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mRNA is an emerging therapeutic platform with applications ranging from vaccines to genome editing. However, there are an exponential number of mRNA sequences to deliver a given payload and the choice in nucleotide sequence largely determines stability and translation efficiency. There exist several computational approaches for optimizing mRNA sequences but these algorithms are limited in performance or the choice of optimization metric. In this work we describe a new mRNA design algorithm that overcomes both of these limitations and is based on differentiable folding, a recently developed paradigm for RNA design in which a probabilistic sequence representation is optimized via gradient-based methods. First, we present major improvements to the original differentiable folding algorithm that drastically reduce the memory overhead of the gradient calculation. Second, we formulate the mRNA design problem in the context of continuous sequences, requiring the generalization of existing metrics and careful treatment of constraints. Given this scaled algorithm and our mRNA design formalism, we then developed a generative deep learning approach that treats our differentiable folding algorithm as a module in a larger optimization pipeline to learn a network that samples optimized sequences. As a demonstration of our method, we optimize mRNA sequences via complex, therapeutically relevant objective functions.