A plasma density model of the inner magnetosphere is important for a variety of applications including the study of wave-particle interactions, and wave excitation and propagation. Previous empirical models have been developed under many limiting assumptions and do not resolve short-term variations, which are especially important during storms. We present a three-dimensional dynamic electron density (DEN3D) model developed using a feedforward neural network with electron densities obtained from four satellite missions. The DEN3D model takes spacecraft location and time series of solar and geomagnetic indices (F10.7, SYM-H, and AL) as inputs. It can reproduce the observed density with a correlation coefficient of 0.95 and predict test data set with error less than a factor of 2. Its predictive ability on out-of-sample data is tested on field-aligned density profiles from the IMAGE satellite. DEN3D's predictive ability provides unprecedented opportunities to gain insight into the 3-D behavior of the inner magnetospheric plasma density at any time and location. As an example, we apply DEN3D to a storm that occurred on 1 June 2013. It successfully reproduces various well-known dynamic features in three dimensions, such as plasmaspheric erosion and recovery, as well as plume formation. Storm time long-term density variations are consistent with expectations; short-term variations appear to be modulated by substorm activity or enhanced convection, an effect that requires further study together with multispacecraft in situ or imaging measurements. Investigating plasmaspheric refilling with the model, we find that it is not monotonic in time and is more complex than expected from previous studies, deserving further attention.