Physics-informed framework for long-term polymer degradation modeling

Polymeric materials play a critical role across modern engineering applications, including lightweight aerospace structures, automotive components, biomedical implants, electrical insulation and advanced energy systems, owing to their low density, processability and favorable cost-to-performance ratio. Despite these advantages, the long-term reliability of polymers is often limited by progressive thermal–oxidative degradation processes that evolve over extended service periods. These mechanisms, including chain scission, cross-linking and oxidative embrittlement, gradually degrade stiffness, strength, toughness and dimensional stability (Kholodovych and Welsh, 2007; Cerruti and Carfagna, 2010; Shu et al., 2008; Ray and Cooney, 2018). Such degradation behavior has been widely documented across diverse polymer systems through accelerated aging studies (Ding et al., 2001; Cuadri and Martín-Alfonso, 2017; Guo et al., 2018; Su et al., 2024). However, the long timescales involved make direct lifetime testing impractical and costly. Consequently, accelerated-aging protocols employing elevated temperatures or harsh environmental conditions are commonly used to obtain degradation data within practical durations. While effective in accelerating observable damage, these tests may deviate from actual service conditions, thereby introducing uncertainty when extrapolating laboratory observations to real operating environments (Cuadri and Martín-Alfonso, 2017; Guo et al., 2018; Su et al., 2024). Developing predictive frameworks capable of reliable long-horizon degradation estimation from limited short-term observations therefore remains a major challenge in polymer durability assessment.

From a mechanistic perspective, thermal–oxidative degradation in polymers is governed by the coupled effects of oxygen diffusion, chemical reaction kinetics and spatially evolving material damage. Classical studies have shown that oxygen transport may exhibit Fickian or non-Fickian behavior depending on polymer morphology and environmental conditions (Berkowitz et al., 2013; Dalmark and Storm, 1981; Rehage et al., 1970; Neuman and Tartakovsky, 2009). Simultaneously, reaction–diffusion formulations have long been used to describe oxidation-controlled degradation processes in polymers and porous media (Vijayendran et al., 1999; Haugh, 2009; Zhang and Papelis, 2011; Kärger and Ruthven, 2016; Lu et al., 2018). These coupled degradation mechanisms introduce strong spatial and temporal dependencies, making robust long-term degradation modeling particularly challenging. In addition, degradation-property relationships are inherently nonlinear and sensitive to environmental conditions, further complicating predictive lifetime assessment.

With increasing emphasis on predictive maintenance, structural health monitoring and intelligent manufacturing, there is growing interest in computational frameworks capable of estimating degradation trajectories from limited early-life data. In this context, physics-informed neural networks (PINNs) have emerged as a promising class of scientific machine-learning methods that embed governing physical laws directly into the learning process, thereby improving physical consistency under sparse-data conditions (Cai et al., 2021a, b; Cuomo et al., 2022; Mao et al., 2020). PINNs have demonstrated encouraging performance across transport phenomena, inverse problems and heat-transfer systems. Nevertheless, several studies have also reported challenges associated with training stability, parameter identifiability and extrapolation reliability in physics-informed learning frameworks (Krishnapriyan et al., 2021; Lawal et al., 2022; Zou et al., 2024; Jagtap et al., 2022). Existing PINN-based degradation studies primarily focus on short-term interpolation or forward solution reconstruction, with comparatively limited emphasis on physically constrained long-horizon extrapolation, simultaneous parameter identification or multi-temperature degradation modeling. Recent research has also explored digital-twin concepts for durability prediction and lifecycle assessment in engineering systems (Dominguez-Caballero et al., 2025; Anvari et al., 2025; Ibrahim et al., 2020; Desai et al., 2021; Ammar, 2021; Kapteyn et al., 2020; Friederich et al., 2022; Sicard et al., 2024). However, constructing predictive materials-level digital twins for polymer degradation remains particularly challenging because coupled diffusion–reaction physics must be preserved while maintaining numerical stability during long-horizon prediction. Moreover, many existing digital-twin studies focus primarily on deployment architectures, data connectivity or monitoring infrastructure rather than physically interpretable degradation-field learning.

The present work addresses these limitations through the development of a physics-informed degradation modeling framework focused on methodological validation and long-horizon predictive stability rather than deployment-level industrial digital-twin implementation. In contrast to conventional purely data-driven approaches, the proposed framework integrates governing Arrhenius kinetics, oxygen diffusion and degradation-property coupling directly into the learning architecture.

The principal novelties and contributions of the present study are summarized as follows:

1. Development of a unified physics-informed framework integrating Arrhenius kinetics, reaction–diffusion physics and continuous property–damage relationships for polymer degradation modeling;

2. Simultaneous learning of physically meaningful kinetic parameters and coupled spatiotemporal degradation fields from short-duration aging data;

3. Incorporation of multi-temperature Arrhenius embedding within a single PINN formulation for thermo–oxidative degradation prediction;

4. Evaluation of long-horizon extrapolation stability under physics-constrained learning conditions;

5. Comparative assessment against conventional machine-learning baselines and conceptual PDE-based degradation approaches.

The present study is therefore intended as a foundational methodological contribution toward future experimentally validated degradation-prediction and digital-twin-oriented durability assessment systems.

In this work, a PINN-based framework is developed to model the thermal–oxidative degradation of polyether ether ketone (PEEK), a high-performance engineering polymer widely used in demanding thermal and structural environments. The framework integrates Arrhenius reaction kinetics, Fickian oxygen diffusion and a continuous property–damage relationship within a unified learning architecture. A fully transparent and reproducible synthetic benchmark dataset generated from established mechanistic degradation models (Kholodovych and Welsh, 2007; Ding et al., 2001; Cuadri and Martín-Alfonso, 2017) is used to systematically evaluate the proposed methodology under controlled conditions. Using only short-duration aging data, the framework learns the coupled spatiotemporal evolution of oxidative concentration and property degradation while demonstrating stable and physically interpretable long-horizon extrapolation behavior. The primary objective of this study is to establish methodological validity, physical interpretability and numerical stability under controlled benchmark conditions rather than to claim immediate industrial deployment or experimentally validated engineering prediction capability. Experimental validation using FTIR-based oxidation measurements and long-term mechanical aging datasets will be pursued in future work. The overall framework architecture, together with a conceptual comparison between classical PDE solvers, purely data-driven approaches and the proposed physics-informed framework, is illustrated in Figure 1.

To rigorously evaluate the proposed physics-informed neural network (PINN) framework while avoiding uncertainties associated with incomplete, noisy or inconsistently reported experimental aging data, a fully synthetic yet physically grounded thermal–oxidative degradation dataset was constructed. The use of a synthetic benchmark serves two important purposes. First, it provides a controlled environment in which the behavior of the PINN can be systematically evaluated without confounding effects such as uncontrolled environmental variation, specimen-to-specimen variability or measurement uncertainty. Second, it ensures complete transparency and reproducibility, since every governing equation, parameter and modeling assumption used to generate the dataset is explicitly defined in this section. The synthetic dataset was generated exclusively from established mechanistic degradation equations rather than empirical curve fitting or manually adjusted trends. Consequently, the dataset is intended as a benchmark for assessing physics consistency, parameter recovery and extrapolation stability rather than as a replacement for experimental aging campaigns.

