Cerebrovascular diseases, in particular ischemic stroke, are one of the leading global causes of death in developed countries. Perfusion CT and/or MRI are ideal imaging modalities for characterizing affected ischemic tissue in the hyper-acute phase. If infarct growth over time could be predicted accurately from functional acute imaging protocols together with advanced machine-learning based image analysis, the expected benefits of treatment options could be better weighted against potential risks. The quality of the outcome prediction by convolutional neural networks (CNNs) is so far limited, which indicates that even highly complex deep learning algorithms are not fully capable of directly learning physiological principles of tissue salvation through weak supervision due to a lack of data (e.g., follow-up segmentation). In this work, we address these current shortcomings by explicitly taking into account clinical expert knowledge in the form of segmentations of the core and its surrounding penumbra in acute CT perfusion images (CTP), that are trained to be represented in a low-dimensional non-linear shape space. Employing a multi-scale CNN (U-Net) together with a convolutional auto-encoder, we predict lesion tissue probabilities for new patients. The predictions are physiologically constrained to a shape embedding that encodes a continuous progression between the core and penumbra extents. The comparison to a simple interpolation in the original voxel space and an unconstrained CNN shows that the use of such a shape space can be advantageous to predict time-dependent growth of stroke lesions on acute perfusion data, yielding a Dice score overlap of 0.46 for predictions from expert segmentations of core and penumbra. Our interpolation method models monotone infarct growth robustly on a linear time scale to automatically predict clinically plausible tissue outcomes that may serve as a basis for more clinical measures such as the expected lesion volume increase and can support the decision making on treatment options and triage.