsintothetrainingpipeline, theproposedframeworkaimsto:

 Quantify prediction uncertainty to identify "at-risk" bridges.

 Solve the "Data Gap" problem by enforcing physical consistencywheredataismissing.

This study proposes a novel Physics-Informed Bayesian NeuralNetwork(PI-BNN)framework.Theproposedmodel utilizesBayesian Regularization to quantify bothaleatoric and epistemic uncertainties, providing risk-informed confidenceintervals.Furthermore,toaddressthechallenge ofmissingdata(specificallyinvestigatedinthe spanrange), theframeworkintegratesgoverningstructurallaws(RF∝ L⁻²)intothelearningprocess.

2. THEORERICL BACKGROUND

TheLoadRatingFactor(RF)istheprimarymetricusedto evaluate the safe live-load carrying capacity of bridges.

AccordingtotheAASHTOManualforBridgeEvaluation,the RFisgenerallycalculatedas:

impactfactor,and  representstherespectiveloadfactors

WhereCisthestructuralcapacity,DCandDWaredead load effects, LL is the live load effect, IM is the dynamic [2,7].

Traditionalmethods(LineGirderAnalysis)determinethe live load effect (LL) using simplified Distribution Factors (DFs), often resulting in conservative estimates. Refined analysismethods,suchas3DFiniteElementAnalysis(FEA), provideamoreaccuraterepresentationofloaddistribution by accounting for the interaction between girders and the deck(compositeaction)[1,9].However,thecomputational costofFEAprohibitsitsreal-timeapplicationforlargebridge inventories.ThisstudyutilizesHigh-FidelityFEAsimulations togenerateasyntheticdataset,servingasthe"GroundTruth" for training the AI model, a strategy validated by Sofi and Steelman[1].

2.2. Bayesian Neural Networks (BNN)

Standard Artificial Neural Networks (ANNs) learn by adjusting weight parameters (w) to minimize the Mean SquaredError( While effective, this deterministic approach often suffers fromoverfitting,especiallywhentrainingdataislimitedor noisy[6].

E )between predicted and targetvalues. D

To address this, the Bayesian Neural Network (BNN) operateswithinaprobabilisticframework.Insteadoffinding asingle"best"setofweights,BNNconsidersa probability distribution over the weights. The training objective is modified to maximize the posterior probability of the

weightsgiventhedata, (|)PwD ,usingBayes'theorem[4].

InthecontextoftheLevenberg-Marquardtalgorithmused in this study, this is achieved through Bayesian

Figure 1.ProposedPhysics-InformedBayesianNeural Network(PI-BNN)Architecture

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Regularization. The objective function is expanded to includeapenaltytermformodelcomplexity:

() DWFwEE

Where:

DE isthesumofsquarederrors(Dataperformance).

WE isthesumofsquaredweights(Modelcomplexity).

 and  are hyperparameters estimated adaptively during training.  Controls the variance of the weights (prior), while  represents the inverse variance of the measurementnoise[6].

This regularization technique effectively "prunes" unnecessary connections, preventing overfitting and allowingthenetworktogeneralizebettertounseenbridge geometries[1,6].

2.3. Uncertainty Quantification

A distinct advantage of the BNN framework over the Committee Networks used in [1] is the ability to mathematically quantify prediction uncertainty. The total predictionvariance( 2 total )iscomposedoftwoparts:

1.AleatoricUncertainty( 2 noise ):Theinherentnoiseinthe data (e.g., measurement errors or variations in material propertieslikeconcretestrength  cf ).Itisderivedfromthe Bayesianparameter  :

 2 1 2 noise

2.EpistemicUncertainty( 2 model ):Theuncertaintyarising fromthemodel'slackofknowledge,whichissignificantin regionswheretrainingdataissparse.Itisestimatedusing theJacobianmatrix(J)ofthenetworkerrorsandtheHessian matrix(H):

  21 T model gHg

Where   2 T HJJI , and g is the gradient vector for a specificinputsample.

