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Modeling cellular responses to genetic perturbations is a significant challenge in computational biology. Measuring all gene perturbations and their combinations across cell types and conditions is experimentally challenging, highlighting the need for predictive models that generalize across data types to support this task. Here we present MORPH, a MOdular framework for predicting Responses to Perturbational cHanges. MORPH combines a discrepancy-based variational autoencoder with an attention mechanism to predict cellular responses to unseen perturbations. It supports both single-cell transcriptomics and imaging outputs and can generalize to unseen perturbations, combinations of perturbations, and perturbations in new cellular contexts. The attention-based framework enables inference of gene interactions and regulatory networks, while the learned gene embeddings can guide the design of informative perturbations. Overall, we envision MORPH as a flexible tool for optimizing perturbation experiments, enabling efficient exploration of the perturbation space to advance understanding of cellular programs for fundamental research and therapeutic applications.