Spatially-Adaptive Calibration for Reliable Uncertainty Quantification in Seismic Response Prediction of RC Frames
Downloads
Data-driven models offer the computational speed needed for rapid post-earthquake assessment, but their uncertainty estimates must be trustworthy to support safety decisions. This study reveals that Monte Carlo dropout uncertainty for RC frame seismic response prediction is severely miscalibrated: 95% prediction intervals capture only 46.6% of actual responses, meaning Immediate Occupancy assessments under ASCE 41-17 would be unconservative in over half of cases. We address this through post-hoc Temperature Scaling calibration. While a global scaling parameter (T* = 4.40) reduces calibration error by 91.4%, we discover that the optimal calibration factor varies systematically across structural locations: T* ranges from 1.94 at fixed-base nodes to 5.52 at mid-height floors—a 2.8-fold variation that single-parameter approaches cannot capture. This spatial variation reflects physical differences in prediction uncertainty: boundary-constrained nodes exhibit lower uncertainty requiring less scaling, while mid-height nodes dominated by higher-mode contributions show greater uncertainty underestimation. Building on this finding, we propose floor-adaptive calibration using location-specific scaling factors. Compared to global calibration, this approach reduces average calibration error by an additional 62%, with improvements of 61-70% at ground and top floors, where global calibration performs worst. The method requires no model retraining—only a lookup table mapping floor levels to optimal scaling factors. Validation across 12 RC frames (3-7 stories), 2,400 analysis cases, and 35,000+ node-level predictions confirms that spatially adaptive calibration provides more reliable uncertainty estimates across all structural locations, enabling trustworthy confidence intervals for performance-based post-earthquake assessment.
Downloads
[1] ASCE/SEI 41-17. (2017). Seismic evaluation and retrofit of existing buildings. American Society of Civil Engineers, Virginia, United States. doi:10.1061/9780784414859.
[2] FEMA 273. (1997). NEHRP guidelines for the seismic rehabilitation of buildings. Federal Emergency Management Agency, Washington, D.C., United States.
[3] EN 1998-1. (2004). Eurocode 8: Design of structures for earthquake resistance – Part 1: General rules, seismic actions and rules for buildings. European Committee for Standardization, Brussels, Belgium.
[4] Xie, Y., Ebad Sichani, M., Padgett, J. E., & DesRoches, R. (2020). The promise of implementing machine learning in earthquake engineering: A state-of-the-art review. Earthquake Spectra, 36(4), 1769–1801. doi:10.1177/8755293020919419.
[5] Abdeljaber, O., Avci, O., Kiranyaz, S., Gabbouj, M., & Inman, D. J. (2017). Real-time vibration-based structural damage detection using one-dimensional convolutional neural networks. Journal of Sound and Vibration, 388, 154–170. doi:10.1016/j.jsv.2016.10.043.
[6] Li, X., Bolandi, H., Masmoudi, M., Salem, T., Jha, A., Lajnef, N., & Boddeti, V. N. (2024). Mechanics-informed autoencoder enables automated detection and localization of unforeseen structural damage. Nature Communications, 15(1), 8136. doi:10.1038/s41467-024-52501-4.
[7] Zhang, R., Chen, Z., Chen, S., Zheng, J., Büyüköztürk, O., & Sun, H. (2019). Deep long short-term memory networks for nonlinear structural seismic response prediction. Computers and Structures, 220, 55–68. doi:10.1016/j.compstruc.2019.05.006.
[8] Oh, B. K., Park, Y., & Park, H. S. (2020). Seismic response prediction method for building structures using convolutional neural network. Structural Control and Health Monitoring, 27(5), 2519. doi:10.1002/stc.2519.
[9] Wu, Z., Pan, S., Chen, F., Long, G., Zhang, C., & Yu, P. S. (2021). A Comprehensive Survey on Graph Neural Networks. IEEE Transactions on Neural Networks and Learning Systems, 32(1), 4–24. doi:10.1109/TNNLS.2020.2978386.
[10] Kipf, T. N., & Welling, M. (2016). Semi-supervised classification with graph convolutional networks. arXiv Preprint, arXiv:1609.02907. doi:10.48550/arXiv.1609.02907.
[11] Hamilton, W. L., Ying, R., & Leskovec, J. (2017). Inductive representation learning on large graphs. Advances in Neural Information Processing Systems, 2017-December, 1025–1035.
[12] Dang, V. H., Vu, T. C., Nguyen, B. D., Nguyen, Q. H., & Nguyen, T. D. (2022). Structural damage detection framework based on graph convolutional network directly using vibration data. Structures, 38, 40–51. doi:10.1016/j.istruc.2022.01.066.