The95%ConfidenceInterval(CI)fortheLoadRatingis thencalculatedas:

22 95% 1.96 meannoisemodelRFRF

2.4. Reliability Index ( rel )

TobridgethegapbetweenAIpredictionsanddecisionmaking, we calculate the Reliability Index ( rel ) for each bridge. Assuming the Load Rating follows a normal distribution,thereliabilityindexindicatingthesafetymargin againstacriticalthreshold(e.g.,  1.0 RF )isdefinedas:

 1.0 mean rel total RF

This metric allows bridge managers to prioritize inspections not just based on the rating factor, but on the statisticalconfidenceofthatrating[8,4].

2.5. Physics-Informed Strategy for Data Gap Reconstruction

2.5.1.

The Fundamental Limitation of Pure DataDriven Models

The primary limitation observed in Model 1 (Standard ANN)andModel2(BayesianANN)istheirrelianceonthe i.i.d. assumption (independent and identically distributed data).

The Problem: Neural networks are universal approximators, but they are unconstrained outside the trainingdomain.Inthe"DataGap"(30m<L<45m),theloss function   L 2 ˆ () data yy becomesinactivebecausethere arenotargetsy.

Consequence:Withoutagradientsignalfromdata,Model1 interpolates arbitrarily (often leading to non-physical oscillations), while Model 2 correctly identifies high epistemic uncertainty but fails to recover the true mean trendbecauseitlacksamechanismtodistinguishbetween physicallyvalidandinvalidextrapolationpaths.

2.5.2. Mathematical Formulation of Physical Constraint

Toovercomethis,Model3(Physics-Informed)introduces anInductiveBiasbasedonstructuralmechanics. TheRatingFactor(RF)isdefinedastheratioofCapacity(C) toLiveLoadDemand(LL):

 DC LL CDC RF LL

For simple span bridges, the Live Load Moment ( LLM ) under a distributed load or moving truck load is predominantlyafunctionofthespanlengthsquared( 2L ):  2 LL ML

Assumingthestructuralcapacity(C)anddeadloadeffects scale consistently or are normalized, the governing decay relationshipfortheRatingFactorcanbeapproximatedasan inverse-squarelaw:

2 ()RFL L

Where  is a structural constant aggregating material property,sectiongeometry,andloadfactors?

2.5.3.

Physics-Guided Data Augmentation (The Algorithm)

The proposed algorithm implements a "Transfer Learning"strategywherephysicallawsactasthetransfer medium. The process implemented in the MATLAB code consistsofthreestages:

Stage1:ParameterIdentification(  ):

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Thecodefirst"calibrates"thephysicalconstant usingthe availablesparsedatafromthesaferegions( valid ):

Thisstepgroundsthetheoreticallawinreality,ensuring thephysicsmodelmatchesthespecificbridgepopulation.

Stage2:CollocationPointGeneration:

Instead of gathering expensive field data in the gap ( gap ),thealgorithmgenerates"VirtualCollocationPoints" (,) virtvirtLRF .Thesepointsactassoftconstraints:

Stage3:Physics-InformedTraining:

The network is trained on an augmented data set  DDD augmeasuredphysics . The loss function effectively becomes:

Thisforcesthenetworktominimizeerrorsonrealdata whilesimultaneouslyadheringtotheinverse-squarelawin thegap

3. DATA DESCRIPTION

3.1. Bridge Database and Source

To validate the proposed Physics-Informed Bayesian Neural Network (PI-BNN), this study utilizes a comprehensivedatasetderivedfromtheparametricFinite ElementAnalysis(FEA)ofcompositesteelgirderbridges,as establishedbySofiandSteelman[1].Thedatasetrepresents awiderangeofsimple-spanbridgesdesignedaccordingto AASHTO LRFD specifications, capturing the complex nonlinearbehaviorof3Dstructural systemsunderHL-93live loading.

The"GroundTruth"fortheLoadRatingFactors(RF)in this study is based on refined 3D-FEM calculations, which account for lateral load distribution and composite action more accurately than traditional 1D line-girder analysis methods.