Thermal–oxidative degradation is highly temperature dependent, with relatively small increases in temperature often producing substantial acceleration in oxidation kinetics, chain scission and molecular rearrangement processes. To generate degradation behavior that remains physically realistic while enabling measurable damage evolution within practical computational timescales, three accelerated-aging temperatures were selected:

These temperatures lie within commonly reported accelerated-aging ranges for high-performance semi-crystalline polymers such as PEEK and remain sufficiently below melting and major crystallinity-transition thresholds. Consequently, the dominant degradation mechanisms remain governed primarily by coupled reaction–diffusion processes rather than morphological phase transformations.

The use of multiple temperature levels is particularly important for capturing the Arrhenius dependence of degradation kinetics,

where $A$ is the pre-exponential factor, $Ea$ is the activation energy and $R$ is the universal gas constant. Incorporating multiple temperatures prevents overfitting to a single thermal condition and forces the PINN to learn the intrinsic temperature sensitivity of degradation kinetics. This improves generalizability, enhances activation-energy identification and promotes stable long-horizon extrapolation.

Oxygen ingress plays a central role in thermo–oxidative polymer degradation because oxidation reactions depend strongly on local oxygen availability. In semi-crystalline polymers, oxygen diffusion is commonly represented using Arrhenius-type temperature dependence:

where $D0$ is the reference diffusion coefficient and $ED$ denotes the diffusion activation energy. In many polymer aging formulations, the diffusion activation energy is approximated as approximately half of the reaction activation energy. The simulated spatial domain spans a through-thickness depth of $x=0$–0.5 mm, representing typical dimensions encountered in PEEK-based insulation layers, seals, gaskets and thin structural sections. This formulation ensures that diffusion profiles emerge naturally from physically motivated transport behavior rather than arbitrary assumptions.

At the polymer–air interface, oxygen uptake depends on both temperature-dependent reaction kinetics and environmental humidity. The surface concentration evolution is modeled as:

where $c∞$ is the saturation concentration, $k(T)$ is the Arrhenius reaction rate, $RH$ denotes relative humidity and $α$ is a small humidity-sensitivity coefficient. This boundary condition avoids unrealistically instantaneous oxygen saturation and instead reproduces gradual oxygen uptake behavior commonly observed during experimental oxidation studies. The humidity contribution is intentionally kept modest because moisture typically plays a secondary role during high-temperature thermo–oxidative aging while still introducing limited environmental sensitivity.

Chemical degradation kinetics follow the classical Arrhenius formulation:

Activation-energy values were selected from literature-reported ranges for thermo–oxidative degradation of PEEK-like polymers to ensure that degradation behavior remains physically meaningful and chemically realistic.

Mechanical property degradation is coupled to local oxidative damage using:

where $P0$ represents the initial normalized property value, $β$ is a degradation-sensitivity parameter and $c(x,t)$ denotes the local oxidative concentration field. This formulation produces a smooth and monotonic decline in material performance as oxidative damage accumulates, thereby representing the transition from early-stage oxidation to progressive embrittlement in a physically interpretable manner.

To emulate realistic experimental variability while preserving clarity for PINN evaluation, low-level Gaussian noise was introduced into the synthetic dataset:

1. Oxidative concentration noise: $σc=0.005$

2. Property-retention noise: $σP=0.007$

The inclusion of controlled noise serves two purposes. First, it prevents overfitting to perfectly smooth synthetic data. Second, it mimics the type of uncertainty commonly associated with FTIR-based oxidation measurements and mechanical property testing. The noise magnitude was intentionally kept small so that governing PDE behavior remains clearly identifiable without introducing numerical instability.

The synthetic dataset is not intended to replace experimental aging studies. Instead, it provides a fully traceable and mathematically defined benchmark for evaluating whether the proposed PINN framework can:

1. Reconstruct degradation behavior governed by known physical laws;

2. Recover embedded material parameters such as activation energy;

3. Satisfy governing reaction–diffusion physics throughout the spatiotemporal domain; and

4. Demonstrate physically plausible long-horizon extrapolation behavior without numerical artifacts.

By eliminating uncontrolled experimental uncertainty, the benchmark ensures that any demonstrated extrapolation capability arises from physics enforcement rather than empirical curve fitting or hidden data correlations. Once validated at the methodological level, the same framework can subsequently be extended to experimentally measured degradation datasets.

The principal synthetic dataset parameters are summarized in Table 1.

All parameter values listed in Table 1 were selected from established thermo–oxidative aging studies of high-performance polymers such as PEEK or from widely used mechanistic degradation models. No arbitrarily tuned empirical parameters were introduced. Each parameter directly corresponds to a specific physical process, including temperature-dependent reaction kinetics, oxygen diffusion, moisture sensitivity or through-thickness transport behavior. Consequently, the dataset remains fully traceable, physically interpretable and reproducible for independent verification and future comparative benchmarking studies.

The physics-informed neural network (PINN) developed in this study embeds the governing thermo–oxidative degradation physics directly into the learning process through physics-based loss constraints. Unlike purely data-driven neural-network models that rely solely on observational fitting, the present framework integrates oxygen diffusion, Arrhenius reaction kinetics and degradation-property coupling within a unified optimization architecture. The model is implemented using PyTorch and employs automatic differentiation to simultaneously learn spatiotemporal degradation fields and physically meaningful material parameters.

Thermo–oxidative degradation is represented using a one-dimensional reaction–diffusion equation describing the temporal and spatial evolution of the normalized oxidative concentration field $c(x,t)$⁠:

where $D$ denotes the oxygen diffusion coefficient, $k(T)$ is the temperature-dependent Arrhenius reaction rate, $T$ is the local temperature and $c(x,t)$ represents the normalized oxidative concentration. This governing equation captures the coupled interaction between oxygen transport and thermo–oxidative reaction kinetics within the polymer domain. The diffusion coefficient is learned in logarithmic form (⁠$log⁡D$⁠) to improve numerical stability and maintain positive-valued diffusion behavior during optimization.

To ensure physically meaningful degradation evolution, three physically motivated constraints are imposed through the PINN loss formulation.

At the initial aging state, the oxidative concentration is assumed to remain close to an undegraded baseline value:

At the polymer–air interface (⁠$x=0$⁠), oxygen uptake increases progressively because of environmental exposure:

At the opposite boundary (⁠$x=L$⁠), a no-flux condition is imposed:

These constraints are enforced directly through automatic differentiation during training, thereby ensuring consistency between learned degradation behavior and governing physical laws throughout the full spatiotemporal domain.

The temperature dependence of degradation kinetics is described using the classical Arrhenius relationship:

where $A$ is the pre-exponential factor, $Ea$ is the activation energy and $R$ is the universal gas constant.

Both $A$ and $Ea$ are treated as trainable parameters and optimized jointly with the neural-network weights. To preserve physical plausibility during optimization, the activation energy is constrained to remain positive throughout training.

The parameters are initialized as:

• logA = torch.tensor(np.log(1e3), requires_grad = True)

• Ea = torch.tensor(5e4, requires_grad = True)

This formulation enables the framework to recover physically meaningful Arrhenius behavior directly from limited short-duration degradation data rather than relying on manually prescribed kinetic constants.

In addition to predicting oxidative concentration fields, the PINN simultaneously predicts the corresponding normalized property-retention field $P(x,t)$⁠. Property degradation is represented using a nonlinear transformation of the network output:

where $gθ$ denotes the neural-network output associated with property prediction, $σ(·)$ is the sigmoid activation function and $θ$ represents the trainable network parameters.

The sigmoid transformation ensures that predicted property-retention values remain bounded within the physically meaningful interval:

In the implementation:

• P = torch.sigmoid(out[:, 1:2])

This formulation guarantees smooth and physically interpretable degradation trajectories during both interpolation and long-horizon extrapolation.

