Physics-Informed Framework for Impact Identification in Aerospace Composites

Ph ysics-Inf ormed Frame w ork f or Impact Identiﬁcation in Aer ospace Composites Nat ´ alia Ribeiro Marinho 1 , Richar d Loendersloot 1 , Jan Willem Wiegman 2 , Frank Grooteman 2 , and Tiedo Tinga 1 Abstract Reliable impact identiﬁcation in aerospace composite str uctures is essential for effectiv e str uctural health monitor ing, as low-v elocity impacts ma y cause inter nal damage without visible surf ace indications . Howe v er , estimating impact energy from measured responses remains challenging because the inv erse problem is ill-posed under sparse, noisy sensor data, nonlinear structural response, and missing e xcitation parameters . T o address these limitations, this paper introduces a nov el physics-inf ormed impact identiﬁcation (Phy-ID) frame work. The proposed method integrates obser v ational, inductiv e, and lear ning biases to combine ph ysical kno wledge with data-dr iven inference in a uniﬁed modelling strategy , achieving physically consistent and numerically stable impact identiﬁcation. The ph ysics-informed approach structures the input space using physics-based energy indicators , constrains admissible solutions via architectural design, and enf orces governing relations via hybrid loss formulations. T ogether , these mechanisms limit non-ph ysical solutions and stabilise inf erence under deg raded measurement conditions . A disjoint inf erence f ormulation is used as a representativ e use case to demonstrate the framew ork capabilities, in which impact velocity and impactor mass are inferred through decoupled surrogate models, and impact energy is computed by enforcing kinetic energy consistency . Experimental ev aluations sho w mean absolute percentage errors below 8% for inf erred impact v elocity and impactor mass and below 10% f or impact energy . Additional analyses conﬁr m stable performance under reduced data av ailability and increased measurement noise , as w ell as generalisation f or out-of-distribution cases across pristine and damaged regimes when damaged responses are included in training. These results indicate that the systematic integration of ph ysics-informed biases enables reliable, ph ysically consistent, and data-efﬁcient impact identiﬁcation, highlighting the potential of the approach f or practical monitoring systems. Ke ywor ds Impact Identiﬁcation, Ph ysics-informed Machine Learning, Scientiﬁc Machine Lear ning, Aerospace Composites, System Identiﬁcation, Structural Health Monitoring Introduction Aerospace composite structures are highly valued for their strength-to-weight ratio, yet the y remain vulnerable to unde- tected internal impact damage that can sev erely compromise structural integrity and operational safety 1–3 . T o address this vulnerability , Structural Health Monitoring (SHM) systems hav e been de veloped, combining passive sensing technolo- gies with adv anced data processing algorithms to detect, locate, and infer the se verity of impact ev ents in real time. Among these tasks, estimating impact ener gy is crucial, as it directly correlates with damage initiation, residual strength degradation, and maintenance thresholds that guide opera- tional decisions. Howe v er , estimating the magnitude of an impact based on measured structural responses is a complex inv erse problem that is not yet fully understood. In practical SHM implemen- tations, the av ailable data are often limited to sparse, and noisy measurements, with direct measurements of e xcitation parameters, such as impact force history , impactor mass, and impact v elocity , frequently absent. As a result, the rela- tionship between sensor signals and impact se verity is non- unique, highly sensitiv e to noise, and inﬂuenced by factors such as structural conﬁguration, sensor placement, and en vi- ronmental conditions 4–6 . These limitations are intensiﬁed under realistic operating conditions, where nonlinear struc- tural beha viour , w av e attenuation, and complex structural geometries further degrade numerical stability and challenge the physical credibility of inferred solutions. Existing methodologies for estimating impact energy can be categorised into three primary groups: physics-based, data-driv en, and probabilistic approaches. Physics-based approaches rely on analytical or numerical models, such as ﬁnite-element formulations 7–9 , modal representations 10,11 , guided-wa ve propagation models 12,13 , and analytical solu- tions 14–18 , to reconstruct impact parameters from measured responses. These methods preserve physical consistency but require detailed knowledge of material properties, boundary conditions, and structural conﬁguration, and they incur high 1 Engineering T echnology F aculty , Depar tment of Mechanics of Solids, Surfaces and Systems, Dynamics Based Maintenance (DBM) Group , University of T wente (UT), Enschede, NL 2 Depar tment of Aerospace V ehicles Integrity and Lif e Cycle Suppor t (A VIL), Ro yal Netherlands Aerospace Centre (NLR), Marknesse, NL Corresponding author: Nat ´ alia Ribeiro Marinho Email: n.ribeiromarinho@utwente.nl 2 computational cost. As a result, their applicability in real- time SHM settings is limited. Data-driv en approaches 19–23 employ machine learning techniques to establish direct mappings from sensor mea- surements to impact energy , achie ving efﬁcient representa- tions of nonlinear relationships. Their performance, howe ver , depends strongly on the representativeness of the training data and shows sensitivity to noise and environmental vari- ability . Moreover , the absence of explicit physical constraints limits interpretability and weakens extrapolation capability , which constrains reliability in safety-critical aerospace set- tings. Probabilistic strategies mer ge physical modelling with data-driv en inference, often incorporating uncertainty quan- tiﬁcation to improv e stability 5,24–26 . While these methods can lead to impro ved performance under controlled conditions, they frequently introduce physical information only implic- itly or locally , with model classes and optimisation objectives often remaining unconstrained. As a result, improvements in predictiv e accuracy do not necessarily translate into physically admissible solutions, which limits interpretability and diagnostic value be yond point estimates. These issues directly af fect generalisation performance and reduce con- ﬁdence in outputs under realistic operational conditions, where en vironmental v ariability and measurement noise are unav oidable. Thus, addressing the limitations identiﬁed in the current body of kno wledge requires strate gies that balance predicti ve performance with physical interpretability and robustness, ensuring that inferred impact energies remain aligned with governing relations while applicable under practical monitoring conditions. In this conte xt, Physics-Informed Machine Learning (PIML) 27–29 offers a systematic approach by embedding physical knowledge within machine learning algorithms. In the PIML framew ork, physical insight is integrated through three complementary mechanisms. First, observa- tional bias shapes the choice of inputs so that the data reﬂect physically meaningful information. Second, inductive bias embeds prior domain knowledge assumptions directly into the model architecture, thereby restricting the range of admissible solutions. Finally , learning bias incorporates gov- erning relations into the optimisation objectiv e to regularise training through explicit constituti ve constraints. Recent adv ances demonstrate that explicitly incorporating physical constraints into learning models improv es accurac y , tolerance to operational variability and performance under distribution shifts. Rautela et al. 30 combined physical mod- els and deep learning, achieving improved behaviour across excitation frequencies and noise lev els compared with purely data-driv en methods. Latent force models further inte grate physics-based representations with probabilistic priors on unknown inputs, enabling robust estimation of structural responses under uncertain loading, as demonstrated in vir - tual sensing applications for offshore wind turbine struc- tures 31 . In addition, multi-ﬁdelity physics-informed models improv e adaptability by integrating low- and high-ﬁdelity simulations, which enhances performance across operating regimes 32 . Despite growing interest in physics-informed learning, its systematic application to impact identiﬁcation in aerospace composites is still underexplored. Existing studies often focus primarily on architectural design choices, ef fecti vely acting only on inductive bias, while physical relations are weakly enforced or applied in isolation. This limits numerical stability , interpretability , and transferability across monitoring scenarios. This work addresses these gaps by proposing Physics- informed Impact Identiﬁcation (Phy-ID), a novel, integrated physics-informed learning frame work that lev erages obser- vational, inducti ve, and