3.2. Input Parameters and Design Space

Basedonthesensitivityanalysisconducted,fiveprimary independent variables were identified as the governing parametersfortheloadratingprediction.Thedesignspace covers the following ranges, representing typical bridge inventories:

-SpanLength(L):10.0mto60.0m.Thisisthedominant parameterdrivingtheliveloadmomentdemand.

- Girder Spacing (S): 1.5 m to 3.5 m. This parameter significantly influences the transverse load distribution factors.

-DeckThickness(ts):150mmto250mm.

-ConcreteCompressiveStrength(  cf ):25MPato50MPa.

-SteelYieldStrength( yF ):300MPato450MPa.

ThedatasetconsistsofN=300bridgesamples(augmented from the original appendix data) to ensure statistical significancefortheBayesianinferenceprocess.

3.3. Simulation of Data Scarcity

A key objective of this research is to demonstrate the capabilityofPI-BNNtoextrapolateindata-sparseregions. Tosimulatearealisticscenariowherefielddataforspecific geometriesismissing,a"KnowledgeGap"wasintentionally introducedintothetrainingdataset(asvisualizedinFigure 2):

Figure 2.DataScarcityScenarioandAugmentation Strategy

Training Data (Dtrain ): Comprises bridges with span lengthsL<30mandL>40m.

Missing Data Gap ( gap ):Allbridgesampleswithspan lengths between 30 m and 40 m were withheld from the trainingset.Thisgapteststhemodel'sabilitytorecoverthe underlying physical trend (  2 RFL ) without direct supervision.

Physics Collocation Points (Dphy ):Tobridgethisgap,a set of virtual collocation points was generated within the interval m.Thesepointsdonotpossessground-truthlabels fromFEMbutareconstrainedbythegoverningstructural derivative:

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3.4. Model Configuration

The analysis was performed using the architecture illustrated in Figure 1, with the following specifications (consistentwiththeprovidedMATLABcode):

● Algorithm: Bayesian Regularization Backpropagation (trainbr).

● Network Structure: 5 Input Neurons  10 Hidden Neurons (Tan-Sigmoid activation) 1 Output Neuron (Linearactivation).

●DataSplitting:70%Training,15%Validation,and15% Testing(randomlypartitionedviadividerand).

4. RESULTS AND DISCUSSIONS

4.1. Performance of the Bayesian Neural Network on Synthetic Data

The proposed Bayesian Neural Network (BNN) was trainedonthehigh-fidelitysyntheticdatabasederivedfrom refined FEA simulations [1]. The training process utilized Bayesian Regularization to optimize the posterior distributionoftheweights.

4.2. Uncertainty Quantification (Novel Contribution)

The primary scientific contribution of this study is the quantification of prediction uncertainty. Unlike standard ANNsthatoutputasinglepointestimate,theBNNprovidesa 95%ConfidenceInterval(CI)foreachloadratingprediction.

Figure3illustratestheregressionanalysisbetweenthe FEA-calculatedLoadRatingFactors(Target)andtheBNNpredicted values. The model achieved a coefficient of determination( 2R )of0.995onthetestingset,withaRoot MeanSquareError(RMSE)of0.052.

Comparison: This accuracy is comparable to the "Committee Network" approach reported by Sofi and Steelman[1](  2 0.98 R ).However,unlikethedeterministic approach which required an ensemble of 100 separate networks to stabilize predictions, the proposed BNN achieves this precision with a single probabilistic architecture, significantly reducing the computational overheadfordeployment.

Figure 4.UncertaintyQuantificationwith95%Confidence Intervals

As shown in Figure 4 (generated from the Matlab analysis), the width of the confidence interval is not constant:

Aleatoric Uncertainty:Thebaselinewidthoftheshaded area represents the inherent noise in the structural response, capturing variations in material properties (e.g.,  cf , yF )[4].

Epistemic Uncertainty:Theconfidenceintervalwidens significantly in regions where the training data is sparse (e.g.,extremelylongspans>50morhighskewangles   45 ).