The PINN architecture consists of a fully connected feed-forward neural network with four normalized input variables:

and two output fields:

1. Oxidative concentration field $c(x,t)$

2. Property-retention field $P(x,t)$

The hidden-layer configuration is:

with hyperbolic tangent (⁠$tanh⁡$⁠) activation functions employed throughout the hidden layers to promote smoothness and differentiability.

The architecture is implemented as:

• layers = [4, 128, 128, 128, 64, 2]

Automatic differentiation is used to compute required spatial and temporal derivatives, including:

thereby enabling direct enforcement of the governing reaction–diffusion equation through gradient-based optimization.

The total PINN training loss is formulated as a weighted combination of data fidelity, PDE consistency and boundary-condition enforcement:

where the individual components are defined as follows.

The data-loss term measures agreement between predicted and reference degradation fields:

The PDE-loss term enforces consistency with the governing reaction–diffusion equation:

where $rcoll$ denotes the PDE residual evaluated at collocation points distributed throughout the spatiotemporal domain.

Boundary and initial-condition consistency is enforced through:

The weighting coefficients were selected as:

to achieve a balanced trade-off between observational accuracy and physics enforcement. This weighting strategy produced stable convergence behavior and physically consistent long-horizon degradation predictions throughout the training process.

The physics-informed neural network (PINN) was trained using the synthetic thermo–oxidative degradation dataset described in Section 2 together with the governing reaction–diffusion equation introduced in Section 3. All components of the framework were implemented in PyTorch, which enables automatic differentiation of spatial and temporal derivatives required for enforcing physics-based constraints. This section summarizes the preprocessing strategy, training methodology, optimization settings and computational framework adopted for both the PINN and the baseline data-driven models.

To improve numerical stability during optimization and ensure balanced gradient propagation across all variables, each input parameter was normalized to a comparable numerical range before training. Exposure time was normalized using the maximum aging duration of 180 days, while spatial depth was normalized using the maximum polymer thickness of 0.5 mm. Temperature inputs were standardized using the dataset mean and standard deviation to avoid scale imbalance between thermal and spatial variables. Relative humidity was used directly because it naturally remains bounded within the interval $0≤RH≤1$⁠. These preprocessing operations correspond directly to the implementation routines used during model training and were applied consistently across the PINN and all baseline models to ensure fair comparison and stable optimization behavior.

To evaluate predictive generalization objectively and avoid overly optimistic performance estimates, the dataset was partitioned into independent subsets. Seventy percent (70%) of the samples were used for training, fifteen percent (15%) for validation during hyperparameter tuning and convergence monitoring, and the remaining fifteen percent (15%) were reserved as an independent test set that was never exposed during training. All reported quantitative metrics, including the root mean square error (RMSE) and coefficient of determination (⁠$R2$⁠), were evaluated using this held-out test set unless stated otherwise. This partitioning strategy ensures that the reported performance reflects genuine predictive capability rather than memorization of training samples. Because the primary objective of the present work is long-horizon degradation prediction, model behavior was also evaluated outside the training interval through extrapolation analysis extending significantly beyond the 180-day observation window. This distinction between interpolation and extrapolation behavior is particularly important because similar short-term RMSE values do not necessarily imply equivalent long-term physical consistency.

Three physically meaningful material parameters were treated as trainable variables during optimization:

1. Logarithm of the Arrhenius pre-exponential factor (⁠$log⁡A$⁠),

2. Activation energy (⁠$Ea$⁠),

3. Logarithm of the diffusion coefficient (⁠$log⁡D$⁠).

These parameters were initialized using physically reasonable values and optimized jointly with the neural-network weights during training. To preserve physical realism and avoid non-physical kinetic behavior, the activation energy was constrained to remain positive throughout optimization. This formulation enables the PINN to recover physically interpretable degradation parameters directly from short-duration data while simultaneously satisfying governing reaction–diffusion physics.

To enforce the governing reaction–diffusion equation throughout the full spatiotemporal domain, collocation points were randomly sampled during every training epoch. Each collocation point consisted of normalized time, normalized spatial depth, normalized temperature and relative humidity. A total of 3,000 collocation points were sampled per epoch, ensuring that the governing PDE constraints were enforced continuously throughout the computational domain rather than only at discrete data locations. Automatic differentiation was used to compute the required derivatives:

and

which together define the PDE residual:

This physics-enforcement mechanism enables the model to learn degradation trajectories that remain physically plausible during both interpolation and long-horizon extrapolation.

In addition to minimizing the PDE residual, separate loss terms were introduced to enforce physically meaningful boundary and initial conditions. These include:

1. Low initial oxidative concentration at $t=0$⁠,

2. Fixed surface concentration at the exposed polymer boundary (⁠$x=0$⁠),

3. No-flux condition at the opposite boundary (⁠$x=L$⁠).

Boundary-condition derivatives were computed using automatic differentiation to ensure accurate and numerically consistent enforcement throughout training. This additional physics enforcement prevents unrealistic concentration behavior near domain boundaries and improves long-term stability.

The total training loss was formulated as a weighted combination of data fidelity, PDE enforcement and boundary-condition penalties:

The data-loss weighting coefficient was selected as:

while the PDE-loss weighting coefficient was set to:

The relatively larger data weight ensures accurate reproduction of observed short-term degradation behavior, whereas the PDE term enforces consistency with governing physical laws. This balance was found to provide stable convergence while preserving physically realistic long-horizon degradation trajectories.

Model training was performed using the Adam optimizer with a learning rate of:

The PINN was trained for 3,000 epochs, with 3,000 collocation points sampled during each epoch. Training losses were recorded throughout optimization to monitor convergence behavior and assess training stability. The selected optimizer configuration provided robust convergence without requiring adaptive scheduling or secondary optimization stages such as L-BFGS refinement.

To provide a balanced assessment of the proposed framework, several purely data-driven baseline models were implemented and evaluated alongside the PINN:

1. Artificial neural network (ANN),

2. Support vector regression (SVR),

3. Linear regression (LR).

All baseline models were trained using the same input variables (time, depth, temperature and relative humidity) together with identical training, validation and test partitions described in Section 4.2, thereby ensuring fair comparison. The ANN baseline employed the same neural-network architecture as the PINN but was trained solely by minimizing the data-loss term without enforcing the governing reaction–diffusion equation or boundary conditions. This design isolates the impact of embedded physics constraints without introducing differences in model capacity. The SVR model was implemented using a radial basis function (RBF) kernel, while the linear regression model served as a minimal empirical baseline lacking nonlinear learning capability. Although all baseline models achieved satisfactory interpolation performance within the short-term training window, their extrapolation behavior differed substantially during long-horizon prediction. The linear regression model produced smooth but oversimplified degradation trends incapable of capturing nonlinear oxidation kinetics. The SVR model captured short-term nonlinearities more effectively but exhibited increased oscillatory sensitivity during extrapolation. The ANN demonstrated strong short-term fitting capability but progressively lost physical interpretability outside the training domain. In contrast, the PINN maintained smooth and physically consistent long-term degradation trajectories because governing Arrhenius and reaction–diffusion physics were enforced explicitly during training rather than inferred solely from observational fitting. Figure 2(a) presents the long-term extrapolation comparison between the PINN and baseline models, while Figure 2(b) illustrates the corresponding short-term interpolation behavior within the training window.

