Fourier multiplier for the heat equation
Question
Consider the 1D heat equation on a periodic domain: partial_t u = alpha * partial_xx u, with alpha > 0. Take the Fourier transform u_hat(k, t) of u(x, t) in the spatial variable. Given an initial Fourier amplitude u_hat(k, 0), what is u_hat(k, t)?
Why this matters
This single Fourier multiplier is the gold standard against which every PINN, neural operator, and diffusion-model forward process is benchmarked on parabolic problems. The exact form encodes that high-frequency modes decay quadratically faster than low-frequency modes, which is why diffusion models destroy texture before destroying coarse structure and why DDPM samplers can recover coarse structure long before fine detail.
Common mistake
Confusing the wave equation (oscillation in t) with the heat equation (exponential decay in t). The classification matters: a PINN that works for one will silently fail on the other because gradient descent's information propagation does not match hyperbolic dynamics.
Source anchor
content/topics/pde-fundamentals-for-ml.mdx#six-pdes-that-matter-for-machine-learning