learning biases to estimate impact energy in aerospace composites. Phy-ID anchors inference to kno wn physical relations, mitigating the ill-posedness of the inv erse problem. Rather than treating impact ener gy as a purely statistical output, the frame work infers physically meaningful parameters gov erning the excitation source. T o demonstrate these capabilities, the Phy-ID frame work is implemented for impact identiﬁcation on a relev ant composite target structure. The impactor mass and impact velocity are estimated independently and combined through a known relation to obtain impact energy , for which a disjoint neural network formulation is used. The Phy-ID framework also remains adaptable to dif ferent monitoring objectives and can be extended to additional impact attributes by incorporating appropriate physical descriptors and go verning relations. The framew ork is ev aluated using experimental impact datasets representati ve of operational SHM conditions, including a geometrically comple x and realistic target struc- ture, limited data av ailability , measurement noise, and transi- tions between damage states. The assessment combines pre- dictiv e analytics to quantify estimation accuracy , robustness analyses with respect to data av ailability and noise levels, and generalisation studies addressing out-of-distrib ution impact scenarios. T ogether , these analyses provide a comprehensi ve ev aluation of both performance and alignment with domain- knowledge principles. Contributions The main contributions of this work are as follows: • A uniﬁed physics-informed frame work, Phy-ID, that integrates observational, inductive, and learning biases to achie ve physically meaningful and numerically stable impact energy estimation grounded in known physical relations. • The capability to inform advanced impact analyses through the inference of physically meaningful parameters, supporting ev ent classiﬁcation, source attribution, and impact characterisation. • Demonstration of rob ust and generalisable perfor - mance under representati ve SHM measurement con- ditions, including limited data av ailability , measure- ment noise, and out-of-distribution impact scenarios, supported by a comprehensi ve experimental ev alua- tion conducted under controlled laboratory conditions speciﬁcally designed to emulate practical monitoring challenges. • Fle xibility and adaptability of the framework that can be tuned to speciﬁc monitoring objecti ves and extended to additional impact attributes through 3 the inclusion of rele vant physical parameters and constraint formulations. The following sections detail the inte gration of observ a- tional, inducti ve, and learning biases; present a case study illustrating the functionality of the Phy-ID frame work; anal- yse physics-based energy indicators; and report predictiv e, robustness, and generalisation performance. Finally , the con- cluding section summarises the main ﬁndings and outlines directions for future work. General framew ork f or Physics-Inf ormed Impact Identiﬁcation (Phy-ID) The proposed Phy-ID methodology applies principles of Physics-Informed Machine Learning (PIML) to the impact energy estimation task. In this approach, observational, inductiv e, and learning biases are embedded in a machine learning architecture to guide training to wards solutions aligned with underlying physics 27 . As sho wn in Figure 1, the framew ork consists of four parts: Observational Bias (P art I), Inductiv e Bias (P art II), Learning Bias (Part III), and Physics Integration (P art IV). In Part I, observational bias deﬁnes ho w the physical system is represented through the av ailable data. It gov erns the selection, organisation, and synthesis of input information to ensure that the model observes realistic structural behaviour through signal representations that capture a comprehensiv e description of the system state. In Part II, inductive bias governs how prior knowledge or constraints shape the model architecture. It embeds assumptions that determine how information is processed within the network structure. Broadly , this bias deﬁnes how inputs and outputs relate through physically-motiv ated mappings. In Part III, learning bias regulates the optimisation process by enforcing agreement between model predictions and governing physical principles via the design of the loss function. It deﬁnes the learning objectiv es that guide con ver gence towards solutions consistent with both empirical observations and underlying physical laws during training. In Part IV , physics integration operationalises the Phy- ID method by combining all three forms of bias within a uniﬁed framew ork for impact energy estimation. This stage characterises the method by integrating data, model structure, and training objecti v es within a single optimisation process. W ithin this integration step, the three forms of bias interact coherently to shape the model and limit the emergence of non-physical predictions that may otherwise satisfy purely data-driv en objecti ves. First, observational bias is introduced through the way physical information enters the learning process via the input space. At this stage, measured data are transformed into multi-domain features (OB.1 in Figure 1) that describe the structural response in time, frequenc y , and time- frequency domains. These features act as physics-based energy indicators that capture key information on the ef fect of impact loading in the measured signals. Although often ov erlooked in machine learning studies, a well-designed observational bias provides an optimised and efﬁcient input space, leading to measurable improv ements in prediction accuracy and interpretability , as demonstrated in Marinho et al. 33 . Subsequently , the inductive bias incorporates prior physical kno wledge into the model architecture. It is implemented through surrogate models (IB.1 in Figure 1), appropriate activ ation functions (IB.2 in Figure 1), and structural assumptions informed by the system behaviour (IB.3 in Figure 1). In the current implementation, surrogate models within Neural Network (NN) architectures infer rele v ant parameters that characterise the impact ev ent, maintaining consistency through known physical relationships. Realistic initial conditions constrain the model response at the onset of impact, while smooth activ ation functions ensure stable differentiation during training, thereby improving numerical stability and reliable gradient computation. Finally , the learning bias governs the optimisation process that driv es model training. It operates through a hybrid loss function (LB.1 in Figure 1) that mer ges data-based and physics-moti vated terms. The data term enforces agreement between predictions and experimental measurements, while the physics term constrains the solution to satisfy gov erning physical laws. Model parameters are iterati vely updated until con ver gence, balancing agreement with measured data and imposed physical relations. Although presented here for impact energy estimation, the proposed framework remains adaptable to other monitoring objectiv es and application domains. Additional physical knowledge or alternati ve model formulations can be incorporated to estimate different physical quantities or to support subsequent stages of structural health monitoring. Implementation of the Phy-ID frame w ork This section presents an implementation strate gy of the Phy-ID framework designed to demonstrate its capabilities rather than to deﬁne a unique or prescriptiv e solution. The implementation serv es as a concrete e xample of how observ ational, inductiv e, and learning biases can be combined within a coherent formulation to support physically meaningful inference. The focus extends beyond prediction accuracy , showing how gov erning ph ysical relations can guide training and enable the estimation of latent physical parameters associated with the excitation process. In this implementation, the kinetic energy relation provides a direct link between the measured responses and the parameters inferred by the model. Its analytical form is compact and well established in mechanics, yet sufﬁciently expressi ve to structure the training process around a physically grounded constraint. In classical mechanics, the kinetic energy of a rigid impactor is gi ven by E = 1 2 mv 2 0 , (1) where m is the impactor mass and v 0 is the impact velocity at the instant of contact. Enforcing this relation during optimisation constrains the predictions to admissible combinations of mass and velocity and ensures consistency with the measured impact energy . Building on this structural prior, the impactor mass and impact velocity are deﬁned as the target outputs for 4 I … Ν Ε ( x, t , Θ ) ≈ w ( t ) Ν Ε ( x, t , Θ ) Phy - ID IB x t … σ … … σ σ σ σ σ σ σ σ E σ … OB LB Physical loss L p Data loss L d T otal loss L No < ε ? Parameter update Θ Y es IB Physics Integration: Physics - informed Impact Identification ( Phy - ID) Combine biases in a machine learning framework for impact energy