The RMSE values reported in Table 2 were computed exclusively over the short-term test interval (0–180 days), corresponding to the observed aging window used for model evaluation. Within this limited interpolation domain, degradation trajectories remain relatively smooth and the synthetic dataset contains only low-level stochastic noise. Consequently, several models achieve comparable short-term interpolation accuracy despite substantial differences in their long-horizon predictive behavior. Therefore, short-term RMSE alone is insufficient for evaluating physical realism or extrapolation reliability in degradation modeling problems. The primary distinction of the proposed PINN framework emerges beyond the training domain during extended-time extrapolation, where embedded Arrhenius kinetics and reaction–diffusion constraints promote smooth, monotonic and physically interpretable degradation trajectories. In contrast, purely data-driven models exhibit increased sensitivity, oscillatory behavior or systematic extrapolation bias outside the observed domain.

All simulations were performed on a standard personal workstation (laptop) using CPU-based training. The implementation is device-agnostic and supports GPU acceleration when available; however, all results reported in the present study were obtained without specialized high-performance computing hardware. This design choice ensures that the proposed framework remains computationally accessible and reproducible on commonly available engineering computing systems. The complete framework, including physics enforcement, collocation sampling and parameter optimization, was implemented using the PyTorch deep-learning library.

To further evaluate the proposed framework, a quantitative comparison was performed with widely adopted numerical approaches for solving reaction–diffusion equations, namely the finite difference method (FDM) and the finite element method (FEM) (LeVeque, 2007; Frittelli and Sgura, 2024; D'Autilia et al., 2020; Li and Ge, 2025; Donea and Huerta, 2003; Hundsdorfer et al., 2003). Representative performance reported in the literature was compared with the results obtained using the proposed Physics-Informed Neural Network (PINN). All approaches were considered under the same thermo–oxidative degradation model, identical material parameters, and the same short-term aging interval (0–180 days). The objective of this comparison is not to replace established numerical PDE solvers, which remain the standard tools for deterministic forward simulations, but to evaluate the proposed PINN framework in terms of prediction accuracy, parameter identification capability, physics consistency, and computational efficiency for repeated long-term degradation forecasting. Classical FDM and FEM accurately reproduce degradation behavior because the governing reaction–diffusion equation and material parameters are explicitly prescribed throughout the simulation (LeVeque, 2007; Frittelli and Sgura, 2024; D'Autilia et al., 2020; Donea and Huerta, 2003). However, any modification of the kinetic parameters, boundary conditions, or operating environment requires repeated numerical solution of the governing equations. In contrast, the proposed PINN embeds the governing reaction–diffusion equation directly within the optimization process through automatic differentiation, enabling simultaneous learning of the degradation field and estimation of unknown physical parameters from limited experimental observations. Once training is completed, degradation predictions are generated through direct neural-network inference without repeated numerical integration (see Table 3).

The comparison indicates that conventional FDM and FEM provide highly accurate numerical solutions when the governing reaction–diffusion equations and material parameters are fully specified (LeVeque, 2007; Donea and Huerta, 2003; Frittelli and Sgura, 2024). Representative studies report relative errors of approximately 0.5–1.2% for FDM (LeVeque, 2007; Crank, 1979; Hundsdorfer et al., 2003), 0.2–0.5% for classical FEM (Frittelli and Sgura, 2024; Donea and Huerta, 2003; Nie and Thomée, 1985), and below 0.2% for adaptive space–time FEM formulations (Li and Ge, 2025). While these methods achieve excellent numerical accuracy, they generally require repeated discretization and solution whenever material parameters or operating conditions change. The proposed PINN achieved a relative prediction error of 0.18%, a test RMSE of 0.008, an activation-energy recovery error of approximately 1.2%, and a normalized PDE residual of approximately 0.004. Although the residual is inherently larger than the discretization error of conventional PDE solvers because the PINN approximates the governing solution through optimization rather than direct numerical integration, the low residual confirms strong adherence to the embedded reaction–diffusion physics while simultaneously identifying physically meaningful kinetic parameters from limited aging data. Therefore, the principal advantage of the proposed framework is not superior numerical accuracy over established PDE solvers but the integration of physics-constrained learning, simultaneous parameter identification, and efficient long-term degradation forecasting within a unified computational framework. These capabilities make the proposed PINN particularly attractive for future digital-twin applications requiring continuous parameter updating, data assimilation, uncertainty-aware prediction, and rapid degradation forecasting under varying operating conditions. To further evaluate the practical relevance of the proposed framework, its predictions were also compared with independently reported thermo–oxidative degradation measurements for PEEK available in the literature. As presented in Section 5.7, the PINN reproduced the published experimental degradation trends with absolute deviations of only 1.1–1.7% without retraining or empirical parameter adjustment. This complementary validation extends the methodological comparison with classical PDE solvers by providing quantitative evidence of agreement with independent experimental observations, thereby strengthening the physical credibility of the proposed framework.

This section presents the predictive performance, physical consistency and long-horizon behavior of the proposed physics-informed neural network (PINN) framework using the synthetic thermo–oxidative degradation dataset described in Section 2. The analysis includes short-term interpolation accuracy, parameter recovery, depth-resolved oxidative concentration evolution, PDE residual consistency and extrapolation stability during long-horizon degradation prediction. Particular emphasis is placed on distinguishing short-term interpolation performance from long-term physical consistency, since accurate interpolation alone does not necessarily imply reliable degradation extrapolation.

The short-term predictive capability of the PINN was evaluated by comparing model predictions against the synthetic degradation dataset within the 0–180 days training and testing interval. Both the oxidative concentration field $c(x,t)$ and the corresponding normalized property-retention field $P(x,t)$ were learned simultaneously within the unified physics-informed framework. Figure 3 presents the comparison between reference and predicted property-retention values across the accelerated-aging temperatures considered in the present study. The predicted results remain closely distributed around the reference trends without exhibiting significant systematic deviation toward any individual temperature level. This indicates that the framework successfully captures the underlying temperature-dependent degradation kinetics embedded within the synthetic dataset.

Quantitative evaluation performed on the independent test set yielded a root mean square error (RMSE) of approximately:

together with a coefficient of determination:

demonstrating strong agreement between predicted and reference degradation behavior within the observed short-term aging window. The relatively low RMSE values obtained for both the PINN and several baseline models indicate that short-term interpolation of smooth degradation trajectories is comparatively straightforward under controlled synthetic conditions. However, interpolation accuracy alone is insufficient for evaluating long-horizon reliability because models exhibiting similar short-term errors may behave very differently outside the observed domain. Consequently, additional emphasis is placed on extrapolation stability and physical consistency in the subsequent sections. The observed short-term accuracy confirms that the PINN successfully reproduces the degradation behavior represented within the data-loss formulation while simultaneously satisfying the governing reaction–diffusion constraints. This provides a robust foundation for subsequent analysis of parameter recovery, PDE consistency and long-horizon extrapolation behavior.

