In many engineering applications, material models with physical properties that describe bodies or systems exhibiting inherent variability must be identified in the absence of exact a priori knowledge. When modelling anisotropic behavior, such as that observed in batteries, increasing importance is placed on the representation of uncertainty in material models. These uncertain material models must preserve the symmetry and positive definiteness of the associated material tensor. Recently, a method has been proposed for modelling uncertain material tensors, that enables the separation of magnitudinal and orientational variations through spectral decomposition. In the original approach, stochastic realizations are generated using finite element method samples to construct a surrogate model via Monte Carlo simulations, which results in high computational cost. To reduce the number of required samples, surrogate modelling based on neural networks is employed instead. Due to spatial variability in material parameters and the presence of far-field dependencies, the resulting problem is high-dimensional and computationally expensive to compute. This challenge is addressed by decomposing the spatial domain. It is further assumed that interactions between distant domains are less important than those between close domains, allowing communication between local domains to be restricted to the interfaces between domains. This domain decomposition strategy improves local accuracy while reducing computational cost by enabling parallel computation of local domains, rather than relying on an over-parameterized global surrogate model. The proposed domain decomposition neural network model is applied to a battery thermal conductivity problem, in which the thermal conductivity tensor is treated as uncertain across the material.