estimation Signal representations that act in the input Define input space OB Observational Bias (OB) Prior assumptions that act in the model Adapt model ar chitectur e Inductive Bias (IB) IB Embedded physics that act in the training Design loss functions Learning Bias (LB) LB Multi - domain featur es Surr ogate models Hybrid data – physics losses ( L p and L d ) Structural assumptions: • Initial conditions • Physical parameters • Known physical relationships OB.1 IB.1 IB.3 LB.1 I I I I I II I III I IV IB Activation Functions ( σ ) IB.2 Figure 1. General framew ork of Physics-Inf ormed Impact Identiﬁcation (Phy-ID), showing ho w obser v ational, inductive , and learning biases are combined in a str uctured integration step to achie v e physics-inf ormed impact energy estimation. inferring impact energy and characterising the impact event. This choice is deliberate. Although impact ener gy could be predicted directly using a single surrogate model, such a formulation would treat energy as an isolated target and would not require the model to reco ver physically interpretable combinations of mass and velocity . By instead identifying mass and velocity and combining them through the kinetic energy relation, the formulation preserves the physical structure of the impact process and ensures mechanical consistenc y between the inferred parameters and the resulting energy estimate. Estimating these quantities also enriches the analysis of the impact ev ent by providing more information on whether damage may ha ve formed and what type of damage is likely . Howe v er , estimating both parameters within a single multi-output network leads to unbalanced optimisation because they inﬂuence the measured impact response in fundamentally dif ferent w ays. V elocity dominates the optimisation, as it gov erns the local and transient dynamics of the impact e vent. It controls the amplitude, duration, and frequency content of the induced stress wa ves and strongly affects measurable signal features such as peak amplitude, rise time, and dominant frequency , which are highly sensitive to variations in impact ener gy 33–35 . In contrast, the effect of mass is more diffuse, inﬂuencing the overall energy transfer and global dynamic properties. Its ef fect on the stress wav es is weaker and often masked, especially when sensor arrays are sparse 36–38 . Consequently , information about mass appears in the measured signals with lower observability and in a more indirect manner than that of velocity . During joint training, the optimisation therefore tends to prioritise velocity learning and underestimates mass. Moreov er , the kinetic energy relation can further accentuate this effect because velocity enters the expression quadratically . Although this is not the primary source of the imbalance optimisation behaviour described above, it increases the sensitivity of the energy-based loss to velocity errors relativ e to mass errors, reinforcing the dominance of velocity during joint optimisation. T o mitigate this imbalance, the Phy-ID framework utilises a disjoint PINN that sequentially estimates mass and velocity using decoupled networks, allowing each parameter to be learned in a more balanced manner , thereby improving optimisation stability and maintaining physical consistency . 5 Using a disjoint architecture, the problem is formulated as a sequential training algorithm. The concept of stabilising training through sequential optimisation of separate networks has been explored in other contexts, such as the thermochemical curing of composite-tool systems 39 . Although the application, gov erning equations, and basis formulation in Amini Niaki et al. 39 differ fundamentally from the present work, the principle of decoupled networks trained in sequence moti v ates the architectural choice adopted here. Accordingly , the Phy-ID method uses two decoupled surrogate models: a displacement network w θ and a mass network m ϕ . Both are implemented as Fully Connected Neural Networks (FCNNs). The input feature matrix x ∈ R N × F represents N impact events described by F features deﬁning the input space. The displacement surrogate model w θ : R N × F × R → R maps x and a time vector t to the transverse displacement w θ ( x , t ) . The mass surrogate model m ϕ : R N × F → R > 0 operates on the same feature matrix to predict a strictly positiv e mass m ϕ ( x ) through a softplus output layer with a small offset ε = 10 − 6 . Enforcing positiv e mass ensures physical consistency , as negati v e or zero mass would lead to non-physical, impossible conditions, including undeﬁned or negati ve kinetic energy . In contrast, displacement may assume positi ve or negati ve values depending on the direction of motion relative to a reference axis. This variability reﬂects real structural behaviour and does not compromise physical consistency , because the model does not rely on the chosen reference frame. The displacement output is then used to infer the impact velocity , which is obtained by differentiating the displacement with respect to time and ev aluating it at t = 0 , v 0 ( x ) = ∂ w θ ( x , t ) ∂ t     t =0 . (2) Embedding impact velocity as the initial time deriv ativ e of a displacement ﬁeld anchors the formulation in structural dynamics, where displacement and its ﬁrst deriv ati ve deﬁne the system state at the instant of contact. This representation also allows the physically meaningful initial condition to be enforced during training through the loss formulation introduced later in this section. In addition, by parameter- ising both the zero-displacement condition and the initial velocity within the same surrogate ﬁeld, the formulation av oids introducing additional independent quantities and improv es numerical conditioning. Because the deri v ativ e is obtained through automatic differentiation, incorporating this relation introduces only minimal additional complex- ity . As a result, the learning problem remains physically grounded and numerically stable while retaining the capacity to accommodate additional gov erning relations if required. Because the impact velocity enters the kinetic energy expression as a squared term, its sign does not affect the energy calculation; therefore, enforcing positivity is unnecessary . T ogether, the predicted mass and inferred velocity are linked through the kinetic energy relation in Equation (1), ensuring physical compatibility between the two surrogate networks. This coupling forms the basis of the sequential training process summarised in Figure 2, which provides a high-le vel ov erview of how the two networks exchange information during alternating optimisation. At each iteration, the parameters of the mass model are ﬁxed, and the displacement network w θ is optimised with respect to its loss function L disp . In the ﬁrst iteration, the measured mass m obs is used to initialise the displacement model. Once updated, the displacement parameters are held constant, and the mass network m ϕ is optimised with respect to its loss L mass . This alternating process continues until conv ergence, ensuring balanced accuracy and adherence to underlying ph ysical constraints between the coupled quantities. The loss functions for both L disp and L mass combines measurement-based and physics-based components, corre- sponding to the data loss ( L d ) and physics loss ( L p ) illus- trated in Figure 1. In general form, hybrid losses can there- fore be expressed as L = L d + L p . (3) The displacement loss is deﬁned as L disp = λ v 0 L v 0 | {z } L d + λ IC L IC | {z } L p + λ KE L KE | {z } L p , L d , (4) where the weighting factors λ v 0 , λ IC and λ KE control the relativ e contribution of the velocity , initial condition, and kinetic energy terms in this phase. The individual components are L v 0 = ∥ v 0 , obs − v 0 ( x ) ∥ 2 L I C = ∥ w θ ( x , 0) ∥ 2 L K E = ∥ E meas − 1 2 m v 2 0 ( x ) ∥ 2 . (5) Here, m denotes the ﬁxed mass (either the measured m obs in the ﬁrst iteration or the predicted mass in later iterations). These terms penalise, respectively , de viations from the observed impact v elocity v 0 , obs , violations of the zero-displacement initial condition (IC), and inconsistencies with kinetic energy (KE). The mass network loss follo ws an analogous form, L mass = λ m L m | {z } L d + λ KE ,m L , KE ,m | {z } L p , L d , (6) with L m = ∥ m obs − m ϕ ( x ) ∥ 2 L KE ,m = ∥ E meas − 1 2 m ϕ ( x ) v 2 0 ( x ) ∥ 2 . (7) The ﬁrst term enforces agreement between predicted and measured mass ( m ϕ and m obs ), while the second ensures consistency between the predicted mass m ϕ , ﬁxed velocity v 0 resulting from the displacement NN, and observed ener gy E meas . The f actors λ m and λ KE ,m act as the corresponding weighting coefﬁcients for the mass phase. The weighting factors introduced abo ve re gulate the inﬂu- ence of measurement terms and physics-based terms during training, prev enting dominance by any single component. For this study , the values are assigned manually , a common and ef fecti ve approach for demonstration purposes