Prior knowledge-infused neural network for efficient performance assessment of structures through few-shot incremental learning

2.1 Prior knowledge-infused neural network

For the original structure, the physical model shown in Figure 1 and Equation (1) are theoretical and simplified. As a result, the solution $$\bm {u}(t)$$ of Equation (1) would inherently contain a deviation from the actual response of the structure under the ground acceleration $$a_g(t)$$. The structural and seismic parameters $$\bm {x}$$ and corresponding $$\bm {u}(t)$$ data are incapable of training a surrogate model to predict accurate structural seismic response. However, $$\bm {u}(t)$$ is still a reasonable prior estimation of actual structural response and could be useful for pretraining a neural network and making it get close to the demanded accurate mapping relationship between $$\bm {x}$$ and corresponding structural response. This is like finding a suitable initial point that would be beneficial for an optimization task.

Accordingly, as shown in Algorithm 1, in stage 1, the physical model represented by the equation of motion $$\mathcal {F}(\bm {u}(t)|\bm {M},\bm {C},\bm {f},a_g(t))=0$$ would be used to generate cheap samples $$(\bm {x}_i^E, y_i^E)\nobreakspace i=1,2\cdots,n$$, for initially training the neural network's weights $$\rm {w}$$. Superscript “E” of $$(\bm {x}_i^E, y_i^E)$$ represents these samples are generated by the equation of motion of the physical model. As illustrated above, solving $$\mathcal {F}(\bm {u}(t)|\bm {M},\bm {C},\bm {f},a_g(t))=0$$ is cost-less in comparison with NLTHA of a refined finite element model, though obtained $$(\bm {x}_i^E, y_i^E)$$ cannot be used to establish a neural network–based surrogate model for predicting accurate structural responses corresponding to given structural, seismic attributes. Instead, the neural network would initially establish a less accurate relationship between them. The main purpose of this step is to initialize the network and make it closer to the authentic function for predicting structural responses. This process is similar to finding a good initial point of the learning weights and hyperparameters $$\bm {w}_0, \bm {\theta }_0$$ for optimizing the network $$\mathbb {N}(\bm {x}|\bm {w}, \bm {\theta })$$ instead of random start. The obtained neural network is defined as PKNN (Figure 1).

2.2 Incremental learning on few-shot

Then, in the second stage, a small number of the numerical simulation samples would be used for fine-tuning the pretrained PKNN via incremental learning. At this time, with the help of the prior knowledge infused in the first stage, the overfitting issue could be alleviated even though the training data are small. In this stage, one critical point is about how to learn new knowledge within new data without forgetting the physical knowledge learned in the first stage.

In general, it can be seen that in the proposed two-stage training strategy, the introduction of the equation of motion of a physical simplified model in the training process is just like adding a soft physical regularization to the neural network. As a result, both the trending of overfitting and the requirement of numerous NLTHA data could be effectively suppressed. This is why it is named a “prior knowledge-infused neural network.”

It should be mentioned that the PKNN proposed here is technically different from the famous PINNs, though their core ideas are similar, which aim at promoting the generalization capacity of a neural network by introducing knowledge. In PINNs, mainly the precise knowledge present in the form of ODEs or PDEs can be utilized, because they need to be added into the loss function of a PINN as a hard regularization term. As a result, PINNs are usually used for finding the solution of ODEs and PDEs (Rasht-Behesht et al., 2022; Zhang & Yuen, 2022). In comparison, the PKNN has no specific requirement, because in a PKNN, the prior knowledge is flexibly infused into the neural network by learning relevant data in a two-stage training strategy instead of directly adding the physical model to the neural network. Thus, a PKNN might be applicable for more issues.

Next, the proposed PKNN would be implemented into the seismic performance assessment of structures.

3.1 Traditional fragility analysis method

It can be observed that in this procedure, the precision of the derived fragility curves mainly depends on the number of IM-EDP samples. In previous studies, hundreds, even thousands, of deterministic NLTHAs were carried out for this purpose (Cao et al., 2023; de Felice & Giannini, 2010; Rezaei et al., 2023). For step 2, refined finite element models (based on solid or fiber elements) are typically used. Therefore, the computational burden of deriving fragility curves is usually high (Feng et al., 2019). Various prior studies have proposed to use ML as a surrogate to the numerical model to decrease the computational cost, but the number of NLTHA samples (say $$N$$) needed for ML training is not much less than that for the conventional approach (Ferrario et al., 2017). Therefore, the proposed PKNN is a viable alternative in step 2 to achieve a meaningful level of computational efficiency.

