Constitutive Manifold Neural Networks

Abstract: Anisotropic material properties like electrical and thermal conductivities of engineering composites exhibit variability due to inherent material heterogeneity and manufacturing uncertainties. As a tensorial quantity, they are represented as symmetric positive definite tensors, which live on a curved Riemannian manifold, and accurately modelling their stochastic nature requires preserving both their symmetric positive definite properties and spatial symmetries. To achieve this, uncertainties are parametrised into scale (strength) and rotation (orientation) components, modelled as independent random variables on a manifold structure derived from the maximum entropy principle. Further, the propagation of such stochastic tensors through physics-based simulations necessitates computationally efficient surrogate models, like neural networks. However, feedforward neural network architectures are not well-suited for SPD tensors, as directly using the tensor components as inputs fails to preserve their geometric properties, often leading to suboptimal results. To address this, we introduce the Constitutive Manifold Neural Network (CMNN), which is equipped with a preprocessing layer to map the SPD tensor from the curved manifold to the local tangent space, a flat vector space, preserving statistical information in the dataset. A case study on a steady-state heat conduction problem with stochastic anisotropic conductivity demonstrates that geometry-preserving preprocessing, such as logarithmic maps for scale information, significantly improves learning performance over conventional MLPs. These findings underscore the importance of manifold-aware techniques when working with tensor-valued data in engineering applications.