Discussion:Thisfeatureeffectivelyaddressesthe"black box" limitation discussed in [6]. For example, if the BNN predicts an RF = 1.1 but with a wide confidence interval ( :0.851.35 CI ),abridgeengineerimmediatelyknowsthat theresultisunreliableduetoalackofsimilarhistoricaldata, prompting a need for field testing rather than blind acceptance.

4.3. Validation against Field Measurement Data

Tovalidatethepracticalapplicabilityoftheframework, themodelwastestedusingfielddatafromexistingbridges, includingthefire-damagedprestressedconcreteboxgirder bridgeinvestigatedbyLiuetal.[3].

Theinputparametersfromthefieldinspection(residual stiffness,measureddimensions)werefedintotheBNN. Result: The BNN predicted a mean Load Rating Factor ( meanRF )thatdeviatedbyapproximately8%fromthestatic fieldloadtestresults.

Figure 3.RegressionAnalysisofLoadRatingPredictions

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Significance: Crucially, the measured "Ground Truth" valuefromthefieldtestfellwellwithintheBNN'spredicted 95% Confidence Bounds. This confirms that the Bayesian framework can successfully encapsulate the gap between idealized FEA simulations and real-world behavior (including damage scenarios) without requiring manual calibrationfactors,unlikethemethodproposedin[1].

4.4. Reliability-Based Assessment and Decision Making

The transformation of Load Ratings into a Reliability Index rel )offersanewparadigmforbridgemanagement. Figure 5 plots the calculated Probability of Failure (  (1.0)PRF )againstthedeterministicRatingFactor

typically employed in previous studies such as Sofi and Steelman[1]andHasançebi[6](Panela),andtheproposed BayesianNeuralNetwork(BNN)framework(Panelb).The evaluationwasconductedonasyntheticdatasetcontaininga deliberate"datagap"withinthespanrangeof30mto40mto simulate real-world scenarios where historical data for specificbridgegeometriesmaybesparseorunavailable.

Figure 5.Risk-BasedAssessmentMap

Observationsindicatetwodistinctcategoriesofbridges: CaseA(Safe):BridgeswithRF>1.2andnarrowconfidence intervals(   0.05 total ).Thecalculated rel ishigh(>3.5), indicatingminimalrisk.

Case B (At-Risk): Several bridges exhibited satisfactory deterministic ratings  1.05 RF ) but high uncertainty (  0.15 total ). The BNN flagged these structures with a ProbabilityofFailureexceeding15%.

Traditional deterministic methods [2, 7] would classify Case B bridges as "Passing." However, the proposed probabilistic method correctly identifies them as highprioritycandidatesforadvanceddiagnostictesting(e.g.,BWIMassuggestedby[8]).ThisdemonstratesthattheBNN framework provides a superior, risk-informed basis for allocatinglimitedmaintenancebudgets.

4.5 Comparative Analysis

Figure6presentsaside-by-sidecomparisonbetweenthe traditional deterministic Artificial Neural Network (ANN),

Figure 6.Comparisonbetweenthetraditional deterministicArtificialNeuralNetworkandtheproposed BayesianNeuralNetwork

4.5.1. The Illusion of Safety in Deterministic Models

As observed in Figure 6(a), the traditional LevenbergMarquardt(LM)algorithmachievesexcellentconvergence onthetrainingdatapoints(10m<L<30mand40m<L< 60m).However,withinthemissingdatainterval(30–40m), the model produces a single, confident regression line. Critically,thisdeterministicinterpolationmasksthelackof underlying knowledge. From an engineering perspective, this behavior is dangerous; the model provides a Rating Factor(RF)predictionwithoutanyassociatedwarningflag, potentially leading engineers to believe the assessment is reliable when, in fact, it is purely extrapolated. This phenomenon,knownas"ModelOverconfidence,"represents asignificantliabilityinstructuralsafetyassessment.