3.2 Fragility analysis via PKNN

The proposed PKNN can be used for predicting the seismic response of a given structure and significantly accelerate the fragility analysis. As shown in Figure 3 and Figure 2, in the conventional approach, the development of PSDM requires plenty NLTHAs. In our approach, the proposed PKNN can be trained with only a few NLTHAs to generate the IM-EDP pairs. Another major difference between the proposed approach and the other relevant prior work featuring ML-based surrogate models is that, as illustrated in the previous section, the PKNN used here requires far fewer NLTHA samples for training to achieve the same accuracy. A case study is designed and carried out next to comprehensively validate the effectiveness of this proposed approach.

4.1 Structural modeling

For initially validating the proposed approach, a representative three-span six-story RCF is chosen as the specimen for fragility assessment (Figure 4). The span length of the RCF is 6.3 m. The height for the first story is 4.2 m, and 3.5 m for all other stories. The dimensions of the beam and column cross sections are 350$$\times$$550 mm and 650$$\times$$650 mm, respectively. For the material, type-C30 concrete (14.3 MPa compressive strength) is used. HRB335 steel bars (300 MPa yield strength) and HPB300 bars (270 MPa yield strength) are selected for longitudinal reinforcement and stirrups, respectively. The reinforcement details for beams and columns are given in Figure 4. It should be mentioned that the main purpose of this study is to initially validate the concept of the proposed PKNN-based assessment of a structure. Thus, the simulated RCF here is a commonly used regular frame structure without considering the damage within existing structures (Ning & Xie, 2023; Park et al., 2018). The performance of the proposed approach would be further investigated on more complex scenarios such as an existing RC structure with planar or vertical irregularities in the following studies considering the degradation damages such as corroded reinforcements (Ding et al., 2023; Di Sarno & Pugliese, 2020).

For NLTHA, commonly used fiber element–based modeling approaches are selected and the finite element model is established in well-developed open-source software “OpenSees” (Mazzoni et al., 2006). The model is illustrated in Figure 5. The force-based fiber element with four integration points is adopted for modeling each beam–column component. The constitutive behaviors of unconfined, confined concrete and reinforcing steel are considered in the fiber section (Feng & Ding, 2018). The Joint2D is adopted for modeling beam–column joints. The shear behavior of the joint and the bond-slip behavior of the interface are considered. The strength and stiffness degradation behavior is described by the “Pinching4” model. For the behavior of the bond-slip stress-slip relationship, it is considered in the moment-rotation curve of the corresponding fiber section. More specific details about modeling can be referred to Feng et al. (2018).

For stochastic seismic response prediction and fragility assessment, uncertainties within 13 key structural parameters are considered, and their distribution characteristics are listed in Table 1 in which “Cov.” stands for “Coefficient of Variation.”' These parameters are determined based on previous studies (Ellingwood, 1980; Mirza & MacGregor, 1979a, 1979b; Mirza et al., 1979). Based on this, 320 parameter sets are sampled in order to generate 320 structural models.

4.2 Ground acceleration selection

For uncertainties within earthquakes, 160 horizontal ground accelerations containing far-field, near-fault pulse-like ground accelerations recorded at diverse soil conditions are selected from the Pacific Earthquake Engineering Research Center database (Baker et al., 2011). The response spectra are given in Figure 6. After scaling these ground accelerations by factors of 1 and 2, a total of 320 ground acceleration pairs are generated for conducting NLTHA of previously generated 320 structural models.

4.3 Database generation for model training

The NLTHA is carried out by an unconditionally stable explicit solving algorithm “KR-$$\alpha$$” to alleviate convergence issue (Kolay & Ricles, 2014). Then, the 320 input–output data $$(\bm {x}^N, y^N)$$ needed for the fragility analysis and ML model training are extracted from the obtained structural response in NLTHA. The input includes the structural attributes listed in Table 1 and amplitude- and duration-based IM parameters containing PGA, peak ground velocity (PGV), effective design acceleration (EDA), effective peak velocity (EPV), sustained maximum velocity (SMV), velocity spectrum intensity (VSI), characteristic intensity (CI), and Housner intensity (HI). Hence, the input $$\bm {x}$$ is a 21-dimensional vector. The output of the database is the EDP. For an RCF, EDP is commonly quantified by the MIDR. Meanwhile, based on the theoretical procedure in Figure 5, physical models of the RCFs with 320 structural parameter sets can be established and 320 equations of motion pairing with 320 ground accelerations can be solved via the Newmark algorithm. The similar 320 input–output data $$(\bm {x}^E, y^E)$$ can be extracted from the corresponding results and would provide the prior knowledge for the neural network.

