Modern experimental technologies can assay large numbers of biological sequences, but engineered protein libraries rarely exceed the sequence diversity of natural protein families. Machine learning (ML) models trained directly on experimental data without biophysical modeling provide one route to accessing the full potential diversity of engineered proteins. Here we apply deep learning to design highly diverse adeno-associated virus 2 (AAV2) capsid protein variants that remain viable for packaging of a DNA payload. Focusing on a 28-amino acid segment, we generated 201,426 variants of the AAV2 wild-type (WT) sequence yielding 110,689 viable engineered capsids, 57,348 of which surpass the average diversity of natural AAV serotype sequences, with 12-29 mutations across this region. Even when trained on limited data, deep neural network models accurately predict capsid viability across diverse variants. This approach unlocks vast areas of functional but previously unreachable sequence space, with many potential applications for the generation of improved viral vectors and protein therapeutics.