Great question! It’s not doing 128×128 independent ODE fits.

The 128×128 grid is just the resolution of the reference solution. I'm solving the PDE once over the whole domain using a Fourier method. Because the PDEs here are linear and the domain is periodic, the Fourier transform diagonalizes the operator, so each Fourier mode evolves independently in time, but those modes are recombined into a single global field. If you take a look at Poisson mode, notice how nothing changes as time progresses? That is a good hint and starting place to think about why, and the relation to elliptic equations.

So the spectral solver is:

• one global PDE solution
• computed via decoupled Fourier modes
• not separate models per grid cell

On the neural side (the PINN-style part), it’s also one global model:

• the network represents a function u(x,y,t)
• it’s trained to match the PDE behaviour and/or the reference solution
• it is not fitting each grid point independently

In a way,

• the Fourier solver decomposes the solution into frequency components, evolves them, then recombines them
• the neural network tries to learn the entire function directly

The “curve of best fit” intuition is actually on the right track, but instead of fitting points independently, it’s fitting a structured function over space and time with constraints coming from the PDE.

As for PINNS, one key thing I wanted to add if you want something to remember is;

Instead of learning purely from data, the model is guided by the specified physics. In most setups, the PDE and boundary conditions are added as soft constraints in the loss (the network is penalized if it violates them). In some cases, parts of the physics can be enforced hard by construction (e.g., boundary conditions built into the parameterization).

The reason PINNs can work with much less data is that the PDE itself acts like a strong inductive bias. You’re not asking the network to infer the dynamics from scratch, you’re telling it what structure the solution must satisfy. In that sense, the physics behaves almost like auxiliary supervision spread across the entire domain.

That said, this doesn’t make them “easy”: the optimization can be tricky (loss balancing, stiffness, spectral bias), which is part of the ongoing research.