Constitutive Manifold Neural Networks

TL;DR

This paper introduces the Constitutive Manifold Neural Network (CMNN), a geometry-aware neural network designed to accurately model stochastic anisotropic material properties represented as SPD tensors, improving predictive performance in engineering applications.

Contribution

The paper presents the CMNN architecture that preserves the geometric structure of SPD tensors by mapping them to the tangent space, enhancing modeling accuracy over traditional neural networks.

Findings

CMNN outperforms conventional MLPs in modeling stochastic anisotropic conductivities.

Geometry-preserving input layers improve neural network learning with tensor data.

Case study demonstrates enhanced accuracy in heat conduction simulations.

Abstract

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 modeled 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. This is achieved through the spectral decomposition of tensors, which enables the parameterization 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. 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 enhances 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.