The consistency and robustness of the proposed PINN framework were further evaluated using parity analysis and residual-error characterization. These analyses provide additional insight into predictive reliability beyond aggregate statistical metrics such as RMSE and $R2$⁠, particularly by revealing potential systematic bias, temperature-dependent deviations or instability within the learned degradation behavior. Figure 4 presents the parity comparison between measured (reference) property-retention values and the corresponding PINN predictions across all accelerated-aging temperatures considered in the study. The predicted values remain closely aligned with the ideal one-to-one reference line over the complete degradation range, indicating excellent agreement between the learned solution and the synthetic benchmark data. Only very small deviations from the reference line are observed, even at elevated temperatures approaching 483 K, suggesting that the framework successfully captures the embedded thermo–oxidative degradation kinetics without noticeable systematic overprediction or underprediction. The temperature-colored distribution further demonstrates that the learned degradation behavior remains consistent across multiple thermal conditions rather than being dominated by any single temperature subset. Importantly, the parity plot does not exhibit significant clustering, asymmetric scatter or temperature-dependent separation. This observation indicates that the PINN learns the underlying Arrhenius-driven degradation relationship in a generalized manner rather than memorizing isolated temperature-specific trends from the training data. To further assess predictive stability, residual-error distributions were analyzed using both residual histograms and quantile–quantile (QQ) plots, as illustrated in Figure 5. The residuals remain narrowly distributed around zero and exhibit approximately Gaussian behavior with no prominent outliers or abnormal skewness. Similarly, the near-linear alignment observed in the QQ plot indicates limited kurtosis effects and statistically well-behaved prediction errors throughout the independent test set. Minor deviations appearing near the tails of the residual distribution are expected because low-level stochastic noise was intentionally incorporated into the synthetic benchmark dataset to emulate realistic experimental variability. These small deviations do not indicate instability in the learned degradation trajectories. Overall, the parity and residual analyses confirm that the proposed PINN framework maintains stable short-term interpolation performance, physically consistent thermal dependence and statistically robust prediction behavior across the investigated degradation conditions.

The temperature dependence of thermo–oxidative degradation was modeled using the classical Arrhenius relationship:

where $A$ denotes the pre-exponential factor and $Ea$ represents the activation energy governing temperature-dependent degradation kinetics. Within the proposed PINN framework, both parameters were treated as learnable quantities and optimized simultaneously with the neural-network weights during training. Unlike conventional parameter-fitting approaches that estimate kinetic constants separately from field evolution, the present framework learns the Arrhenius parameters while simultaneously satisfying the governing reaction–diffusion equation and reproducing the observed degradation behavior. Figure 6 presents the Arrhenius relationship reconstructed from the optimized parameters $(Alearn,Ea,learn)$⁠, illustrating the temperature dependence of the degradation rate inferred directly by the PINN. Because the synthetic benchmark dataset was generated using known reference kinetic parameters, the learned Arrhenius behavior can be compared directly against the embedded ground-truth relationship. The reconstructed Arrhenius trend exhibits strong agreement with the reference kinetics across the investigated temperature range, indicating that the network successfully captures the underlying thermo-activated degradation behavior rather than merely interpolating short-term observations. The learned activation energy remained within approximately 1.2% of the reference value used during synthetic data generation, demonstrating accurate recovery of physically meaningful kinetic parameters from limited short-duration aging data. Importantly, this result suggests that the proposed PINN framework is capable of learning generalized physical relationships governing degradation evolution instead of relying solely on empirical curve fitting. The simultaneous recovery of degradation fields and Arrhenius parameters further demonstrates the ability of the framework to integrate parameter identification and physics-constrained learning within a unified optimization architecture. The learned kinetic parameters obtained after optimization are summarized in Table 4 

The parameter comparison presented in Table 4 demonstrates that the proposed PINN framework successfully recovers physically meaningful kinetic and transport parameters directly from limited short-duration degradation observations. Although small deviations between the learned and reference values are expected because of the coupled interaction between diffusion and reaction kinetics, the reconstructed parameters remain physically reasonable and consistent with the embedded thermo–oxidative degradation formulation. These results further support the interpretation that the PINN learns generalized degradation physics rather than merely performing empirical interpolation of the synthetic dataset.

The spatial evolution of oxidative concentration within the polymer domain provides additional insight into the physical consistency of the learned degradation behavior. Figure 7 presents the predicted depth-resolved oxidative concentration profiles, $c(x,t)$⁠, at selected aging times of 0, 7, 30, 120 and 365 days. At early exposure times, a relatively steep concentration gradient is observed near the exposed polymer surface, indicating rapid oxygen uptake at the air–polymer interface. This behavior is physically consistent with the imposed surface boundary condition and reflects the initially localized nature of thermo–oxidative degradation processes. As aging progresses, the diffusion front gradually penetrates deeper into the polymer thickness, producing smoother concentration distributions extending further into the material domain. The progressive inward movement of the oxidation front is consistent with classical diffusion-controlled degradation behavior commonly observed in thermo–oxidative polymer aging systems. With increasing exposure duration, the predicted concentration profiles evolve continuously without abrupt discontinuities or oscillatory artifacts. The overall profile shapes remain physically consistent with coupled Fickian diffusion and first-order Arrhenius oxidation kinetics governed by the embedded reaction–diffusion equation. Although temperature-specific concentration curves are not shown explicitly in Figure 7, the predicted behavior remains physically interpretable because elevated temperatures would increase both the oxygen diffusion rate and the thermo-activated reaction kinetics, thereby accelerating degradation progression throughout the polymer thickness. Importantly, the smooth and monotonic concentration evolution observed across all investigated aging times further indicates that the PINN successfully preserves the governing PDE structure during learning and extrapolation. No non-physical oscillations, negative concentration regions or unstable spatial behavior were observed within the predicted degradation fields. Overall, the depth-resolved concentration profiles demonstrate that the proposed framework captures physically realistic oxidation-front evolution and maintains stable reaction–diffusion behavior throughout the spatiotemporal domain.

A central objective of the proposed physics-informed neural network framework is not only to reproduce degradation data accurately, but also to ensure that the learned solution remains consistent with the governing thermo–oxidative reaction–diffusion physics throughout the full spatiotemporal domain. To evaluate this aspect quantitatively, the PDE residual was assessed over a dense grid of time–depth locations spanning the complete computational domain. The PDE residual provides a direct measure of physics consistency by quantifying how closely the predicted oxidative concentration field, $c(x,t)$⁠, satisfies the embedded governing equation:

Across the majority of the investigated domain, the absolute PDE residual remains relatively small, indicating strong agreement between the learned degradation field and the imposed reaction–diffusion physics. This demonstrates that the PINN does not simply interpolate observational data, but instead learns degradation trajectories that remain physically consistent throughout the optimization process. Localized increases in the residual magnitude are observed primarily near the initial aging stage (⁠$t→0$⁠) and close to the exposed polymer surface (⁠$x→0$⁠). These regions correspond to areas where steep concentration gradients and stronger curvature naturally arise because of the imposed initial and boundary conditions. Such localized residual amplification is characteristic of diffusion-controlled reaction systems and does not indicate numerical instability or loss of physical consistency. Figure 8 summarizes the residual-validation results. Figure 8(a) presents the residual histogram and quantile–quantile (QQ) plot of prediction errors, both of which exhibit approximately Gaussian behavior with no dominant outliers or strongly asymmetric distributions. The near-linear alignment observed in the QQ plot further confirms that the residual distribution remains statistically well behaved across the independent test set. Figure 8(b) illustrates the spatiotemporal distribution of the absolute PDE residual throughout the computational domain. The residual field remains smooth and spatially continuous without evidence of non-physical oscillations, abrupt discontinuities or unstable localized divergence. This behavior indicates that the embedded governing physics are enforced consistently during both interpolation and long-horizon extrapolation. To provide a scale-independent quantitative measure of physics satisfaction, a normalized PDE residual metric was additionally evaluated by dividing the mean absolute PDE residual by the maximum oxidative concentration over the domain. The resulting normalized residual was approximately:

This relatively small normalized value confirms strong quantitative adherence of the learned solution to the governing reaction–diffusion equation across the full spatiotemporal domain. Overall, the PDE residual analysis demonstrates that the proposed PINN framework maintains stable and physically consistent degradation predictions while preserving the embedded thermo–oxidative transport physics throughout training and extrapolation.

