Essential Math for AI

Chapter 13. Artificial Intelligence and Partial Differential Equations

I want to model the whole world.

H.

The first scene in the movie Top Gun: Maverick (2022) shows Maverick (Tom Cruise) manning an experimental military aircraft and pushing it to 10 times the speed of sound (10 Mach) before losing its stability at around 10.2 Mach. The fastest nonfictional manned aircraft so far can reach 6.7 Mach (Figure 13-1). Real speed or unreal (yet), it is mesmerizing to watch physics, math, and engineering come together to put these planes in the air, especially with their spectacular midair maneuvering.

Figure 13-1. Fastest manned aircraft ever made (image source)

These are a few of the partial differential equations (PDEs) that come to mind while watching Maverick’s awesome dogfight and 10 Mach scenes:

The wave equation for wave propagation

Think of the speed of sound, the propagation of a sound wave in the air, and the variations of the sound of speed at different altitudes due to the variations in temperature and air density.

Navier-Stokes equations for fluid dynamics

Think of the fluid flow, air tunnels, and turbulence.

The G-equation for combustion

Think of the combustion in the aircraft’s engine and the flames coming out of the aircraft’s exhausts.

Material elasticity equations

Think of the aircraft wing panel, the lift force, the buckling of the wing panel (the process ...