There are distinctions between direct injections systems from different manufacturers, however. While Audi/VW's FSI (Fuel Stratified-or Straight-Injection, depending on which continent you're on) takes advantage of this by injecting fuel before the intake valve closes, so more air can come in as the intake charge contracts. Some systems close the intake valve prior to injecting the fuel. This doesn't let more air into the cylinder as the charge cools, it just means the charge inside the cylinder becomes cooler. Mazda's DISI engine injects fuel while the valve is still open. The intake charge is hot enough that the cooling contraction can more than offset the displacement of air by fuel.
A lot of it also has to do with when the fuel is injected during the compression stroke and how many times injection takes place in that one stroke. It's much like using fuel to cool the pistons in turbo cars, but you don't have to add excess fuel and run the car pig-rich.
Added power comes not only from the increase in volumetric efficency, but also many other factors. By lowering air/fuel charge temperatures inside the combustion chamber, engineers can be much more aggressive in tuning ignition timing, since the knock threshold is so much higher. Chances of detonation are also less, since the fuel is sprayed toward the plug tip, making that area just slightly richer and easier to ignite intentionally.
This is how Mazda can offer a warranty on a turbo engine running 9.5:1 compression, while happily soaking up 15.6psi of boost all day. The Audi/VW 2.0-liter FSI engine uses a 10.3:1 compression ratio. Advancing the ignition timing also lowers exhaust gas temperatures (EGT), which lessens stress on the turbo. This opens up turbo sizing constraints and allows engineers to select turbos with wider operating thresholds.
Gasoline direct injection systems also remove flow obstruction within the intake runners. Since the injectors are mounted on the cylinder head, there's no need for injector bosses in the intake manifold. More importantly, intake port shapes are no longer affected by the need to get the injector as close to the port as possible. Low-port heads with tight radius bends just before the valve are a result of having to put the injector close to the valves and having it aimed directly at them. With this packaging restriction removed, most GDI engines have straight intake ports and runners for better flow. The DISI turbo has a high port head, with a straight shot into the combustion chamber.
While the principle of GDI is the same, the management strategy each manufacturer employs varies. For example, Mazda's DISI system takes advantage of the cooling effects for added power and efficiency. DISI uses only a homogeneous air/fuel charge, meaning the fuel delivery is timed and metered so the entire cylinder is filled with an even distribution of air and fuel.
European FSI engines from Audi/VW use a tricky strategy for added fuel economy. At mid and high loads, the FSI system mixes fuel with air in much the same way as any other gasoline engine. But under super-light loads, FSI will switch to a stratified lean injection mode, where a small amount of fuel is injected just as the spark plug is about to fire. Since the injector is right next to the plug tip, the localized air/fuel ratio is virtually stoichiometric (since there isn't enough time for the spray to spread). This allows it to properly ignite, while the rest of the cylinder is filled with almost pure air.