that pro- vided stable training and adequate performance. Alternative 6 L disp 𝜃 ; 𝑚 𝒙 𝑡 𝑤 𝑡 L IC 𝑤 𝜃 𝒙 , 𝑡 Derivative terms Loss terms 𝑚 … 𝒙 𝑤 𝑚 ∅ 𝒙 Derivative terms Loss terms 𝑚 … 𝑤 𝑚 L v0 L KE L mass ∅ ; 𝑤 L 𝑚 L KE Figure 2. High-lev el ov erview of the sequential disjoint training process: the displacement network (top) generates a v elocity estimate f or the impactor mass network (bottom), which in tur n produces a mass estimate fed back to the displacement model. schemes, such as adaptiv e weighting 40 or uncertainty-based scaling 41,42 , may beneﬁt more complex or sensitive applica- tions and could be explored in future work. Building on the loss formulations, the two optimisation objectiv es summarise the sequential training process, θ ∗ = arg min θ L disp ( θ ; m ) and ϕ ∗ = arg min ϕ L mass ( ϕ ; v 0 ) , (8) where θ and ϕ are updated alternately according to the displacement- and mass-phase to ensure balanced accuracy and physical consistency . Algorithm 1 provides a detailed description of the sequential training steps. The algorithm takes as input a feature matrix x ∈ R N × F , measured energy E meas ∈ R N × 1 , measured mass m obs ∈ R N × 1 , and measured impact velocity v 0 , obs ∈ R N × 1 , corresponding to N impact ev ents. The ﬁnal outputs ˆ m and ˆ v 0 are combined through the classical kinetic energy relation introduced in Equation (1) to predict the impact energy . It is important to distinguish between training and inference phases. During training, the model recei ves both input features and the measured physical quantities (measured energy E meas , measured mass m obs , and measured impact velocity v 0 , obs ), which act as supervised targets for optimising the surrogate models through the hybrid loss. During inference, the trained models take only the feature matrix x as input. The learned model is then used to estimate the physical parameters and compute the impact energy from the predicted mass ˆ m and impact velocity ˆ v 0 . Algorithm 1 Physics-Informed Impact Identiﬁcation (Phy-ID) Sequential training 1: Input: x , E meas , m obs , v 0 , obs , t . 2: Models: w θ ( x , t ) (displacement FCNN), m ϕ ( x ) (mass FCNN). 3: for cycle c = 0 , . . . , C − 1 do 4: Phase A: Displacement optimisation 5: Fix m = m obs if c = 0 , else m = m ϕ ( x ) . 6: Solve θ ∗ = arg min θ L disp ( θ ; m ) with Adam optimiser . 7: Phase B: Mass optimisation 8: Compute v 0 from ﬁxed w θ . 9: Solve ϕ ∗ = arg min ϕ L mass ( ϕ ; v 0 ) with Adam optimiser . 10: end for 11: Output: ˆ m, ˆ v 0 . Notation: ( · ) denotes a ﬁxed value; ( · ) ∗ marks the optimal solution; and ˆ ( · ) indicates predicted quantities. The implementation presented in this section opera- tionalises the bias structure introduced in General framework for Physics-Informed Impact Identiﬁcation (Phy-ID). The observational bias (OB.1) is realised through the feature matrix, which contains the multi-domain signal represen- tations describing the measured structural response. The inductiv e biases are embedded in the model formulation: IB.1 corresponds to the surrogate neural network models used to represent displacement and mass; IB.2 arises from the use of smooth activ ation functions that enable stable differentiation and consistent parameter estimation while 7 T able 1. Hyper parameter search space explored f or the displacement and mass FCNNs . Parameter Displacement network, w θ ( x , t ) Mass network, m ϕ ( x ) fully connected layer size, n h 32, 64, 128, 256 16, 32, 64 number of hidden layers, L h 2, 3, 4 2, 3 learning rate, lr 10 − 2 , 10 − 3 10 − 2 , 10 − 3 activ ation function Sigmoid, Softplus, T anh Sigmoid, Softplus, T anh T able 2. Implementation settings for the disjoint PINN architecture. Parameter V alue Displacement network w θ ( x , t ) architecture FCNN, 3 hidden layers of size 256, activ ation tanh output impact velocity , ˆ v 0 learning rate 1 × 10 − 2 loss function L disp = λ v 0 L v 0 + λ ic L I C + λ ke L K E weighting factors λ v 0 = 1 × 10 − 4 , λ IC = 1 × 10 − 6 and λ KE = 1 × 10 − 4 Mass network m ϕ ( x ) architecture FCNN, 3 hidden layers of size 64, activ ation softplus output impactor mass, ˆ m learning rate 1 × 10 − 3 loss function L mass = λ m L m + λ ke ,m L K E ,m weighting factors λ m = 1 × 10 − 6 and λ KE ,m = 1 × 10 − 4 T raining procedur e epochs max 10000 per phase outer cycles max C = 10 batch size full-batch optimiser Adam criterion MSELoss patience 500 aligning with the expected behaviour of the physical ﬁelds; and IB.3 is introduced through structural constraints such as the kinetic relation between displacement and veloc- ity , the initial condition, the positivity constraint on mass, and the kinetic energy relation. Finally , the learning bias (LB.1) is implemented through the hybrid loss functions and the sequential optimisation procedure, which combine measurement-based terms and physics-based constraints to guide training to wards predictions grounded in structural mechanics. Architecture details The architecture for the disjoint PINN supports the sequential training strategy described earlier in this section. It consists of two separate Fully Connected Neural Networks (FCNNs), each dedicated to a speciﬁc regression task: the displacement network estimates the impact velocity , and the mass netw ork predicts the impactor mass. Although the subnetworks are trained separately , they share the same feature matrix as input, and their outputs are coupled in the loss function via a kinetic-ener gy relation to recover the physical quantities required for energy prediction. T o select the network architecture, a grid search was performed using ﬁv e-fold cross-validation. For each fold, the dataset was split into 80% for training and 20% for testing. The search tested different combinations of hidden layer numbers ( L h ), layer sizes ( n h ), learning rates ( lr ), and activ ation functions ( σ ) for both surrogate models. The search space for both networks is summarised in T able 1. The acti v ation functions tanh (hyperbolic tangent), softplus, and sigmoid introduce inductive biases into the network architecture. Their continuous and smooth behaviour supports stable differentiation, which is required to ev aluate the deriv ati ves used in the physics-based constraints. In addition, the softplus output layer ensures that the predicted mass is non-negati ve. These choices, therefore, mainly control the regularity and boundedness of the learned representations while remaining compatible with the physical formulation. The performance of each architectural conﬁguration was assessed using the mean coef ﬁcient of determination R 2 across the test folds, with the corresponding validation trends shown in Figure 3. The colour scale shows the av erage R 2 for each parameter combination and highlights the most ef fectiv e settings within the explored search space. Guided by these trends, a ﬁnal set of hyperparameters was selected. These values, together with the remaining implementation details used in the sequential scheme, are listed in T able 2, and this conﬁguration was adopted for all subsequent analyses. 8 L h n h l r σ (a) Displacement network w θ ( x , t ) . L h n h l r σ (b) Mass network m ϕ ( x ) . Figure 3. Par allel coordinate plots showing the perf ormance of the displacement and mass networks across hyperparameter combinations. ( n h : fully connected la yer size; L h : number of fully connected la yers; lr : lear ning rate; and σ : activation functions.) T able 3. Impact locations and sensor network coordinates. ID Coordinates [mm] x y PZT1 255 428 PZT2 252 818 PZT3 250 945 PZT4 187 945 PZT5 125 945 PZT6 125 427 IE1 124 816 IE2 182 818 IE3 123 491 IE4 252 559 Dataset The proposed method is v alidated using experimental data from intermediate-mass 43 impact tests on a stiffened ther- moplastic composite structure. This element-level conﬁgu- ration 44 is geometrically complex, offering a realistic set- ting for assessing the method. The composite structure fea- tures three stiffeners and three distinct thickness regions, with a quasi-isostatic lay-up, and nominal dimensions of 1600 mm × 370 mm. The schematic of the test conﬁguration is illustrated in Figure 4, while the impact locations and sensor coordinates are summarised in T able 3. Impact tests follow the ASTM D7136M-15 47 standard, ensuring consistent loading and measurement conditions throughout the experimental campaign. Drop-weight impacts were applied on the outboard side of the composite component. T wo hemispherical impactors, each with a diameter of 16 mm but dif fering in mass of 2.238 kg, 2.356 kg and 5.510 kg, deli vered the impacts. A network T able 4. Impact energy statistics. Number of unique targets 73 Minimum energy [J] 3.74 Maximum energy [J] 80.95 Mean energy [J] 35.92 Median energy [J] 29.83 Standard deviation [J] 24.04 of six piezoelectric (PZT) sensors recorded the impact responses. Additional details on the material system, sensor conﬁguration, and data-collection procedure are av ailable in Marinho et al. 46 . The resulting dataset comprises 73 distinct impact events, with energies ranging from 3.74 J to 80.95 J, including both pristine and damaged conditions. An ov erview of the impact ener gy distrib ution statistics is presented in T able 4. Phased-array ultrasonic inspections were performed after ev ery impact event to identify the energy level associated with the damage onset. The measurements were performed using an Olympus Omniscan M-P A16-128 system with a phased-array transducer containing 128 elements, sampled at 100 MHz, operating with 256 focal laws and an 80 V pulse v oltage per element, with automatic probe recognition. A water-based couplant assured proper signal transmission. The scans provide complementary vie ws, including backwall attenuation, reﬂection maps, and cross-sectional images, that rev eal the progression of subsurface damage. As shown in Figure 5, the ultrasonic response captured damage initiation under progressive loading conditions. Analysis at IE1 and IE2 (Figure 5a) showed that impacts of 50 J resulted in only minor changes in the backwall signal, with subtle irregularities more pronounced at IE2 . 