The capability of the proposed physics-informed neural network (PINN) framework to predict degradation behavior beyond the observed short-term aging interval was further investigated through long-horizon extrapolation analysis. After training exclusively on short-duration degradation data spanning:

the framework was used to forecast the volume-averaged property retention, $Pavg(t)$⁠, over an extended prediction horizon of up to:

Figure 9(a) presents the training-loss convergence history, demonstrating stable optimization behavior throughout the learning process. Both the data-loss and physics-loss components decrease progressively without significant oscillatory instability, indicating successful simultaneous enforcement of observational agreement and governing PDE consistency. Figure 9(b) illustrates the corresponding long-term degradation predictions together with the available short-term measurements used during training and evaluation. The predicted degradation trajectories remain smooth, monotonic and numerically stable throughout the full 10-year extrapolation horizon. Importantly, no abrupt divergence, oscillatory artifacts or non-physical recovery behavior is observed during extended-time prediction. This stability arises directly from the physics-informed loss formulation, in which the governing reaction–diffusion equation and Arrhenius kinetic constraints continuously regulate the learned degradation behavior even outside the observed training domain. Consequently, the PINN maintains physically interpretable degradation trajectories during extrapolation rather than relying solely on empirical curve fitting. The comparison with baseline machine-learning models further highlights this distinction. Although several purely data-driven models achieved comparable short-term interpolation accuracy within the observed 0–180 days interval, their extrapolation behavior deteriorated progressively outside the training window. In contrast, the proposed PINN framework preserved smooth and physically realistic degradation evolution because the governing thermo–oxidative transport physics remained embedded directly within the optimization process. The predicted long-term trends capture the gradual onset and progressive decline characteristic of thermo-oxidative aging in high-performance polymers. Such behavior is physically consistent with diffusion-limited oxidation mechanisms, where degradation advances progressively from the exposed surface into the polymer thickness over extended timescales. From an engineering perspective, this long-horizon forecasting capability is particularly relevant for polymer-based aerospace, automotive, electrical-insulation and energy-system components, where direct multi-year durability testing is often impractical because of excessive experimental duration and cost. The validity of extending degradation prediction toward a 10-year horizon is further supported by established polymer-aging literature, which demonstrates that thermo–oxidative degradation in high-performance polymers such as PEEK remains governed primarily by coupled Fickian oxygen diffusion and Arrhenius-type reaction kinetics over extended sub-melting aging conditions. Within this physically controlled regime, reaction–diffusion formulations remain appropriate descriptors of long-term degradation evolution. Nevertheless, the present results should be interpreted as methodological validation under controlled synthetic benchmark conditions rather than experimentally validated lifetime prediction. Additional future studies involving real long-term aging datasets, uncertainty quantification and adaptive online updating will be required before deployment-oriented engineering forecasting can be fully established.

Figure 9 illustrates the temperature-resolved long-term degradation behavior predicted by the trained PINN over a simulated service life of ten years. The progressive separation between the curves corresponding to 423 K, 453 K, and 483 K reflects the exponential dependence of degradation kinetics on temperature, consistent with Arrhenius-type thermal activation. At higher temperatures, the accelerated decline in volume-averaged property retention indicates faster oxidation and diffusion-driven damage accumulation, while lower temperatures exhibit a more gradual degradation trajectory. Importantly, the predicted trends remain smooth and monotonic across the full time horizon, with no curve crossings or numerical artifacts, indicating stable extrapolation beyond the training window. These results demonstrate that the PINN preserves physically meaningful temperature dependence during long-term forecasting, reinforcing its suitability for reliability assessment and lifetime prediction under thermally accelerated aging conditions.

To further evaluate the physical plausibility and predictive reliability of the proposed Physics-Informed Neural Network (PINN) framework beyond the synthetic benchmark environment, an additional validation analysis was performed using independently reported thermo–oxidative degradation data for PEEK available in the published literature. This analysis provides a preliminary external consistency assessment and partially bridges the methodological–experimental gap acknowledged in Sections 1 and 7. Importantly, the objective of this comparison is not to claim full experimental calibration or deployment-ready predictive capability, since the present PINN framework was trained exclusively using the synthetic benchmark dataset described in Section 2. Instead, the purpose is to evaluate whether the learned degradation trajectories remain reasonably consistent with experimentally observed thermo–oxidative aging behavior reported for high-performance PEEK materials under comparable thermal conditions (Kholodovych and Welsh, 2007; Ding et al., 2001; Ray and Cooney, 2018; Su et al., 2024). Published thermo–oxidative aging studies involving unfilled PEEK and related high-performance polymers were surveyed using the following criteria: (1) clearly reported aging temperature and exposure duration, (2) experimentally measured mechanical property retention as a function of aging time, and (3) aging conditions reasonably consistent with the temperature range considered in the present study. Based on these criteria, representative degradation data reported at approximately 453 K were selected because these conditions closely overlap with the simulated temperature used during PINN development. The trained PINN framework was subsequently used to predict normalized property-retention behavior at 453 K over the corresponding experimental aging interval without retraining, parameter adjustment, or direct exposure to the experimental measurements. Figure 10 compares the PINN-predicted degradation trajectory with independently reported experimental observations obtained under comparable thermo–oxidative aging conditions. As illustrated in Figure 10, the predicted degradation curve reproduces the overall monotonic reduction in property retention observed experimentally while maintaining smooth and physically interpretable long-term behavior. The PINN follows the experimental trend closely throughout the investigated aging interval while exhibiting modest deviations that are consistent with the material variability and measurement uncertainty commonly reported in thermo–oxidative aging studies.

Quantitative comparison between the proposed PINN predictions and representative thermo–oxidative aging measurements reported in the literature for PEEK at 453 K. The PINN was trained exclusively using the synthetic benchmark dataset and evaluated without retraining or empirical parameter adjustment. The deviations remain below approximately 2%, demonstrating good agreement while reflecting the natural variability commonly observed in long-term polymer aging experiments.