4.4 Scenario design for case study

According to the conventional approach given in Figure 2, a baseline PSDM would be derived by pairing the 320 IM-EDP (PGA-MIDR here) pairs via NLTHA for comparison. Then, to illustrate the effectiveness of the proposed approach, only 20% (64) NLTHA generated data are randomly selected for training a PKNN. Then, the trained neural network would be used as a surrogate model for predicting seismic response and facilitating fragility assessment. For ablation tests and comparisons, the conventional neural network–based surrogate model approach is also conducted via the identical training database. In addition to the neural network, the XGboost algorithm is also selected to generate a classic surrogate model to investigate the consequence of algorithm selection, because according to previous studies about predicting seismic response and assessment safety of a structure, the decision tree-based ML algorithm represented by XGBoost mostly outperforms other ML algorithms (Aladsani et al., 2022; Mangalathu et al., 2020; Mangalathu & Jeon, 2019). As for deep learning algorithms, they were proven to be not more adept on tabular data than decision tree–based models. Mostly the gradient-boosted tree ensembles still outperform deep learning models on supervised learning tasks on tabular data (Borisov et al., 2022; Shwartz-Ziv & Armon, 2022). The data size treated in this case is small, which is also not suitable for a deep learning model. Thus, the deep learning algorithms are not adopted in the case study.

The scenarios for the entire case study are summarized in Table 2 where “PK” stands for prior knowledge. It should be noticed that S1 is the baseline for comparison and S4 represents that the neural network is only trained via stage 1 of Algorithm 1 without any NLTHA data in order to illustrate the pure effect of a physical model. For all ML models trained in this study, their hyperparameters would be optimized via mature Bayesian optimization algorithms. For specific processes, one can be referred to Akiba et al. (2019).

5.1 Seismic response prediction of a structure

The accuracy of the predicted response within each scenario is discussed first. The comparison between actual and predicted seismic MIDR of the RCF is displayed in Figure 7. In Figure 7a,bd, the points circled out represent the training set for the model. The accuracy index commonly used is also given in Figure 7 where R$$^2$$ represents the coefficient of determination and MSE is defined as before.

According to the distribution of the scatter of points, it can be seen that among the four models, the PKNN model used in S5 exhibits the best regression performance, which can also be proved by the accuracy indexes. The improvement over the other model is evident while the least improvement ratio on R$$^2$$ and MSE reaches 13.94%, and 22.38%, respectively. Though the neural network trained in S2 has a high accuracy on the training data, the overall prediction accuracy is poor, which implies the model has been overfitted. A similar overfitting phenomenon can also be seen in S3, though the value of the accuracy indexes in S3 is close to the index performance in S5, which mainly owing to its good prediction when MIDR $$<$$0.25. However, the prediction of the XGBoost model in S3 seems to be restrained under a certain value, and for MIDR larger than.05 its prediction error is extremely high. This is mainly caused by the nature of the XGBoost algorithm, as a tree-based ML model that inherently cannot make predictions outside the valuing range of the training database, and most of the MIDR in the training data is less than. 25 (Arik & Pfister, 2021). Thus, when the training data are not adequate, a conventional ML-based surrogate model may not work as usual no matter which algorithm is used. As for S4, the whole prediction resulting from the physical model is poor and entirely underestimates the MIDR. Based on these results, it can be proved that on one hand the good performance of PKNN is not simply contributed by the small training database from NLTHA or the data from the physical model, and on the other hand, the effectiveness of the proposed two-stage training strategy of PKNN on response prediction is validated.

Although the performance of PKNN on seismic response prediction exceeds all other models, its absolute accuracy is not ideally high. Its performance on the downstream task such as fragility assessment is still unknown. Then, the performance of applying the PKNN in fragility assessment is further discussed.

5.2 Fragility assessment

Based on four generated response prediction models in the last section, the PGA-MIDR (IM-EDP) pair data for fragility assessment can be generated. The generated PGA-MIDR data and corresponding PSDMs are illustrated in Figure 8 where “LR” stands for “linear regression.” Similarly, in Figure 8, the data points circled in black represent the training data for S2, S3, and S5. One interesting thing that needs to be pointed out is that the ln(PGA) value of all training data is larger than −3, which means the extrapolating capacity of the trained surrogate models would be examined when the prediction falls in the range ln(PGA)$$<$$ −3.

