Many engineering problems in multi-physics simulations are characterized by anisotropic and heterogeneous material behavior, often governed by constitutive laws that are not fully understood. To address this lack of knowledge, the corresponding high-dimensional partial differential equations (PDEs) are parameterized by material constituents that are modelled by positive definite stochastic tensors on the corresponding manifolds. However, this approach significantly increases the dimensionality of the PDEs, making real-time solutions impractical for applications such as design optimization and control. Given that the primary objective in these problems is to predict uncertainty in the quantity of interest (QoI), we propose utilizing deep neural networks as efficient surrogate models. To further reduce the computational cost of constructing the neural network that maps input features to the QoI, we introduce the use of domain decomposition methods. This approach partitions the neural network vertically or horizontally into smaller sub-domains, allowing the global neural network model to be replaced by several local networks that are computationally less demanding. In this presentation, we will explore several methodologies for implementing this framework, accompanied by numerical examples relevant to mechanical engineering applications.