9 impactor composite h (a) IE4 (0,0) x y t 1 transition Legend: t 2 t 3 PZT1 PZT2 PZT3 PZT4 PZT5 PZT6 IE3 IE1 IE2 stif feners (b) impactor guiding wires steel fixing assembly steel fixing assembly 3-stif fener panel (c) Figure 4. Schematic impact test conﬁguration; • piezoelectric sensors (PZT) and ⋆ Impact Locations (IE): (a) drop-tow er assembly with impact height ( h ) set by impact energy; (b) stiff ened thermoplastic composite str ucture ( 1600 × 370 mm) with var ying thicknesses t 1 = 8 . 26 mm, t 2 = 6 . 30 mm, and t 3 = 5 . 46 mm; (c) experimental set-up , with stiff eners at the bottom-side of the panel. Adapted from 45 , 46 . Howe v er , at 55 J, scanning re v ealed signiﬁcant delamination and surface dents, indicating the onset of damage. Further observations at IE3 (Figure 5b) indicated limited variation at 55 J, while the scan at 60 J displayed distinct delamination patterns and localised surface marks. For IE4 (Figure 5c), views at 45 J exhibited no signs of damage, while the 50 J inspection revealed visible delamination and surface dents. These results suggest an average damage onset energy of approximately 55 J for the stif fened composite panel. This threshold is used to deﬁne and interpret various impact scenarios and data av ailability conditions in the following analyses. Data processing T o systematically ev aluate the proposed method, a structured data processing workﬂo w has been implemented, focusing on three ke y objecti ves: predictive performance, robustness against limited and noisy data, and generalisation to unseen impact situations. While all modelling choices remain ﬁxed across objecti ves, the optional processing steps are tailored to address data conditioning requirements for each analysis. A detailed summary of the examined data processing cases is presented in T able 5. This table outlines the different scenarios for predicti ve analytics (P1), robustness analyses (R1–R3), and generalisation assessments (G1), each characterised by variations in av ailable impact data, noise lev els, and the nominal impact energy range employed during training. The general data processing workﬂow is illustrated in Figure 6, highlighting where case-speciﬁc adaptations apply . All cases follow the same core processing pipeline. Initially , raw impact responses are processed to yield time signals of consistent quality , as previously outlined by Marinho et al. 45 . Feature extraction subsequently transforms each processed response into a ﬁxed-length vector , forming the feature matrix ( x ), in accordance with the methodology outlined in Marinho et al. 33 . T o ensure uniformity across features, min-max normalisation is applied, constraining each input dimension to [0, 1]. The dataset is partitioned according to the guidelines in Architecture details, allocating 80% of the samples to training and 20% to testing. Model training and ev aluation occur within the disjoint PINN framew ork. Elaborating on this foundational pipeline, the speciﬁc data structuring for each analytical objectiv e is detailed. Case P1 explores predictive analytics with a focus on ov erall predictiv e capability under realistic measurement variability encountered in SHM applications. In this scenario, the complete dataset is used, augmented with 1% Gaussian noise, yielding a total of 146 impact responses for modelling. The augmented dataset, therefore, contains equal proportions of original and noise-perturbed signals. Here, the term pr ediction refers to model-estimated outputs ev aluated on samples not provided as labelled targets during the training phase. As such, all quantities inferred from test samples are considered predictions, as they are estimated using the learned surrogate models rather than direct observ ation. Cases R1-R3 assess robustness against data limitations and increasing noise, which commonly arise in practical SHM deployments due to limited sensor cov erage, con- strained testing campaigns, and variable signal quality . Case R1 explores data av ailability ranging from 25% to 100% using only original signals, thereby isolating the effects of reduced sample size. The percentages denote fractions of the total number of impact tests, which consists of 73 impact samples. After the initial 80/20 split, the test set is ﬁxed across all dataset sample sizes, while subsetting is applied only to the training set. In Case R2, the complete sample set (100%) is retained while noise levels are incrementally increased from 0% to 10 backwall attenuation r eflection view cr oss - sectional view IE1 IE2 delamination delamination backwall frontwall delamination early delamination early delamination stiffener dent stiffener 55 J 50 J (a) Impact locations IE1 and IE2. backwall attenuation r eflection view cr oss - sectional view IE3 backwall frontwall delamination delamination dent 60 J 55 J (b) Impact location IE3. b ackwall attenuation r eflection view cr oss - sectional view IE4 delamination delamination dent backwall frontwall 50 J 45 J (c) Impact location IE4. Figure 5. Phased array ultr asonic scans for diff erent impact locations acquired after the impact e v ents that precede and produce damage, highlighting the impact energy associated with damage onset. Each scan sho ws backwall atten uation, reﬂection view and cross-sectional view . 5%, thereby assessing the robustness against uncertainties in measurement. Case R3 modiﬁes both data availability and noise le vels, mirroring the approach of R1, which subsamples the training dataset while keeping the test set ﬁxed. Figure 7 shows an example of a time series signal with the maximum 5% noise le vel introduced in Cases R2 and R3, conﬁrming the challenging measurement conditions represented in these cases. Performance metrics across all robustness cases are computed as the average error over ﬁv e folds, ensuring that the reported trends reﬂect consistent behaviour rather than relying on a single data split. Finally , Case G1 inv estigates generalisation through out-of-distribution testing samples. This process in volv es deﬁning training ev ents within a speciﬁed nominal energy range while keeping impacts outside this range for testing only . Such an approach is crucial for SHM systems operating 11 T able 5. Summar y of data processing cases for predictiv e analytics, rob ustness analysis and generalisation assessment. Case a,b Description Impact data T raining scenario Samples Data av ail. [%] Noise [%] Energy range c [J] P1 Predictiv e analytics 100 1.0 4–80 146 R1 Data av ailability sweep 25–100 0.0 4–80 variable R2 Noise sweep 100 0–5 4–80 73 R3 Combined av ailability and noise 25–100 0–5 4–80 variable G1 Out-of-distribution training ranges 100 1.0 variable 146 a All cases use the same input feature set. b All models use the same FCNN architecture for displacement and mass subnetworks (see Architecture details). c Energy limits refer to nominal impact energy , which may differ slightly from measured values recorded by the drop tower instrumentation. data pr ocessing cases optionally adds Gaussian noise P1 R2 R3 G1 load and process raw sensor data feature extraction and normalisation evaluation and metrics model training optionally defines data subsets R1 R3 data conditioning and splitting 80% training data 2 0% test data trained model 5 - fold Figure 6. Data-processing workﬂow highlighting the associated analysis cases: P1 (predictive analytics), R1 (data a vailability s weep), R2 (noise s weep), R3 (combined av ailability and noise), and G1 (out-of-distribution training r anges). original signal noisy signal (5.0 %) 1 time [s] 0 -10 0 10 amplitude [V] Figure 7. Comparison between the or iginal and noisy signals recorded by PZT5 due to an impact e v ent of 10 J at impact location IE1. PZT: piez oelectric in en vironments where new , yet physically related, scenarios may arise without immediate access to additional training data. Results and Discussion The results are presented in the same