Data-driven reduced-order modeling of chaotic dynamics can result in systems that either dissipate or diverge catastrophically. Leveraging nonlinear dimensionality reduction of autoencoders and the freedom of nonlinear operator inference with neural networks, we aim to solve this problem by imposing a synthetic constraint in the reduced-order space. The synthetic constraint allows our reduced-order model both the freedom to remain fully nonlinear and highly unstable while preventing divergence. We illustrate the methodology with the classical 40-variable Lorenz '96 equations and with a more realistic fluid flow problem-the quasi-geostrophic equations-showing that our methodology is capable of producing medium-to-long-range forecasts with lower error using less data than other nonlinear methods.