PINNs Against Finite Differences, Spectral Grids and Monte Carlo, from One to Eight Dimensions

A matched-accuracy benchmark on two ground states with exact answers. The dimension at which a physics-informed network starts to beat a grid depends on which grid it is compared with.

The case for physics-informed neural networks in high dimensions is usually made in one line. A tensor-product grid with N points per axis has N^d unknowns, a network has a fixed number of weights, and so somewhere in dimension the network has to win. The line is correct and it is incomplete, because it treats N as if it were a constant of nature. N is set by the discretisation, and it is the one quantity in the argument that a numerical analyst controls.

Take the ground state of the harmonic oscillator and ask for an energy within one part in 1⁰⁴ of the truth. A second-order finite-difference stencil needs about 200 points per axis, a fourth-order stencil about 30, and a spectral grid 8. In five dimensions those numbers become 3.2 × 1⁰¹¹, 2.4 × 1⁰⁷ and 3.3 × 1⁰⁴ unknowns, so a single vector of the grid’s size occupies 2.6 terabytes, 194 megabytes or 262 kilobytes. The dimension at which “the grid runs out” is a property of the grid, and a claim that a network beats classical methods from some dimension onward is a claim about one classical method in particular.