* Liu et al. (2021b) J. Liu, Z. Hu, P. Cui, B. Li, and Z. Shen.  Kernelized heterogeneous risk minimization.  _arXiv preprint arXiv:2110.12425_ , 2021b.
  * Liu et al. (2021c) W. K. Liu, S. Li, and H. Park.  Eighty years of the finite element method: Birth, evolution, and future.  _arXiv preprint arXiv:2107.04960_ , 2021c.
  * Liu et al. (2021d) X. Liu, C. E. Athanasiou, N. P. Padture, B. W. Sheldon, and H. Gao.  Knowledge extraction and transfer in data-driven fracture mechanics.  _Proceedings of the National Academy of Sciences_ , 118(23), 2021d.
  * Logg and Wells (2010) A. Logg and G. N. Wells.  Dolfin: Automated finite element computing.  _ACM Transactions on Mathematical Software (TOMS)_ , 37(2):1–28, 2010.
  * Logg et al. (2012) A. Logg, K.-A. Mardal, and G. Wells.  _Automated solution of differential equations by the finite element method: The FEniCS book_ , volume 84.  Springer Science & Business Media, 2012.
  * Lookman et al. (2019) T. Lookman, P. V. Balachandran, D. Xue, and R. Yuan.  Active learning in materials science with emphasis on adaptive sampling using uncertainties for targeted design.  _npj Computational Materials_ , 5(1):1–17, 2019\.
  * Lu et al. (2018) J. Lu, A. Liu, F. Dong, F. Gu, J. Gama, and G. Zhang.  Learning under concept drift: A review.  _IEEE Transactions on Knowledge and Data Engineering_ , 31(12):2346–2363, 2018.
  * Lu et al. (2020) L. Lu, M. Dao, P. Kumar, U. Ramamurty, G. E. Karniadakis, and S. Suresh.  Extraction of mechanical properties of materials through deep learning from instrumented indentation.  _Proceedings of the National Academy of Sciences_ , 117(13):7052–7062, 2020.
  * Mahajan et al. (2021) D. Mahajan, S. Tople, and A. Sharma.  Domain generalization using causal matching.  In _International Conference on Machine Learning_ , pages 7313–7324. PMLR, 2021.
  * Mianroodi et al. (2021) J. R. Mianroodi, N. H Siboni, and D. Raabe.  Teaching solid mechanics to artificial intelligence—a fast solver for heterogeneous materials.  _npj Computational Materials_ , 7(1):1–10, 2021\.
  * Ming et al. (2022) Y. Ming, H. Yin, and Y. Li.  On the impact of spurious correlation for out-of-distribution detection.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 36, pages 10051–10059, 2022.
  * Mohammadzadeh and Lejeune (2022) S. Mohammadzadeh and E. Lejeune.  Predicting mechanically driven full-field quantities of interest with deep learning-based metamodels.  _Extreme Mechanics Letters_ , 50:101566, 2022.
  * Morgan and Jacobs (2020) D. Morgan and R. Jacobs.  Opportunities and challenges for machine learning in materials science.  _Annual Review of Materials Research_ , 50:71–103, 2020\.
  * Mozaffar et al. (2019) M. Mozaffar, R. Bostanabad, W. Chen, K. Ehmann, J. Cao, and M. Bessa.  Deep learning predicts path-dependent plasticity.  _Proceedings of the National Academy of Sciences_ , 116(52):26414–26420, 2019.
  * Nagarajan et al. (2020) V. Nagarajan, A. Andreassen, and B. Neyshabur.  Understanding the failure modes of out-of-distribution generalization.  _arXiv preprint arXiv:2010.15775_ , 2020.
  * Ni and Gao (2021) B. Ni and H. Gao.  A deep learning approach to the inverse problem of modulus identification in elasticity.  _MRS Bulletin_ , 46(1):19–25, 2021.
  * Park et al. (2006) H. S. Park, K. Gall, and J. A. Zimmerman.  Deformation of fcc nanowires by twinning and slip.  _Journal of the Mechanics and Physics of Solids_ , 54(9):1862–1881, 2006.
  * Peng et al. (2021) G. C. Peng, M. Alber, A. Buganza Tepole, W. R. Cannon, S. De, S. Dura-Bernal, K. Garikipati, G. Karniadakis, W. W. Lytton, P. Perdikaris, et al.  Multiscale modeling meets machine learning: What can we learn?  _Archives of Computational Methods in Engineering_ , 28(3):1017–1037, 2021.