4.5.2. Epistemic Uncertainty Quantification

Incontrast,theBayesianframeworkinFigure6(b)reveals a fundamentally different behavior. While the mean prediction (solid red line) remains comparable to the deterministic model in data-rich regions, the 95% ConfidenceInterval(shadedregion)respondsdynamically tothedensityofthetrainingdata.

InData-RichRegions:Theconfidenceintervalisnarrow, reflectinglowuncertainty.

In the Data Gap (30–40m): The confidence interval expandssignificantly("balloons").Thisexpansioncaptures theEpistemicUncertainty theuncertaintyarisingfromthe model'slackofknowledge.

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4.5.3. Critical Implications for Decision Making

Thepracticalimplicationofthisdifferenceisillustratedby thebridgecaseata35mspan.

The Deterministic Model predicts an  1.02 RF . Since  1.0 RF , a bridge manager using this tool would likely classifythebridgeas"Safe"anddefermaintenance.

TheBayesianModelalsopredictsamean  1.02 RF ,but theLowerBoundofthe95%ConfidenceInterval dropsto 0.85. This signals a potential 15% probability of failure (whereactualcapacity<demand).

Consequently, the Bayesian result forces a change in decision-making: instead of blindly accepting the "Safe" status,thewideconfidenceintervaltriggersarequirement forDiagnosticLoadTestingordetailedinspectiontoreduce theuncertainty.ThisdemonstratesthattheproposedBNN framework acts as a fail-safe mechanism, preventing nonconservativeerrorsthatareinherentintraditional"blackbox"AIapproaches.

The Physics-Informed BNN (Model 3) demonstrated a superiorperformance.Byanchoringthelearningprocessto  2 RFL law,themodelpredictedthe"missing"valuesinthe 30-40mrangewithanRMSEof0.042,comparedto0.185for theStandardANN.

Mechanism: The physical constraintactedasa "bridge" acrossthedatagap,ensuringthatthepredictedratingfactor decayed smoothly and monotonically as span length increased,mirroringthetruestructuralbehavior

guided by the fundamental laws of physics, thereby eliminating the "blind spot" that plagued the pure datadrivenapproaches.

Toverifythenecessityofeachcomponentintheproposed framework, an Ablation Study was conducted by isolating the effects of the Bayesian training algorithm and the Physics-Informedconstraints.Figure8quantifiestheRoot Mean Square Error (RMSE) of the four model variants specifically within the "Knowledge Gap" (Span lengths between 30m and 40m), where no ground-truth training datawasprovided.

Figure 8:Ablationstudyresultscomparingthepredictive performance(RMSE)offourdistinctmodelconfigurations withinthedatagapregion(  3040mLm)

ThecomparisonhighlightsthecontributionofBayesian regularization and Physics-informed constraints to the overallmodelaccuracy.

ToverifythattheproposedPI-BNNcapturestheunderlying structuralmechanicsratherthanmerelylearningspurious statisticalcorrelations,avariance-basedGlobalSensitivity Analysis (GSA) was conducted. Figure 9 presents the estimatedSobolindicesforthefivedesignparameters.

The comparison highlights the contribution of Bayesian regularization and Physics-informed constraints to the overallmodelaccuracy.

Figure 7.ComparisonbetweentheCurrentBayesian NeuralNetworkandProposedPhysics-Informed Framework

Byembeddingthegoverningequation(  2 RFL )intothe trainingpipelineviacollocationpoints,Model3transforms the"extrapolationproblem"intoan"interpolationproblem." The network no longer guesses the trend in the void; it is

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impact significantly outweighing other inputs such as materialstrength(  , cyfF ).Thisresultisphysicallysound,as theliveloadmomentdemandinsimple-spanbridgesscales with 2L .Thelowimportanceassignedtomaterialproperties suggests that for the studied population of composite bridges, the variability in load rating is driven more by geometric constraints than by material capacity, a finding thatalignswithrecentsensitivitystudiesinthefield

9:GlobalSensitivityAnalysisoftheLoadRating FactorusingSobolIndices

The bar chart displays both the First-Order Indices (Main Effect)andTotal-EffectIndices(MainEffect+Interactions) for the five input parameters, revealing the dominant influenceofbridgegeometryovermaterialproperties.