[13] Tang, A., Li, C., Yang, J., Zhang, H., Zheng, Q., & Zhang, J. (2025). Training and application of graph neural networks for predicting structural responses targeted at tall building structures. Journal of Building Engineering, 103, 112131. doi:10.1016/j.jobe.2025.112131.
[14] Veličković, P., Cucurull, G., Casanova, A., Romero, A., Liò, P., & Bengio, Y. (2017). Graph Attention Networks. arXiv Preprint, arXiv:1710.10903. doi:10.48550/arXiv.1710.10903.
[15] Brody, S., Alon, U., & Yahav, E. (2022). How Attentive Are Graph Attention Networks? arXiv Preprint, arXiv: 2105.14491. doi:10.48550/arXiv.2105.14491.
[16] Fey, M., & Lenssen, J. E. (2019). Fast Graph Representation Learning with PyTorch Geometric. arXiv Preprint, arXiv: 1903.02428. doi:10.48550/arXiv.1903.02428.
[17] Liu, F., Xu, Y., Li, J., & Wang, L. (2025). Graph Neural Network–Based Spatiotemporal Structural Response Modeling in Buildings. Journal of Computing in Civil Engineering, 39(2), 6229. doi:10.1061/jccee5.cpeng-6229.
[18] Chou, Y. T., Kuo, P. C., Li, K. Y., Chang, W. T., Huang, Y. N., & Chen, C. S. (2025). Inductive graph-based long short-term memory network for the prediction of nonlinear floor responses and member forces of steel buildings subjected to orthogonal horizontal ground motions. Earthquake Engineering and Structural Dynamics, 54(2), 491–507. doi:10.1002/eqe.4264.
[19] Shen, Y., Ma, G., Hwang, H. J., Kim, D. J., & Zhang, Z. (2025). Prediction of seismic response of building structures using a CNN-LSTM-ATT network with transfer learning. Advances in Structural Engineering, 28(14), 2710–2725. doi:10.1177/13694332251340730.
[20] Abdar, M., Pourpanah, F., Hussain, S., Rezazadegan, D., Liu, L., Ghavamzadeh, M., Fieguth, P., Cao, X., Khosravi, A., Acharya, U. R., Makarenkov, V., & Nahavandi, S. (2021). A review of uncertainty quantification in deep learning: Techniques, applications and challenges. Information Fusion, 76, 243–297. doi:10.1016/j.inffus.2021.05.008.
[21] Kendall, A., & Gal, Y. (2017). What uncertainties do we need in Bayesian deep learning for computer vision? Advances in Neural Information Processing Systems, 2017-December, 5575–5585.
[22] Gal, Y., & Ghahramani, Z. (2016). Dropout as a Bayesian approximation: Representing model uncertainty in deep learning. 33rd International Conference on Machine Learning, ICML 2016, 3, 1651–1660.
[23] Srivastava, N., Hinton, G., Krizhevsky, A., Sutskever, I., & Salakhutdinov, R. (2014). Dropout: A simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 15, 1929–1958.
[24] Lakshminarayanan, B., Pritzel, A., & Blundell, C. (2017). Simple and scalable predictive uncertainty estimation using deep ensembles. Advances in Neural Information Processing Systems, 2017-December, 6403–6414.
[25] Ovadia, Y., Fertig, E., Ren, J., Nado, Z., Sculley, D., Nowozin, S., Dillon, J. V., Lakshminarayanan, B., & Snoek, J. (2019). Can you trust your model’s uncertainty? evaluating predictive uncertainty under dataset shift. Advances in Neural Information Processing Systems, 32, 13969–13980.
[26] Kim, J., & Wang, Z. (2025). Uncertainty quantification for seismic response using dimensionality reduction-based stochastic simulator. Earthquake Engineering and Structural Dynamics, 54(2), 471–490. doi:10.1002/eqe.4265.
[27] Choi, B., Yi, S. ri, & Kim, T. (2025). Seismic Structural Response and Loss Estimation for Dense Urban Districts Using Neural Network Parameterized Gaussian Process. Earthquake Engineering and Structural Dynamics, 55(2), 397–412. doi:10.1002/eqe.70087.
[28] Xie, Y. (2025). Deep Learning in Earthquake Engineering: A Comprehensive Review. ASCE OPEN: Multidisciplinary Journal of Civil Engineering, 3(1), 03125001. doi:10.1061/aomjah.aoeng-0080.
