The Other Paradigms
What if stepping through time is the wrong idea entirely?
Everything so far marched forward in small steps. These do not. Spectral methods solve in frequency space; variational integrators discretise the action instead of the equations; Monte Carlo abandons determinism to beat dimensionality; differentiable simulation makes the solver itself something you can optimise through.
Stop approximating derivatives locally. Represent the whole solution in a Fourier basis, where differentiation is multiplication and accuracy is exponential.
Every deterministic quadrature rule degrades exponentially with dimension. Monte Carlo error is O(N^-1/2) in any number of dimensions, which eventually makes it the only option.
Integrate enough particles for long enough and macroscopic quantities emerge from the trajectory. This is where symplectic integration stops being elegant and becomes mandatory.
Make the solver itself differentiable and simulation becomes something you can optimise through - inverse problems, control, and learned dynamics all follow.