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 or aim to yield interpretable constitutive models. In this work, we couple 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，基于虚拟场方法（VFM）的无监督本构模型发现框架（尤其是EUCLID方法）已受到广泛关注。然而，包括EUCLID在内的VFM方法性能易受测量噪声与数据稀疏性的影响，这在实际应用中难以避免。统计有限元方法（statFEM）提供了一个互补视角：它通过贝叶斯框架融合含噪稀疏测量数据，重建全场位移响应并量化不确定性。尽管statFEM能在不确定条件下重建位移场，但并未严格保证本构关系的一致性，也未以获取可解释本构模型为目标。本研究将statFEM与EUCLID框架中的无监督本构模型发现方法相结合，提出statFEM--EUCLID框架。该框架在各向同性超弹性材料中得以验证。结果表明，该融合方法在确保重建场同时满足平衡方程与本构定律的前提下，有效降低了对噪声与数据稀疏性的敏感度。