Deep Material Network (DMN) has recently emerged as a data-driven surrogate model for heterogeneous materials. Compared to other neural networks, DMN distinguishes itself by its capability to directly encode the morphology of a particular microstructure through its fitting parameters. After an offline training solely based on linear elastic data generated by computational homogenization, the trained model is able to accurately extrapolate to history-dependent complex inelastic behaviors such as plasticity. In this work, the rotation-free DMN formulation is extended for multiscale materials with a varying parameterized microstructure. A single-layer feedforward neural network is used to account for the dependence of DMN parameters on the microstructural ones. Micromechanical constraints are prescribed both on the architecture and the outputs of this new neural network. Offline training is performed by minimizing a loss function that aggregates data generated across various morphologies. The proposed parametric DMN model [1] has been tested numerically for different fiber composites. It is able to accurately predict the effective linear and nonlinear behaviors and successfully capture the structure-property relationships, demonstrating satisfying generalization capabilities. In addition, significant speed-up can be achieved compared to full scale finite element simulations. Parametric DMN can also be recast in a multiphysics framework, through an appropriate redefinition of the DMN laminate homogenization function (building block of the neural network). We demonstrate that, in addition to mechanical properties, other physical properties like thermal conductivity and coefficient of thermal expansion can also be predicted using the same model previously trained on isothermal mechanical data. This shows that parametric DMN learns the parameterized microstructure per se, and not a physical property in particular. [1] Li, T. (2024). Micromechanics-informed parametric deep material network for physics behavior prediction of heterogeneous materials with a varying morphology. Computer Methods in Applied Mechanics and Engineering, 419, 116687.