structure as the data processing w orkﬂow deﬁned in the pre vious section. First, feature e xtraction and selection are applied to the impact test data to deﬁne physics-based energy indicators. Next, results from the predicti ve analytics case (P1) quantify the accurac y of the estimated mass, velocity , and impact energy at test samples. The analysis then examines robustness through Cases R1–R3, followed by an assessment of generalisation in Case G1 using out-of-distribution scenarios. Ph ysics-based energy indicators This subsection addresses the feature extraction and selection from the impact responses in the case study described in Dataset. The selection follows the framew ork reported in Marinho et al. 33 on energy indicators for impact ener gy estimation in composites. W ithin this framework, the multi- domain candidate set is assessed using three criteria that characterise reliable indicators: sensitivity to variations in impact energy , robustness under noisy measurements and independence from other descriptors. These criteria ensure that the selected features reﬂect measurable responses linked to impact mechanics, embedding observ ational bias within the Phy-ID frame work (refer to Part I in Figure 1). T able 6 summarises the resulting scores and determines the ﬁnal set of indicators used in this study . Based on these scores, the selected set includes the signal features: Root Mean Square (RMS) 48 , Transmitted Energy (TE) 22 , Peak Amplitude (P A) 49 , Energy Peak Ratio (EPR) 50 , Peak Centroid Ratio (PCR) 51 , W eighted Peak Frequency (WPF) 52 , Peak Frequency (PF) 53 , Approximation Max Energy (AME) 54,55 and Approximation Max (AM) 54,55 . These indicators constitute the inputs, i.e., the feature matrix, for the predictive models assessed in the subsequent analyses. While this section focuses on the ﬁnal results and 12 T able 6. Feature ranking and selection f or deﬁning the ph ysics-based energy indicators. Domain Group ID w m r m s m evaluation Time T1 RMS 0.58 0.97 0.57 ∗∗ / •• T2 TE 0.59 0.95 0.56 ∗∗ / •• T3 P A 0.61 0.99 0.60 ∗∗ / •• T4 EPR 0.61 0.95 0.58 ∗∗ / •• T5 RA 0.34 0.80 0.28 ∗∗ / ⋄ Frequency F1 PCR 0.34 0.81 0.28 ∗∗ / •• F2 WPF 0.55 0.81 0.45 ∗∗ / •• F3 PF 0.52 1.00 0.52 ∗∗ / •• Time- W1 AME 0.59 0.99 0.59 ∗∗ / •• Frequency W2 AM 0.61 0.99 0.60 ∗∗ / •• W3 DM 0.46 0.56 0.26 ∗∗ / ⋄ Legend Scores : w ( m ) importance score; r ( m ) robustness score; s ( m ) selection score. Evaluation : ⋄ not stable under noisy conditions; ∗∗ independent; •• relev ant and stable feature. the chosen feature set, further methodological details are av ailable in Marinho et al. 33 . Predictiv e analytics (Case P1) Case P1 ev aluates the predictiv e capabilities of surrogate models in the context of real-w orld measurement variability , focusing on inferring impact velocity , impactor mass, and impact energy . Model performance is quantiﬁed using the mean absolute percentage error (MAPE) 56 and further assessed qualitativ ely by examining the concentration of predictions within the ± 10% tolerance bands, adopted here as a practical engineering threshold for SHM capability in line with recent aerospace studies reporting prediction errors within this range as acceptable performance 57 . As illustrated in Figure 8a and Figure 8b, parity plots depict the surrogate predictions for impact velocity and impactor mass, respectiv ely . The displacement surrogate, which infers impact v elocity , achiev es a MAPE of 4.97%, while the mass surrogate reports a MAPE of 7.75%. Most predictions lie within the shaded ± 10% tolerance band around the ground truth for test samples. For the mass surrogate, howe ver , a few predictions at the 2.238 kg lev el lie outside the ± 10% band. This deviation likely reﬂects the limited number of samples av ailable at this mass lev el and the small difference between the 2.238 kg and 2.356 kg masses. Under certain impact conditions, these two masses generate similar structural responses, increasing the sensitivity of the in verse mass estimation to measurement noise and optimisation trade-offs. Overall, the deviations remain limited and do not alter the general trend observed in the parity plot. Figure 8c presents the results for impact energy . Impact energy is inferred from the kinetic energy relation (Equation (1)) using the predicted impactor mass and impact velocity . The resulting estimates yield a MAPE of 9.85%. Importantly , 88.9% of the test samples (23 out of 27) fall within the ± 10% tolerance band, conﬁrming that the accuracy of the indi vidual surrogates translates into reliable energy estimation when kinetic consistenc y is enforced. T o isolate the contribution of the physics-moti vated biases, Figure 9 reports the impact energy predictions obtained after removing the physics-informed components while retaining the same FCNN architecture and identical test samples used in the Phy-ID ev aluation. In this conﬁguration, the observational bias (OB.1) remains unchanged, whereas inductiv e biases (IB.1 and IB.3) and learning bias (LB.1) are remov ed by replacing the physics-informed formulation with a single FCNN trained using a standard mean-squared error loss to directly predict impact energy . Under these conditions, the MAPE increases to 12.66%. Despite the relativ ely small difference in average error compared to the physics-informed design (9.85%), the accuracy lev els drop signiﬁcantly; only 51.8% of predictions (14 out of 27) remain within the ± 10% threshold. This disparity highlights the increased noise in the predictions when prior kno wledge constraints are absent. Overall, these results indicate that including physics- motiv ated constraints narro ws the feasible solution space and supports more stable energy estimates, which is associated with improved accuracy . Speciﬁcally , including prior knowledge increases the proportion of predictions within a deﬁned target tolerance band, demonstrating a positiv e effect on predicti ve accuracy under the same architecture and test conditions. Robustness studies (Cases R1-R3) The rob ustness e v aluation of the proposed framew ork is detailed in Cases R1–R3 with a particular focus on impact energy prediction. These analyses in v estigate the performance of the model under conditions of reduced data av ailability and degraded measurement quality . The mean absolute percentage error (MAPE) is utilised to quantify robustness, with the a verage computed across ﬁve folds to ensure reliable performance. Figure 10 summarises the results, presenting heatmap plots that illustrate MAPE v alues across ener gy bins for different scenarios, and measured average error for all energy lev els in combined scenarios. Each plot features numerical annotations that specify corresponding error metrics. In Cases R1 and R2, adjacent plots sho w error metrics av eraged ov er all energy lev els, enabling a comprehensive assessment of ov erall performance behaviour . In Case R1, illustrated in the left panel of Figure 10, predicti ve performance exhibits limited sensiti vity to 13 (a) Surrogate prediction of impact velocity . (b) Surrogate prediction of impactor mass. (c) Impact energy inferred via kinetic energy consistency using predicted mass and velocity . Figure 8. Predictive analytics results f or Case P1. MAPE: Mean Absolute P ercentage Error Figure 9. Impact energy inferred after remo ving architectural priors, surrogate components, and h ybrid loss ter ms. reductions in data av ailability . MAPE values, av eraged across all energy levels, remain closely aligned at approximately 10% as data a v ailability decreases from 100% to around 62.5%. A slight increase in average error is observed as training sample size decreases, particularly between 50% and 25% data availability . Notably , ev en at 25% data av ailability , representing only 15 impact samples for training, the av erage MAPE sustains at approximately 15%. From a SHM perspecti ve, such accuracy remains signiﬁcant and practical for warning systems, depending on the context and risk tolerance 58–60 . When examining individual energy ranges, the analysis rev eals more pronounced variations in MAPE values at lower impact energies, which are inﬂuenced by the formulation of the error metric. As MAPE accounts for percentage error relativ e to true values, small absolute residuals in low- energy ranges can inﬂate percentage errors considerably 61,62 . Thus, the relativ e increase in MAPE at limited sample sizes reﬂects both reduced training information and greater metric sensitivity in lo wer -magnitude regions. Case R2, depicted in the central panel of Figure 10, isolates sensitivity to measurement noise, using the complete training samples while varying noise lev els from 0% to 5%. Under these conditions, the av erage MAPE remains close to 10%, with no clear systematic dependence on impact energy . During input space design (see previous section Physics-based energy indicators), the selected descriptors were ev aluated for both energy sensitivity and robustness under controlled noise conditions. Because robustness