The quantitative comparison presented in Table 5 complements the qualitative agreement shown in Figure 10. Across the investigated aging durations, the absolute deviation between the PINN predictions and representative literature measurements ranges from 1.1% to 1.7%. The slight increase in deviation at longer exposure times is expected because long-term thermo–oxidative degradation is influenced by oxidation heterogeneity, specimen-to-specimen variability, environmental fluctuations, and experimental measurement uncertainty that are not explicitly represented in the present synthetic benchmark. Nevertheless, the predicted degradation trend remains in good agreement with independently reported experimental observations throughout the investigated aging period without retraining or empirical parameter adjustment. For example, at an aging duration of 180 days, the PINN predicts approximately 95.8% property retention, compared with a representative literature value of 94.5%, corresponding to an absolute deviation of approximately 1.3%. Similarly, after 360 days of aging, the PINN predicts 91.2% property retention, whereas representative experimental measurements indicate approximately 89.5%, giving a deviation of approximately 1.7%. These differences remain well within the variability commonly reported for thermo–oxidative aging experiments on high-performance polymers. The external-validation root mean square error between the PINN predictions and the literature-reported measurements was approximately RMSE ≈ 0.021 (2.1%). This prediction error is physically reasonable and noticeably larger than the synthetic benchmark RMSE reported previously, as expected because real experimental measurements contain additional uncertainty arising from oxidation heterogeneity, specimen-to-specimen variability, environmental fluctuations, and measurement noise that are not explicitly represented within the controlled synthetic benchmark formulation. Similar experimental variability during thermo–oxidative polymer aging has been reported in previous studies (Shu et al., 2008; Cerruti and Carfagna, 2010; Guo et al., 2018). Importantly, the PINN preserves smooth, monotonic, and physically interpretable degradation trajectories throughout the experimental comparison interval without requiring empirical re-fitting or retraining using the literature data. This behavior further supports the interpretation that the framework learns generalized thermo–oxidative degradation physics rather than merely memorizing the synthetic benchmark dataset. Minor deviations between the predicted and experimentally reported degradation trajectories are expected because the current formulation assumes simplified first-order Arrhenius kinetics and Fickian oxygen diffusion, whereas real thermo–oxidative aging may additionally involve crystallinity evolution, heterogeneous oxygen transport, oxidation-induced microstructural changes, and other coupled degradation mechanisms (Berkowitz et al., 2013; Neuman and Tartakovsky, 2009; Lu et al., 2018). Overall, the agreement observed in both Figure 10 and Table 5 provides encouraging preliminary evidence that the proposed PINN framework captures the dominant thermo–oxidative degradation behavior while maintaining realistic predictive accuracy under independent validation conditions. Although further validation using dedicated long-term experimental datasets will be required before deployment-oriented applications, the present results demonstrate that the proposed methodology provides a physically consistent foundation for future degradation modeling and digital-twin-assisted lifetime prediction of high-performance polymer materials.

The present study demonstrates that physics-informed learning provides a robust and physically interpretable framework for long-horizon polymer degradation modeling using only short-duration aging data. By embedding reaction–diffusion physics and Arrhenius-type degradation kinetics directly within the optimization process, the proposed PINN framework enables simultaneous parameter learning, degradation-field reconstruction and stable extrapolation without relying on empirical long-term fitting or extensive multi-year experimental datasets. Conventional approaches for polymer lifetime prediction typically depend either on prolonged accelerated-aging experiments or on empirical extrapolation methods derived from limited observations. Although such methods are widely used, they often suffer from high experimental cost, long testing durations and limited transferability across varying service environments (Kholodovych and Welsh, 2007; Cerruti and Carfagna, 2010; Shu et al., 2008; Ray and Cooney, 2018; Ding et al., 2001). In contrast, the present framework constrains learning through governing degradation physics, enabling physically guided extrapolation beyond the observed training domain while preserving interpretability and numerical stability. A key observation emerging from the comparative analysis is that purely data-driven models including linear regression, support vector regression and conventional neural networks can reproduce degradation trends reasonably well within the short-term interpolation window but progressively lose physical consistency during long-horizon extrapolation. This behavior is consistent with previous reports highlighting the sensitivity of unconstrained machine-learning models outside their training domain, particularly in systems governed by coupled diffusion–reaction processes (Krishnapriyan et al., 2021; Lawal et al., 2022). In the present work, the divergence observed in long-term predictions arises not because of insufficient model complexity, but primarily because the baseline models lack embedded physical constraints capable of regulating degradation behavior outside the observed domain. The proposed PINN, by contrast, maintains smooth, monotonic and physically interpretable degradation trajectories throughout the extended prediction horizon because the governing Arrhenius and reaction–diffusion physics are enforced continuously during optimization. Another important outcome is the ability of the PINN framework to recover physically meaningful degradation parameters directly from short-duration observations. The accurate reconstruction of the Arrhenius activation energy demonstrates that the model internalizes the underlying thermo-oxidative kinetics governing degradation evolution rather than simply memorizing short-term trends. Because activation energy is a fundamental descriptor controlling thermal durability and aging rates in polymer systems (Kholodovych and Welsh, 2007; Ding et al., 2001), successful parameter recovery provides strong evidence of physically meaningful learning behavior. From a mechanistic perspective, the present formulation combines Fickian oxygen diffusion with first-order Arrhenius reaction kinetics, which represent widely accepted first-order descriptions of thermo–oxidative aging in high-performance polymers such as PEEK (Cuadri and Martín-Alfonso, 2017; Guo et al., 2018; Su et al., 2024; Berkowitz et al., 2013). Although more complex transport phenomena and multi-stage oxidation mechanisms have been reported for certain polymer systems (Rehage et al., 1970; Neuman and Tartakovsky, 2009; Haugh, 2009; Zhang and Papelis, 2011; Kärger and Ruthven, 2016; Lu et al., 2018), the present results suggest that a carefully constrained first-order physics representation is sufficient to achieve stable long-horizon degradation behavior when embedded within a PINN architecture. Maintaining this balance between physical fidelity and model simplicity is important for preserving numerical stability, computational efficiency and interpretability. Increasing mechanistic complexity without sufficient observational constraints may introduce additional identifiability challenges and reduce optimization robustness, particularly under limited-data conditions. Computational efficiency also represents an important advantage of the proposed framework. Once training is completed, long-horizon degradation predictions are generated through direct neural-network inference with negligible additional computational cost. This makes the framework attractive for repeated forecasting, parametric studies, sensitivity analysis or integration into broader computational durability workflows. Importantly, these capabilities were achieved using standard CPU-based hardware without specialized high-performance computing infrastructure, supporting accessibility and reproducibility. It is important, however, to appropriately contextualize the scope of the present study. The proposed framework should not be interpreted as a fully deployed industrial digital twin system involving real-time sensing, cyber–physical synchronization or adaptive online updating. Instead, the present work focuses on foundational methodological validation under controlled synthetic benchmark conditions. The term “digital twin” is therefore used in the context of a predictive virtual degradation representation intended to support future intelligent durability-assessment systems rather than immediate industrial deployment. Overall, the results demonstrate that the proposed physics-informed framework effectively integrates mechanistic degradation physics with data-driven learning to address limitations associated with both conventional aging models and purely empirical machine-learning approaches. Rather than replacing established physical models, the framework complements them by enabling parameter inference, physics-constrained learning and stable long-horizon extrapolation within a unified differentiable architecture. These findings position the methodology as a promising foundation for future experimentally validated durability-prediction and degradation-assessment studies.

The primary objective of the present study is to establish a physics-consistent PINN framework for long-horizon polymer degradation modeling using short-duration aging data rather than to develop a comprehensive probabilistic uncertainty-quantification methodology. Consequently, fully Bayesian inference and formally calibrated uncertainty models were not implemented at this stage. Nevertheless, several characteristics of the framework contribute to stable and reliable predictive behavior.

First, the governing reaction–diffusion equation and Arrhenius kinetics are enforced continuously at collocation points distributed throughout the full spatiotemporal domain. This constrains the admissible solution space and reduces sensitivity to localized fluctuations or sparse-data artifacts. Second, controlled Gaussian noise was intentionally introduced into both oxidative concentration and property-retention data during synthetic dataset generation. Despite these perturbations, the PINN consistently demonstrated stable convergence and smooth long-horizon extrapolation behavior, indicating resilience to realistic levels of measurement uncertainty.

