Recently, unsupervised constitutive model discovery has gained attention through frameworks based on the Virtual Fields Method (VFM), most prominently the EUCLID approach. However, the performance of VFM-based approaches, including EUCLID, is affected by measurement noise and data sparsity, which are unavoidable in practice. The statistical finite element method (statFEM) offers a complementary perspective by providing a Bayesian framework for assimilating noisy and sparse measurements to reconstruct the full-field displacement response, together with quantified uncertainty. While statFEM recovers displacement fields under uncertainty, it does not strictly enforce consistency with constitutive relations. In this work, we integrate statFEM with unsupervised constitutive model discovery in the EUCLID framework, yielding statFEM-EUCLID. The framework is demonstrated for isotropic hyperelastic materials. The results show that this integration reduces sensitivity to noise and data sparsity, while ensuring that the reconstructed fields remain consistent with both equilibrium and constitutive laws.

翻译：近年来，无监督本构模型发现通过基于虚拟场方法（VFM）的框架（最突出的是EUCLID方法）获得了关注。然而，包括EUCLID在内的基于VFM方法的性能受到测量噪声和数据稀疏性的影响，这在实际中不可避免。统计有限元方法（statFEM）提供了一个互补的视角，它提供了一个贝叶斯框架，用于融合含噪声的稀疏测量数据以重建全场位移响应，同时量化不确定性。尽管statFEM能在不确定性下恢复位移场，但它并未严格强制本构关系的一致性。在本工作中，我们将statFEM与EUCLID框架中的无监督本构模型发现相结合，提出了statFEM-EUCLID。该框架针对各向同性超弹性材料进行了演示。结果表明，这种集成降低了对噪声和数据稀疏性的敏感性，同时确保重建的场保持与平衡定律和本构定律的一致性。