to noise was one of the criteria guiding the construction of the input space, the limited variation in error under moderate noise lev els is consistent with the feature design strategy adopted in this work. Comparison of cases R1 and R2 indicates that reductions in data av ailability hav e a greater impact on model performance than higher noise le vels. This trend aligns with established insights regarding learning-based models, where less training information has a more pronounced ef fect on robustness than measurement noise 63–65 . Case R3, illustrated in the right panel of Figure 10, integrates the combined effect of reduced data av ailability and increased noise le vels. Performance remains stable, with a verage MAPE values around 15% across most combinations. It is only under the most extreme conditions, 25% data av ailability paired with 5% noise, that MAPE escalates to 25.4%. This observ ation indicates a gradual, expected performance decline rather than a sudden failure of the model’ s capabilities. From a general perspectiv e, the robustness analyses sho w that the proposed Phy-ID framew ork preserves predictiv e performance under varying lev els of data a vailability and measurement noise. Generalisation assessment (Case G1) The generalisation assessment examines the ability of the proposed Phy-ID framework to extrapolate impact energy predictions beyond the training domain. Models are trained on restricted nominal ener gy intervals and assessed exclusi vely on out-of-distribution (OOD) samples that lie outside these intervals. Figure 11 summarises the e xtrapolation behaviour for four progressiv ely extended training intervals. Each subﬁgure 14 less data higher noise Figure 10. Impact energy robustness analysis. Measured error across energy bins f or (left) v ar ying data av ailability at zero noise (Case R1), (centre) increasing noise le vels at full data a vailability (Case R2), and (right) combined reductions in data av ailability and increases in noise (Case R3). MAPE: Mean Absolute P ercentage Error shows parity plots of predicted v ersus true impact energy for OOD samples. Predictions at low impact energies (0–20 J) consistently demonstrate accurac y and stability , independent of the selected training range. This consistency reﬂects the inherent stability of the underlying impact response in the pristine regime, indicating that the framework generalises reliably within this linear , damage-free domain. In contrast, extrapolations at higher impact energies exhibit a marked dependence on the proportion of damaged data included in the training set. In Figure 11a, where the training range spans 20–65 J with approximately 25% of samples damaged, predictions at higher energies are widely scattered and poorly aligned with the parity line. This may indicate that insuf ﬁcient information exists to capture the altered response associated with damage progression. W ith an extended training range of 20–70 J, corresponding to approximately 40% of damaged samples (Figure 11b), predictions at higher energies become more clustered and trend to wards the ground-truth line. Despite this improv ement, noticeable dispersion persists, and many predictions remain outside the ± 10% bounds, suggesting incomplete learning of the damaged-state response. A signiﬁcant qualitative shift occurs when the training set exceeds 50% of damaged samples. In Figure 11c, where the training interval is 20–75 J and contains approximately 60% of damaged samples, OOD predictions at higher energies align more closely with the parity line. This beha viour is maintained with further inclusion of damaged data, as illustrated in Figure 11d. The predictions qualitatively improv e extrapolation performance compared with narro wer training intervals, although some points remain outside the ± 10% band. This indicates that increasing the proportion of damaged samples improv es trend consistency through better data coverage, but does not eliminate residual scatter . The remaining dispersion likely reﬂects either the increased nonlinearity of the damaged regime, measurement variability , or the limited physical description of damage progression in the present model. This trend directly relates to the change in structural response associated with damage onset. As identiﬁed in Dataset through ultrasonic inspections, damage onset occurs at around 55 J. Belo w this threshold, the impact response remains predominantly linear , whereas abo ve it, the presence of damage introduces a more nonlinear relationship between impact energy and the measured response. The Phy-ID frame work incorporates observational, inductive, and learning biases to encode physically meaningful relationships for impact energy inference. Howe ver , it does not explicitly inte grate models of damage mechanics or failure progression. Consequently , the increased nonlinearity associated with damage must be learned solely from the data. T o further assess extrapolation behaviour , Figure 12 shows the OOD ener gy predictions from a model trained over the same 20–80 J interval (and 80% of the damaged samples) but without the physics-informed biases. Comparison with Figure 11d re veals that in the lo w-energy range (0–20 J), the Phy-ID formulation maintains predictions well-aligned with the ground truth, whereas the unconstrained model exhibits greater spread and visible deviations from the parity line. At higher impact energies, both formulations show comparable behaviour once a sufﬁcient proportion of damaged samples is included during training. These observations indicate that incorporating prior physical structure improves prediction consistency in the linear regime, while performance in the damaged regime reﬂects the lev el of data representation in the training set under the present formulation. T aken together , these results indicate that the framework achiev es limited e xtrapolation capability for impact energy estimation, provided that a minimum amount of damaged data is av ailable. Generalisation across damage states, there- fore, requires access to representativ e damaged responses during training. In this context, further improvements may be achiev ed by extending the Phy-ID frame work to incorporate physical constraints on damage mechanisms. Such exten- sions would maybe reduce reliance on damaged samples alone and enhance generalisation across structural states. Discussion The preceding results, together with characteristics reported in the literature, enable a qualitativ e comparison between data-driv en, physics-based, probabilistic, and physics- informed approaches for impact energy estimation under realistic SHM conditions characterised by low-quality measurements and incomplete physical knowledge. The comparison focuses on four criteria that directly affect 15 (a) Energy estimation f or a training range of 20–65 J; including 25% of damaged samples. (b) Energy estimation f or a training range of 20–70 J; including 40% of damaged samples. (c) Energy estimation f or a training range of 20–75 J; including 60% of damaged samples. (d) Energy estimation f or a training range of 20–80 J; including 80% of damaged samples. Figure 11. Generalisation behaviour f or v ar ying training intervals. Predictions are e v aluated on Out-Of-Distribution (OOD) samples outside each training range , illustrating e xtrapolation in impact energy estimation. low-ener gy predictions Figure 12. Generalisation behaviour without ph ysics-motiv ated architectural priors, surrogate components, and h ybrid loss terms. Predictions on out-of-distribution samples illustrate e xtrapolation in impact energy estimation. deployability in aerospace structures: accuracy , rob ustness, generalisation, and interpretability . The interpretation of these ﬁndings draws on established characteristics of each modelling paradigm reported in the literature and contrasts them with the behaviour observed for the physics-informed formulation in this study . The present comparison, therefore, remains qualitativ e and context-speciﬁc. T able 7 summarises this qualitati ve assessment. It reﬂects the practical SHM setting addressed here and should not be interpreted as a general ranking across all applications. The performance le vels assigned to the data-driven, ph ysics- based, and probabilistic approaches are grounded in the literature cited in the table, whereas the le vels attrib uted to the physics-informed formulation follow directly from the results presented in this study . Purely data-driv en approaches learn mappings directly from measured data without embedding go verning rela- tions. The SHM literature consistently links these methods to strong dependence on large and representati ve datasets, sensitivity to measurement noise, and weak extrapolation beyond the training domain 66,67 . Although good accuracy can be achiev ed under controlled conditions 57 , reliability 16 T able 7. Qualitative comparison of impact identiﬁcation strategies under limited and noisy data with incomplete ph ysics. Criterion Data-driven Physics-based