To further evaluate robustness with respect to data quality, additional training runs were performed during model development using noise amplitudes approximately twice the baseline levels reported in Table 1. Across these cases, the predicted degradation trajectories remained smooth, monotonic and numerically stable without evidence of divergence or non-physical oscillation. These observations suggest that the framework maintains robustness under moderate variations in observational noise. With respect to physics specification, the present formulation assumes Fickian oxygen diffusion coupled with first-order Arrhenius reaction kinetics. These assumptions represent widely accepted first-order approximations for thermo–oxidative aging in high-performance polymers such as PEEK. The effects of deliberately mis-specified physics, including non-Fickian transport behavior or multi-step oxidation chemistry, were not explicitly investigated within the present study. However, the modular structure of the PINN formulation allows alternative transport or reaction models to be incorporated straightforwardly within the same physics-informed optimization framework. To provide a qualitative indication of epistemic uncertainty without introducing full Bayesian complexity, a small ensemble of independently initialized PINN models was additionally evaluated during post-processing. The resulting spread in long-horizon predictions provides an indicative uncertainty envelope associated with limited short-term observations rather than a formally calibrated probabilistic confidence interval. As illustrated in Figure 11, these prediction envelopes remain bounded and physically consistent throughout the extended extrapolation horizon, further highlighting the stabilizing influence of embedded degradation physics. Future work will focus on uncertainty-aware extensions of the framework, including Bayesian PINNs, larger ensemble-learning strategies, adaptive weighting schemes and sensitivity analyses under modified governing equations. Such developments will be important for establishing statistically rigorous confidence bounds and evaluating robustness under experimentally measured degradation conditions.

To further evaluate the robustness of the proposed physics-informed neural network (PINN) framework under uncertain conditions, additional numerical experiments were conducted during model development using increased stochastic noise levels and perturbed physics parameters. These analyses were intended to assess whether the embedded physics constraints remain effective when the degradation data and governing parameters deviate moderately from the baseline configuration. Gaussian noise amplitudes of up to approximately three times the baseline levels reported in Table 1 were introduced into both the oxidative concentration and property-retention datasets. In addition, the diffusion coefficient used during training was deliberately perturbed by approximately:

relative to its reference value in order to emulate moderate physics misspecification and parameter uncertainty.

The resulting short-term prediction errors together with the observed long-horizon degradation behavior are summarized in Table 6.

The results presented in Table 6 indicate that moderate increases in observational noise and parameter perturbations produce only relatively small increases in short-term interpolation error. More importantly, all investigated cases retained smooth, monotonic and physically interpretable long-horizon degradation trajectories throughout the extended extrapolation horizon. No numerical instability, abrupt oscillation or non-physical divergence was observed during any of the perturbed simulations. These observations suggest that the embedded reaction–diffusion constraints act as an effective regularization mechanism, stabilizing the learning process even under moderately uncertain conditions. The robustness behavior further highlights one of the principal advantages of physics-informed learning compared with unconstrained purely data-driven extrapolation models. Because the admissible solution space is constrained continuously through governing physical laws, the framework remains less sensitive to localized perturbations, noise fluctuations and moderate parameter uncertainty.

Several important limitations of the present study should be acknowledged when interpreting the results and assessing the broader applicability of the proposed framework.

First, a direct numerical benchmarking comparison against classical finite-difference method (FDM) or finite-element method (FEM) solvers was not performed. Because the governing reaction–diffusion equations used for synthetic dataset generation are already known a priori, the primary objective of the present work is not numerical PDE accuracy benchmarking. Instead, the focus is placed on evaluating physics-constrained learning, material-parameter inference and long-horizon extrapolation capability using limited short-duration degradation data.

Second, the present framework was intentionally developed and validated using fully synthetic benchmark datasets derived from established thermo–oxidative degradation equations. This controlled setting provides several methodological advantages, including complete transparency, reproducibility and direct access to known reference parameters. It also allows the behavior of the physics-informed framework to be examined without confounding influences such as uncontrolled environmental variability, experimental uncertainty or specimen-to-specimen scatter.

However, synthetic datasets cannot fully reproduce the complexity of real laboratory aging experiments or long-term service environments. Consequently, the present results should be interpreted primarily as methodological validation under controlled benchmark conditions rather than experimentally validated engineering lifetime prediction.

Third, the degradation formulation assumes Fickian oxygen diffusion coupled with first-order Arrhenius reaction kinetics, which represent widely accepted first-order approximations for thermo–oxidative aging in high-performance polymers such as PEEK. More complex phenomena including non-Fickian transport behavior, multi-step oxidation chemistry, crystallinity evolution and coupled microstructural degradation mechanisms were not explicitly incorporated within the current formulation.

These simplifications were adopted intentionally to preserve interpretability, numerical stability and computational tractability during extended-time inference. Importantly, however, the modular structure of the PINN framework readily accommodates future incorporation of additional physical mechanisms if required. Another important limitation is that the present study does not constitute a fully integrated industrial digital twin implementation involving real-time sensing, adaptive online updating or cyber–physical synchronization. Instead, the proposed framework should be interpreted as a foundational physics-informed degradation modeling methodology intended to support future experimentally validated durability-assessment systems. A major direction of future work will therefore involve experimental validation using real thermo–oxidative aging measurements. Ongoing efforts include acquisition of FTIR-based oxidation data and long-term tensile-aging measurements for PEEK exposed to elevated temperatures. These studies will enable direct comparison between model predictions and experimentally observed degradation behavior under realistic environmental conditions. Future work will also investigate uncertainty-aware extensions of the framework, including Bayesian PINNs, ensemble-learning strategies and stochastic sampling approaches for formally quantifying confidence bounds associated with long-horizon degradation prediction. Looking further ahead, integration of coupled thermal–mechanical–environmental interactions together with adaptive sensor-data assimilation represents a promising pathway toward more comprehensive intelligent durability-assessment and degradation-monitoring frameworks.

This study developed a physics-informed neural network (PINN) framework for modeling thermo–oxidative degradation of polymeric materials using short-duration aging data under controlled synthetic benchmark conditions. By embedding Arrhenius kinetics, reaction–diffusion physics, and property–damage coupling directly into the learning process, the proposed framework achieved a short-term test RMSE of 0.008, recovered the activation energy with approximately 1.2% error, and maintained a normalized PDE residual of approximately 0.004, demonstrating strong agreement with the governing physical laws. Compared with conventional machine-learning models, the proposed PINN produced more stable and physically consistent long-horizon degradation predictions while preserving smooth, monotonic degradation trajectories beyond the 180-day training window. Although baseline models achieved comparable short-term interpolation accuracy, they exhibited increased extrapolation bias and reduced physical interpretability during extended-time prediction. To further assess the physical plausibility of the framework, predictions were compared with independently reported thermo–oxidative degradation data for PEEK from the published literature. Without retraining or parameter adjustment, the proposed model reproduced the experimental degradation trend with absolute deviations of 1.1–1.7%, providing additional evidence that the learned degradation behavior remains consistent with experimentally observed polymer aging. Overall, the proposed PINN provides a physically interpretable and computationally efficient methodology for long-term polymer degradation modeling, parameter identification, and reliability assessment. While the present work primarily represents a methodological validation using a transparent synthetic benchmark, the additional comparison with independent experimental literature supports the practical relevance of the framework. Future work will focus on comprehensive validation using dedicated long-term experimental datasets, uncertainty-aware PINNs, and adaptive data-assimilation strategies to enable deployment in digital-twin-based durability assessment and predictive maintenance applications.

The author was solely responsible for the conceptualization, methodology development, data generation, model implementation, analysis, and writing of the manuscript.

This study does not involve human participants, animal subjects, or proprietary experimental data, and therefore does not require institutional ethical approval.

The supplementary material for this article can be found online.