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Molecules with strong two-photon absorption (TPA) are important in many advanced applications such as upconverted laser and photodynamic therapy, but their design is hampered by the high cost of experimental screening and accurate quantum chemical (QC) calculations. Here a systematic study is performed by collecting an experimental TPA database with "900 molecules, analyzing with interpretable machine learning (ML) the key molecular features explaining TPA magnitudes, and building a fast ML model for predictions. The ML model has prediction errors of similar magnitude compared to experimental and affordable QC methods errors and has the potential for high-throughput screening as additionally validated with the new experimental measurements. ML feature analysis is generally consistent with common beliefs which is quantified and rectified. The most important feature is conjugation length followed by features reflecting the effects of donor and acceptor substitution and coplanarity. [ABSTRACT FROM AUTHOR]