  * Peters et al. (2016) J. Peters, P. Bühlmann, and N. Meinshausen.  Causal inference by using invariant prediction: identification and confidence intervals.  _Journal of the Royal Statistical Society: Series B (Statistical Methodology)_ , 78(5):947–1012, 2016.
  * Pilania (2021) G. Pilania.  Machine learning in materials science: From explainable predictions to autonomous design.  _Computational Materials Science_ , 193:110360, 2021.
  * Piratla et al. (2020) V. Piratla, P. Netrapalli, and S. Sarawagi.  Efficient domain generalization via common-specific low-rank decomposition.  In _International Conference on Machine Learning_ , pages 7728–7738. PMLR, 2020.
  * Prachaseree and Lejeune (2022) P. Prachaseree and E. Lejeune.  Learning mechanically driven emergent behavior with message passing neural networks.  _arXiv preprint arXiv:2202.01380_ , 2022.
  * Quade et al. (2018) M. Quade, M. Abel, J. Nathan Kutz, and S. L. Brunton.  Sparse identification of nonlinear dynamics for rapid model recovery.  _Chaos: An Interdisciplinary Journal of Nonlinear Science_ , 28(6):063116, 2018.
  * Raissi et al. (2019) M. Raissi, P. Perdikaris, and G. E. Karniadakis.  Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.  _Journal of Computational physics_ , 378:686–707, 2019\.
  * Raissi et al. (2020) M. Raissi, A. Yazdani, and G. E. Karniadakis.  Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations.  _Science_ , 367(6481):1026–1030, 2020.
  * Sagawa et al. (2019) S. Sagawa, P. W. Koh, T. B. Hashimoto, and P. Liang.  Distributionally robust neural networks for group shifts: On the importance of regularization for worst-case generalization.  _arXiv preprint arXiv:1911.08731_ , 2019.
  * Sagawa et al. (2020) S. Sagawa, A. Raghunathan, P. W. Koh, and P. Liang.  An investigation of why overparameterization exacerbates spurious correlations.  In _International Conference on Machine Learning_ , pages 8346–8356. PMLR, 2020.
  * Saha et al. (2021) S. Saha, Z. Gan, L. Cheng, J. Gao, O. L. Kafka, X. Xie, H. Li, M. Tajdari, H. A. Kim, and W. K. Liu.  Hierarchical deep learning neural network (hidenn): An artificial intelligence (ai) framework for computational science and engineering.  _Computer Methods in Applied Mechanics and Engineering_ , 373:113452, 2021.
  * Sanchez-Lengeling and Aspuru-Guzik (2018) B. Sanchez-Lengeling and A. Aspuru-Guzik.  Inverse molecular design using machine learning: Generative models for matter engineering.  _Science_ , 361(6400):360–365, 2018.
  * Satorras et al. (2021) V. G. Satorras, E. Hoogeboom, and M. Welling.  E (n) equivariant graph neural networks.  In _International conference on machine learning_ , pages 9323–9332. PMLR, 2021.
  * Schölkopf (2022) B. Schölkopf.  Causality for machine learning.  In _Probabilistic and Causal Inference: The Works of Judea Pearl_ , pages 765–804. 2022.
  * Shen et al. (2021) Z. Shen, J. Liu, Y. He, X. Zhang, R. Xu, H. Yu, and P. Cui.  Towards out-of-distribution generalization: A survey.  _arXiv preprint arXiv:2108.13624_ , 2021.
  * Smidt et al. (2021) T. E. Smidt, M. Geiger, and B. K. Miller.  Finding symmetry breaking order parameters with euclidean neural networks.  _Physical Review Research_ , 3(1):L012002, 2021\.
  * Smits et al. (2011) A. J. Smits, B. J. McKeon, and I. Marusic.  High–reynolds number wall turbulence.  _Annual Review of Fluid Mechanics_ , 43:353–375, 2011.
  * Su et al. (2021) M. Su, Q. Zhong, H. Peng, and S. Li.  Selected machine learning approaches for predicting the interfacial bond strength between frps and concrete.  _Construction and Building Materials_ , 270:121456, 2021\.
  * Sui et al. (2021) F. Sui, R. Guo, Z. Zhang, G. X. Gu, and L. Lin.  Deep reinforcement learning for digital materials design.  _ACS Materials Letters_ , 3(10):1433–1439, 2021\.
