  Titanium dioxide has been extensively studied in the rutile or anatase
phases, while its high-pressure phases are less well understood, despite that
many are thought to have interesting optical, mechanical and electrochemical
properties. First-principles methods such as density functional theory (DFT)
are often used to compute the enthalpies of TiO$_2$ phases at 0~K, but they are
expensive and thus impractical for long time-scale and large system-size
simulations at finite temperatures. On the other hand, cheap empirical
potentials fail to capture the relative stablities of the various polymorphs.
To model the thermodynamic behaviors of ambient and high-pressure phases of
TiO$_2$, we design an empirical model as a baseline, and then train a machine
learning potential based on the difference between the DFT data and the
empirical model. This so-called $\Delta$-learning potential contains long-range
electrostatic interactions, and predicts the 0~K enthalpies of stable TiO$_2$
phases that are in good agreement with DFT. We construct a pressure-temperature
phase diagram of TiO$_2$ in the range $0<P<70$~GPa and $100<T<1500$~K. We then
simulate dynamic phase transition processes, by compressing anatase at
different temperatures. At 300~K, we observe predominantly
anatase-to-baddeleyite transformation at about 20~GPa, via a martensitic
two-step mechanism with highly ordered and collective atomic motion. At 2000~K,
anatase can transform into cotunnite around 45-55~GPa in a thermally-activated
and probabilistic manner, accompanied by diffusive movement of oxygen atoms.
The pressures computed for these transitions show good agreement with
experiments. Our results shed light on how to synthesize and stabilize
high-pressure TiO$_2$ phases, and our method is generally applicable to other
functional materials with multiple polymorphs.
