Predictive computational science is an emerging discipline concerned with assessing the predictability of mathematical and computational tools, particularly in the presence of inevitable uncertainty and limited information. In this talk, I will present a new comprehensive predictive methodology embedded in a new hybrid fuzzy-stochastic framework to predict physical events described by partial differential equations (PDEs) and subject to both random (aleatoric) and non-random (epistemic) uncertainty. In the new framework the uncertain parameters will be characterized by random fields with fuzzy moments. This will result in a new class of PDEs with hybrid fuzzy-stochastic parameters, coined fuzzy-stochastic PDEs, for which forward and inverse problems need to be solved. I will demonstrate the importance and feasibility of the new methodology by applying it to a complex problem: prediction of the response of materials with hierarchical microstructure to external forces. This model problem will serve as an illustrative example, one that cannot be tackled by today’s UQ methodologies.