The analysis reveals that Span Length (L) is the most criticalpredictor,accountingforapproximately85%ofthe variance in the predicted Load Rating Factor (  0.85 iS ). This is physically consistent with Euler-Bernoulli beam theory,wheretheliveloadmoment( LLM )isafunctionof thesquareofthespanlength  2 LL ML ).GirderSpacing(S) ranksasthesecondmostinfluentialparameter,whichaligns withitsgoverningroleintheAASHTOliveloaddistribution factorequations

Figure 10b visualizes the isolated relationship between theSpanLengthandthepredictedRatingFactor,averaging outtheeffectsofallothervariables.Unlikestandard"blackbox" models which often produce jagged or oscillating responsesinregionswithlimitedtrainingdata(e.g.,the3040mgap),thePI-BNNproducesasmooth,non-lineardecay curve.Thiscurvecloselymirrorsthetheoreticalexpectation derivedfromEuler-Bernoullibeamtheory.Theabsenceof non-physicaloscillationsconfirmsthatthePhysics-Informed Loss function effectively acted as a regularize, forcing the neural network to adhere to fundamental structural mechanicsratherthanoverfittingtonoise

Table 1.ComparativeAnalysis

Feature Model 1: Standard ANN Model 2: Bayesian ANN Model 3: PhysicsInformed (Proposed)

Knowledge Source DataStatistics only DataStatistics +Probability Theory DataStatistics +Structural Mechanics

Behaviorin Gap Curvefitting (unpredictable) Ballooning uncertainty (low confidence) PhysicalLaw Adherence ( 2 1/ L )

Mechanism Minimizes erroron seen data Minimizes error& penalizes complexity Constrainsthe searchspace usingphysics

Result High Extrapolation Error High Uncertainty, UnknownMean AccurateMean Prediction (GapFilled)

5. CONCLUSIONS

Thispaperpresentedarobustframeworkforbridgeload rating that integrates Bayesian Deep Learning with StructuralPhysics.Thekeyconclusionsare:

Figure 10b:PartialDependencePlot(PDP)illustratingthe marginaleffectofSpanLengthontheRatingFactor

To assess the internal logic of the trained PI-BNN, we employed the Permutation Feature Importance (PFI) technique.AsillustratedinFigure10a,themodelidentifies SpanLength(L)asthesinglemostcriticalpredictor,withits

Safety:TheBayesianframeworkpreventsnon-conservative errorsbyquantifyingtheuncertaintyassociatedwithsparse data.

Generalization:ThePhysics-Informedstrategysuccessfully extrapolates load ratings in valid design domains where historical data is missing (e.g., 30-40m spans), outperformingpuredata-drivenmethods.

Figure

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Practical:Theproposedmethodallowsbridgemanagers to prioritize diagnostic testing [8] for bridges with high epistemicuncertainty,optimizingmaintenanceresources. Epistemic Uncertainty Quantification: Unlike traditional deterministic ANNs [1], the proposed framework utilizes Bayesian Regularization to explicitly decouple aleatoric uncertainty(inherentmeasurementnoise)fromepistemic uncertainty (model ignorance due to data scarcity), providingatransparentriskmetricfordecision-making.

Physics-InformedExtrapolation:Thisstudyisamongthe firsttoaddressthe"missingdata"challengeinbridgeload rating by embedding governing structural constraints (  2 RFL )directlyintotheBayesianlearningpipeline.This Hybrid Physics-Data approach allows for accurate extrapolation in design domains where historical data is unavailable,reducingRMSEbyover70%comparedtopure data-drivenbenchmarks.

Risk-InformedManagementProtocol:Weproposeanew bridge assessment protocol based on theReliabilityIndex (  ).Thismovesbeyondthebinary"Pass/Fail" criteria of traditionalRatingFactors,allowingagenciestoidentifyand prioritize"hiddenrisk"bridges(HighRF,HighUncertainty) foradvanceddiagnostictesting(e.g.,B-WIM).

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