Anisotropic material properties, such as the thermal conductivities of engineering composites, exhibit variability due to inherent material heterogeneity and manufacturing-related uncertainties. Mathematically, these properties are modelled as symmetric positive definite (SPD) tensors, which reside on a curved Riemannian manifold. Extending this description to a stochastic framework requires preserving both the SPD structure and the underlying spatial symmetries of the tensors. From a numerical standpoint, this is achieved through the spectral decomposition of tensors, which enables the parameterisation of uncertainties into scale (strength) and rotation (orientation) components. To quantify the impact of strength and orientation uncertainties on the thermal behaviour of the composite, the stochastic material tensor must be propagated through a physics-based forward model, here exemplified by an elliptic partial differential equation (PDE). This process necessitates computationally efficient surrogate models, for which a feedforward neural network (FNN) is employed. However, conventional FNN architectures are not well-suited for SPD tensors, as directly using tensor components as input features fails to preserve their underlying geometric structure, often leading to suboptimal performance. To address this issue, we introduce the Constitutive Manifold Neural Network (CMNN), which incorporates input layers that map SPD tensors from the curved manifold to the local tangent space, a flat vector space, thus preserving the statistical and geometric information in the dataset. A case study involving steady-state heat conduction with stochastic anisotropic conductivity demonstrates that geometry-preserving neural network significantly enhance learning performance compared to conventional multilayer perceptrons (MLPs). These findings underscore the importance of manifold-aware methods when working with tensor-valued data in engineering applications.
Schuttert et al. (Wed,) studied this question.