Tackling the curse of dimensionality with physics-informed neural networks (opens in new tab)

The way ML people use "dimension", each free parameter is an extra dimension. A high-resolution 2D image is considered to have millions of dimensions - one for each pixel.

Physics uses vectors, which are multi-dimensional and can represent a lot more than 11 dimensions... Machine learning typically uses feature vectors, which are basically just lists of numerical properties.

To add to the other answers: dimensions also appear in higher-order PDE: you typically go from an nth order 1-dimensional equation (an equation mentioning the n-th derivative of your unknown function) to an n-dimensional 1st order equation. There is a general pattern of adding parameters to have several simple problems instead of one complicated.

What? The Schrodinger equation is just a linear differential equation. I can solve it here right now by hand if the Hamiltonian is time-independent, even for millions of dimensions.

|psi(t)> = exp(-iHt/hbar) |psi(t=0)>

No offence but this sounds like you've never actually studied any quantum mechanics. The difficulty of solving such equations is purely just the difficulty of solving any complex set of PDEs, as the article talks about. There is nothing about the Schrodinger equation that means it's not computable on a classical computer (EDIT nor would it be easier on a quantum computer).

The example they try it on is a quantum harmonic oscillator, i.e. the potential V(x) is just the squared norm of x. That's analytically solvable. Also,

> in the case of the Schro ̈dinger equation, due to the separable nature of its network structure, we employ a separate neural network for each dimension.

which doesn't sound very general. Admittedly in my quick skim of this paper I didn't follow much.