  * Süli (2012) E. Süli.  Lecture notes on finite element methods for partial differential equations.  _Mathematical Institute, University of Oxford_ , 2012.
  * Svolos et al. (2022) L. Svolos, H. M. Mourad, G. Manzini, and K. Garikipati.  A fourth-order phase-field fracture model: Formulation and numerical solution using a continuous/discontinuous galerkin method.  _Journal of the Mechanics and Physics of Solids_ , page 104910, 2022\.
  * Tripuraneni et al. (2021) N. Tripuraneni, B. Adlam, and J. Pennington.  Overparameterization improves robustness to covariate shift in high dimensions.  _Advances in Neural Information Processing Systems_ , 34, 2021.
  * Tsipras et al. (2018) D. Tsipras, S. Santurkar, L. Engstrom, A. Turner, and A. Madry.  Robustness may be at odds with accuracy.  _arXiv preprint arXiv:1805.12152_ , 2018.
  * Tsymbal (2004) A. Tsymbal.  The problem of concept drift: definitions and related work.  _Computer Science Department, Trinity College Dublin_ , 106(2):58, 2004.
  * Wang et al. (2021) C. Wang, S. Li, D. Zeng, and X. Zhu.  Quantification and compensation of thermal distortion in additive manufacturing: a computational statistics approach.  _Computer Methods in Applied Mechanics and Engineering_ , 375:113611, 2021.
  * Wang and Sun (2019) K. Wang and W. Sun.  Meta-modeling game for deriving theory-consistent, microstructure-based traction–separation laws via deep reinforcement learning.  _Computer Methods in Applied Mechanics and Engineering_ , 346:216–241, 2019.
  * Wang et al. (2019) K. Wang, W. Sun, and Q. Du.  A cooperative game for automated learning of elasto-plasticity knowledge graphs and models with ai-guided experimentation.  _Computational Mechanics_ , 64(2):467–499, 2019\.
  * Wang and Deng (2018) M. Wang and W. Deng.  Deep visual domain adaptation: A survey.  _Neurocomputing_ , 312:135–153, 2018.
  * Wang and Aitchison (2021) X. Wang and L. Aitchison.  Bayesian ood detection with aleatoric uncertainty and outlier exposure.  In _Fourth Symposium on Advances in Approximate Bayesian Inference_ , 2021.
  * Weichwald and Peters (2021) S. Weichwald and J. Peters.  Causality in cognitive neuroscience: concepts, challenges, and distributional robustness.  _Journal of Cognitive Neuroscience_ , 33(2):226–247, 2021.
  * Winship and Mare (1992) C. Winship and R. D. Mare.  Models for sample selection bias.  _Annual review of sociology_ , 18(1):327–350, 1992.
  * Winship and Morgan (1999) C. Winship and S. L. Morgan.  The estimation of causal effects from observational data.  _Annual review of sociology_ , 25(1):659–706, 1999.
  * Xie et al. (2022a) Y. Xie, S. Li, C. Wu, D. Lyu, C. Wang, and D. Zeng.  A generalized bayesian regularization network approach on characterization of geometric defects in lattice structures for topology optimization in preliminary design of 3d printing.  _Computational Mechanics_ , 69(5):1191–1212, 2022a.
  * Xie et al. (2022b) Y. Xie, C. Wu, B. Li, X. Hu, and S. Li.  A feed-forwarded neural network-based variational bayesian learning approach for forensic analysis of traffic accident.  _Computer Methods in Applied Mechanics and Engineering_ , 397:115148, 2022b.
  * Yang et al. (2021) J. Yang, K. Zhou, Y. Li, and Z. Liu.  Generalized out-of-distribution detection: A survey.  _arXiv preprint arXiv:2110.11334_ , 2021.
  * Ye et al. (2021) H. Ye, C. Xie, T. Cai, R. Li, Z. Li, and L. Wang.  Towards a theoretical framework of out-of-distribution generalization.  _Advances in Neural Information Processing Systems_ , 34, 2021.
  * Zhang and Garikipati (2020) X. Zhang and K. Garikipati.  Machine learning materials physics: Multi-resolution neural networks learn the free energy and nonlinear elastic response of evolving microstructures.  _Computer Methods in Applied Mechanics and Engineering_ , 372:113362, 2020.
  * Zhou et al. (2021) K. Zhou, Z. Liu, Y. Qiao, T. Xiang, and C. C. Loy.  Domain generalization in vision: A survey.  _arXiv preprint arXiv:2103.02503_ , 2021.