Natural science datasets frequently violate assumptions of independence. Samples may be clustered (e.g. by study site, subject, or experimental batch), leading to spurious associations, poor model fitting, and confounded analyses. While largely unaddressed in deep learning, this problem has been handled in the statistics community through mixed effects models, which separate cluster-invariant fixed effects from cluster-specific random effects. We propose a general-purpose framework for Adversarially-Regularized Mixed Effects Deep learning (ARMED) models through non-intrusive additions to existing neural networks: 1) an adversarial classifier constraining the original model to learn only cluster-invariant features, 2) a random effects subnetwork capturing cluster-specific features, and 3) an approach to apply random effects to clusters unseen during training. We apply ARMED to dense, convolutional, and autoencoder neural networks on 4 applications including simulated nonlinear data, dementia prognosis and diagnosis, and live-cell image analysis. Compared to prior techniques, ARMED models better distinguish confounded from true associations in simulations and learn more biologically plausible features in clinical applications. They can also quantify inter-cluster variance and visualize cluster effects in data. Finally, ARMED improves accuracy on data from clusters seen during training (up to 28% vs. conventional models) and generalization to unseen clusters (up to 9% vs. conventional models).