the inverse Fourier transform
If the Fourier transform takes a signal apart into its pure-frequency ingredients, the inverse Fourier transform puts it back together. It is the recipe that, given the amount of every frequency present (the spectrum f-hat), rebuilds the original function f(x). Without an inverse, transforming would be a one-way street and useless for solving anything; the whole method works precisely because the trip is reversible.
The formula is a mirror image of the forward transform: f(x) = (1/(2*pi)) integral from -infinity to infinity of f-hat(xi) e^(i*xi*x) dxi. Read it as a superposition: you add up the pure wave e^(i*xi*x) for every frequency xi, each weighted by how much of that frequency the signal contains, namely f-hat(xi). The only differences from the forward transform are the sign in the exponent (+i*xi*x instead of -i*xi*x) and a normalizing factor of 1/(2*pi). Various books split that 2*pi differently or put a sign convention the other way; the mathematics is the same, only the bookkeeping changes.
In the PDE method, the inverse is the final step that returns you to physical space. After you have solved the easy ODE for u-hat(xi, t), you invert to recover u(x, t). The inversion theorem guarantees you get the original f back (at points where f is continuous) for any reasonable f, which is what makes the transform a genuine change of viewpoint rather than a loss of information. The honest caveat: actually computing the inverse integral is frequently the hard labour of the whole calculation — you lean on the convolution theorem, tables of standard pairs, or contour integration to do it.
A single Gaussian f(x) = e^(-x^2/2) has Fourier transform f-hat(xi) = sqrt(2*pi) e^(-xi^2/2), another Gaussian. Feeding that back through the inverse formula reproduces e^(-x^2/2) exactly — the round trip is the identity. The Gaussian is special because it is its own transform up to scaling.
Forward then inverse returns the original — the Gaussian famously transforms into itself.
The 'inverse equals the original' guarantee holds at points of continuity; at a jump the inversion integral returns the average of the left and right limits, the same Gibbs-style subtlety you meet in Fourier series.