Probabilistic Physics-inf ormed (Phy-ID) Accuracy Low–Medium 66–68 Low–Medium 69 , 70 Medium 71 , 72 High Robustness Low 57 , 67 Low–Medium 70 , 73 Medium 71 , 72 High Generalisation Low 57 , 67 Low–Medium 66 , 70 Medium 71 , 72 Medium Interpretability Low 57 , 67 High 66 , 70 Low–Medium 71 , 72 High decreases when operating conditions vary or data cover - age is restricted. W ithout physical constraints, predictions may remain statistically consistent while lacking mechan- ical plausibility , which limits interpretability and generali- sation 67,68 . These characteristics correspond to the low-to- medium accuracy , low robustness, low generalisation and low interpretability reﬂected in T able 7. In contrast, physics-based approaches formulate the problem through gov erning equations and constitutiv e relations. This structure provides high interpretability and supports extrapolation within the assumed physics. Howe v er , performance depends directly on model ﬁdelity . Revie ws highlight sensiti vity to en vironmental v ariability , modelling simpliﬁcations, and incomplete representation of damage mechanisms 69,73 . In addition, high-ﬁdelity formulations can become computationally demanding, limiting scalability and real-time use 66,70 . Consequently , accuracy and robustness are often characterised as low to medium, and generalisation remains restricted to explicitly modelled phenomena. Probabilistic approaches combine prior assumptions with statistical inference to mitigate these limitations. The literature describes them as intermediate solutions whose effecti v eness depends on how explicitly physical relations are embedded 74 . When prior information enters indirectly through regularisation rather than enforced constraints, interpretability decreases and inconsistencies may arise 75 . Performance remains conditioned by the quality of prior assumptions and data representati veness, particularly under noisy or limited data conditions 76 . Increased computational demand may further limit scalability . These factors place probabilistic methodologies at intermediate le vels of accuracy , robustness, and interpretability , with moderate generalisation shaped by system-speciﬁc assumptions. The proposed Phy-ID framew ork integrates domain knowledge at three levels: input deﬁnition, architectural formulation, and loss design. Physics-informed learning literature associates explicit enforcement of governing relations with improved tolerance to limited and noisy data and with stronger interpretability 77–79 . By restricting admissible solutions to those consistent with known physical descriptions, such formulations moderate solution variability . The behaviour observed for the proposed Phy-ID frame- work aligns with the conceptual e xpectations reported in the literature and demonstrates their practical relev ance. In Case P1, the physics-informed conﬁguration achieves a higher concentration of predictions within the tolerance band than the unconstrained v ariant. In Cases R1–R3, perfor- mance degrades gradually under reduced data and increased noise, reﬂecting enhanced robustness. In Case G1, extrap- olation remains consistent within the linear regime, while behaviour at higher energies depends on the av ailability of representativ e samples, indicating moderate generalisation. In terms of interpretability , the disjoint inference of impactor mass and velocity e xposes physically meaningful interme- diate quantities instead of relying solely on direct energy regression. T ogether , these characteristics support the high accuracy , high rob ustness, high interpretability , and moder- ate generalisation le vels attrib uted to the physics-informed approach in T able 7. Concluding remarks The Phy-ID frame work combines observ ational, inductiv e, and learning biases, ensuring a balanced approach to impact identiﬁcation that prioritises physically admissible predic- tions, data efﬁciency , and interpretability . This integrated method pro vides a cohesive solution for monitoring impact ev ents in composite structures, o vercoming the limitations of reference methods under realistic conditions. The results sho w that the frame work infers impact velocity and mass in a physically coherent manner, enabling impact energy estimation through kinetic consistency . The controlled comparison further indicates a positi ve effect of the imposed physical constraints, reﬂected in a higher proportion of predictions within the tolerance band under identical conditions. Across the ev aluated scenarios, performance remains robust to moderate data reduction and measurement noise. Extrapolation beyond the training range remains feasible in the linear regime. In the damaged regime, howe ver , prediction quality depends on adequate representation of damaged responses in the training set under the present implementation. The conclusions are drawn from a case study designed to ev aluate the frame work under controlled conditions that reproduce key challenges encountered in practical SHM monitoring scenarios. The test campaign inv olves an element-le vel composite structure subjected to multiple impact scenarios under challenging measurement conditions. W ithin this setting, the framework infers physically interpretable quantities, including impact velocity and mass, with quantiﬁed error lev els. The explicit separation of underlying physical parameters within the formulation provides a structured basis that may allow adaptation to different monitoring objecti ves and extension to additional impact attributes through the inclusion of relev ant physical descriptors and gov erning relations. Although the framework demonstrates fav ourable perfor- mance, some aspects require further consideration. First, per - formance depends on the representativ eness of the selected feature set. This assumes that the measured data capture the relev ant physical information. As such, structural conﬁgu- rations outside the training en velope may therefore reduce accuracy . Second, the absence of ground-truth measurements for the inferred physical quantities may af fect direct valida- tion of the parameter estimates in e xperimental campaigns 17 where such measurements are impractical. In addition, the lack of explicit constraints related to failure mechanisms limits performance at higher energy levels, where structural behaviour changes due to damage onset and progression. Lastly , measurement uncertainty is not explicitly repre- sented; consequently , conﬁdence bounds on the inferred quantities are unav ailable, which may limit applicability in safety-critical contexts. Future work should address these limitations along three main directions. T ransfer learning strategies can enhance scalability and data efﬁcienc y . As the frame work relies on domain-in v ariant physical relationships, kno wledge transfer across v arying structures, geometries, and loading regimes becomes feasible, minimising the necessity for extensiv e experimental datasets and reducing computational costs. In this context, the framew ork can further beneﬁt from multi-ﬁdelity formulations, where information from numer- ical simulations complements experimental observations, improving parameter inference when testing data are scarce, costly to obtain, or lack direct measurements of impact parameters. Beyond transfer learning, the frame work can be e xpanded to target additional physical quantities and constraints, such as f ailure criteria, health indicators, damage initiation thresholds, and contact-related parameters, which directly inﬂuence stress distribution and local damage dev elop- ment. Another promising direction in volv es incorporating uncertainty-aware formulations, such as probabilistic sur- rogate modelling or Bayesian parameter inference, which explicitly represent measurement uncertainty and conﬁdence in predicted quantities, thereby supporting informed mainte- nance decisions. Overall, the proposed Phy-ID framew ork provides a robust, interpretable, and versatile foundation for impact identiﬁcation in composite structures. By integrating multiple forms of bias, it deliv ers stable performance under realistic monitoring conditions and facilitates physically meaningful inference that supports engineering analysis and operational decision-making. The identiﬁed limitations and extension pathw ays highlight ho w the frame work can ev olve tow ard scalable impact monitoring solutions within SHM systems. Ackno wledgements This work is part of the PrimaV era Project, which is partly ﬁnanced by the Dutch Research Council (NWO) under grant agreement NW A.1160.18.238. References 1. Ifan Dafydd and Zahra Sharif Khodaei. Analysis of barely visible impact damage se verity with ultrasonic guided lamb wa ves. Structural Health Monitoring , 2019. doi: 10.1177/ 1475921719878850. 2. Nan Y ue, Zahra Sharif Khodaei, and M H Aliabadi. 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Physics-Informed Framework for Impact Identification in Aerospace Composites

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