Physics-informed neural networks (PINNs) have recently prevailed as differentiable solvers that unify forward and inverse analysis in the same formulation. However, PINNs have quite limited caliber when dealing with concentration features and discontinuous multi-material heterogeneity, hindering its application when labeled data is missing. We propose a novel physics-encoded finite element network (PEFEN) that can deal with concentration features and multi-material heterogeneity without special treatments, extra burden, or labeled data. Leveraging the interpretable discretized finite element approximation as a differentiable network in the new approach, PEFEN encodes the physics structure of multi-material heterogeneity, functional losses, and boundary conditions. We simulate three typical numerical experiments, and PEFEN is validated with a good performance of handling complex cases where conventional PINNs fail. Moreover, PEFEN entails much fewer iterations (<10%) than some published improved PINNs (namely the mixed form and domain decomposition method), and the proposed PEFEN does not employ extra variables for stresses or special treatments for subdomains. We further examine PEFEN in hyperelastic multi-layer strata with and without a pile, validating its ability for more practical applications. PEFEN is also tested for inverse analysis. In 3D experiments, transfer learning with PEFEN is validated. PEFEN need much less memory than FEM (<20%), and its training from zero initialization is faster than FEM forward analysis (>1 million dofs). It is also discussed that PEFEN may act like domain decomposition in a refined way, and a simple experiment validates that PEFEN can solve the problem with multi-scale frequency. The PEFEN, thus, proves to be a promising method and deserves further development.

中文翻译：

---

可解释的物理编码有限元网络，用于处理超弹性中的集中特征和多材料异质性


物理信息神经网络（PINN）最近作为可微分求解器而盛行，它在同一公式中统一了正向和逆向分析。然而，PINN 在处理浓度特征和不连续的多材料异质性时能力相当有限，在标记数据丢失时阻碍了其应用。我们提出了一种新颖的物理编码有限元网络（PEFEN），它可以处理浓度特征和多材料异质性，而无需特殊处理、额外负担或标记数据。 PEFEN 利用可解释的离散有限元近似作为新方法中的可微分网络，对多材料异质性、功能损失和边界条件的物理结构进行编码。我们模拟了三个典型的数值实验，验证了 PEFEN 在处理传统 PINN 失败的复杂情况方面具有良好的性能。此外，PEFEN 需要的迭代次数要少得多（<10 id=0>100 万自由度）。还讨论了PEFEN可能以精细的方式充当域分解，并且简单的实验验证了PEFEN可以解决多尺度频率的问题。因此，PEFEN 被证明是一种有前途的方法，值得进一步发展。