[29] Naeini, M. P., Cooper, G. F., & Hauskrecht, M. (2015). Obtaining well calibrated probabilities using Bayesian Binning. Proceedings of the National Conference on Artificial Intelligence, 4, 2901–2907. doi:10.1609/aaai.v29i1.9602.
[30] Nixon, J., Dusenberry, M., Jerfel, G., Nguyen, T., Liu, J., Zhang, L., & Tran, D. (2019). Measuring Calibration in Deep Learning. arXiv Preprint, arXiv:1904.01685.
[31] Guo, C., Pleiss, G., Sun, Y., & Weinberger, K. Q. (2017). On calibration of modern neural networks. 34th International Conference on Machine Learning, ICML 2017, 3, 2130–2143. doi:10.48550/arXiv.1706.04599.
[32] Kull, M., Perello-Nieto, M., Kängsepp, M., Filho, T. S., Song, H., & Flach, P. (2019). Beyond temperature scaling: Obtaining well-calibrated multiclass probabilities with dirichlet calibration. Advances in Neural Information Processing Systems, 32, 12295–12305.
[33] Kuleshov, V., Fenner, N., & Ermon, S. (2018). Accurate uncertainties for deep learning using calibrated regression. 35th International Conference on Machine Learning, ICML 2018, 6, 4369–4377.
[34] Blundell, C., Cornebise, J., Kavukcuoglu, K., & Wierstra, D. (2015). Weight uncertainty in neural networks. 32nd International Conference on Machine Learning, ICML 2015, 2, 1613–1622.
[35] Zhuang, D., Jiang, C., Zheng, Y., Wang, S., & Zhao, J. (2025). Gets: Ensemble Temperature Scaling for Calibration in Graph Neural Networks. 13th International Conference on Learning Representations, ICLR 2025, 71167–71189.
[36] Kingma, D. P., & Ba, J. (2014). Adam: A method for stochastic optimization. arXiv Preprint, arXiv:1412.6980. doi:10.48550/arXiv.1412.6980.
[37] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., Desmaison, A., Köpf, A., Yang, E., DeVito, Z., Raison, M., Tejani, A., Chilamkurthy, S., Steiner, B., Fang, L., … Chintala, S. (2019). PyTorch: An imperative style, high-performance deep learning library. Advances in Neural Information Processing Systems, 32, 8024–8035.
[38] Zhu, M., McKenna, F., & Scott, M. H. (2018). OpenSeesPy: Python library for the OpenSees finite element framework. SoftwareX, 7, 6–11. doi:10.1016/j.softx.2017.10.009.
[39] Kanai, K. (1957). 210) Semi-empirical Formula for the Seismic Characteristics of Ground (Structure). Transactions of the Architectural Institute of Japan, 57.1(0), 281–284. doi:10.3130/aijsaxx.57.1.0_281.
[40] Tajimi, H. (1960). A statistical method for determining the maximum response of a building structure during an earthquake. Proceedings of the 2nd World Conference on Earthquake Engineering, 781–797.
[41] Gasparini, D., & Vanmarcke, E. H. (1976). SIMQKE: A program for artificial motion generation. Department of Civil Engineering, Massachusetts Institute of Technology, Massachusetts, United States.
[42] Li, Q., Han, Z., & Wu, X. M. (2018). Deeper insights into graph convolutional networks for semi-supervised learning. 32nd AAAI Conference on Artificial Intelligence, AAAI 2018, 3538–3545. doi:10.1609/aaai.v32i1.11604.
[43] Oono, K., & Suzuki, T. (2019). Graph neural networks exponentially lose expressive power for node classification. arXiv Preprint, arXiv:1905.10947. doi:10.48550/arXiv.1905.10947.
[44] Chen, D., Lin, Y., Li, W., Li, P., Zhou, J., & Sun, X. (2020). Measuring and relieving the over-smoothing problem for graph neural networks from the topological view. AAAI 2020 - 34th AAAI Conference on Artificial Intelligence, 3438–3445. doi:10.1609/aaai.v34i04.5747.
[45] FEMA P-58. (2018). Seismic performance assessment of buildings. Federal Emergency Management Agency, Washington, D.C., United States.
[46] Mangalathu, S., & Jeon, J.-S. (2019). Machine Learning–Based Failure Mode Recognition of Circular Reinforced Concrete Bridge Columns: Comparative Study. Journal of Structural Engineering, 145(10), 4019028. doi:10.1061/(asce)st.1943-541x.0002402.
- Authors retain all copyrights. It is noticeable that authors will not be forced to sign any copyright transfer agreements.
- This work (including HTML and PDF Files) is licensed under a Creative Commons Attribution 4.0 International License.