According to the distributions of the scatter points shown in Figure 8, it can be seen that the data generated in S2 differs a lot from that in S1 when ln(PGA) is less than −3. A similar scene can also be found in S3 and moreover, the predicted MIDRs are almost constant as ln(PGA) $$<$$ −3. As found in the last section, this phenomenon is also the result of poor extrapolation capacity of the trained surrogate model in S2 and S3. Thus, when the training data are not adequate, conventional ML-based surrogate models still cannot work in fragility assessment. As for S4, according to the slope of the regressed linear model, the data generated in S4 has a global bias, which mainly comes from the theoretical simplifications of the physical model. As for S5, it can be seen that the distribution of data highly resembles the data in S1. Though the PKNN in S5 is trained on the same database used in S2 and S3, the poor extrapolation issue does not appear when ln(PGA) $$<$$ −3. This mainly owes to the prior knowledge from the theoretical model. Despite the physical model not being accurate, as displayed in S4, it can provide good prior knowledge on the entire prediction space to the ML model.

For further quantifying the performance of each approach, the relative difference between the baseline PSDM's parameters in S1 and others is summarized in Figure 9. It can be seen that the PSDM in S5 is the closest one to the baseline with merely 1.18%, 1.55%, and 0.12% relative error on slope, intercept, and $$\beta _{\text{EDP}}$$, respectively. For comparison, it can be seen that S4 is the worst, which also implies that the original physical model cannot accurately describe the seismic behavior of the structure. In the meantime, the PSDMs in S2 and S3 also severely deviate from the baseline PSDM. However, based on the identical database, a quite accurate PSDM could be obtained via the proposed PKNN-based approach. This strongly validates its effectiveness.

Furthermore, the fragility curves derived from five PSDMs are given in Figure 10. For fragility assessment, four widely adopted performance levels provided by federal emergency management agency are used, where 0.2%, 1%, 2%, and 4% MIDR are used to describe slight, moderate, extensive, and complete damage states, respectively (Feng, Sun, et al., 2023; Prestandard, 2000). It can be seen that in four performance levels, the fragility curves derived in S2, S3, and S4 have a large deviation. In contrast, the fragility curves in S5 almost coincide with those baseline curves in S1.

The JS divergence results between baseline fragility curves and others are given in Figure 11. It can be seen that for all scenarios, the difference between baseline fragility results and others would rise under a more severe performance level. Meanwhile, the fragility results in S5 are the closest to the baseline no matter which performance level is considered. With the help of JS divergence as the index, it is proved that the precision of the proposed approach in S5 is significantly better than the others.

All in all, these results strongly demonstrated the effectiveness of the proposed PKNN-based fragility assessment approach. Although the PKNN's accuracy on the seismic response prediction is not perfect, it can still be applied in fragility assessment with rather good performance. It was proved that comparable assessment results could be obtained based on only 20% computational cost.

5.3 Influence of NLTHA data amount used in PKNN

To further investigate the influence of the NLTHA data number on the proposed PKNN-based method, the assessment based on the proposed approach with 10% (32) and 30% (96) NLTHA data is carried out. First, the generated PGA-MIDR data and derived PSDMs from various NLTHA data are present in Figure 12 where superscript “B” represents the baseline model from the previous S1. It can be seen that the difference between the derived PSDM and the baseline would evidently decrease as the amount of NLTHA data for model training increases. Moreover, the degree of improvement would seemingly become saturated, because the gap between cases with 10% and 20% NLTHA data is more dramatic than that between cases with 20% and 30% data. Similarly, the relative error of these three PSDMs in comparison with the baseline is given in Figure 13. It can be seen that the model generated from 30% NLTHA data attains the best relative error performance with only 0.91%, 0.33%, and 0.01% on three PSDM parameters. It is also demonstrated that the improvement caused by introducing more NLTHA data would gradually decrease.

The fragility results are also derived from these PSDMs as given in Figure 14. A similar trend can be observed that with more NLTHA data, the derived fragility curves would be closer to the baseline. To better quantify this trend, JS divergence between them is displayed in Figure 15. It is illustrated more clearly that more NLTHA data could effectively promote the precision of the fragility results but the gain from the increased data would lessen. Therefore, when applying the proposed approach, an optimal balance between the demanded accuracy and the acceptable computational cost should be carefully considered.