This is a research project to study Donaldson's geometric flow on the space of symplectic structures by means of computational solutions. In particular, I use PINN's (physically informed neural networks) to approximate geometric quantities arising in the flow. The hope is to gain a better intuition on the existence of critical points, singularities and the structure of the space of symplectic forms.
An overview of Donaldson's geometric flow on the space of symplectic structures can be found here.
The reasoning for the local coordinate expression of the flow is contained in
local_coordinates.tex
.
There is a 2-dimensional analog to the Donaldson flow on the space of volume
forms of a closed surface. Its equation is given by
The following are a few solutions on the 2-torus.
sol1.mp4
sol2.mp4
sol3.mp4
A few insights:
- The maximum principle can be clearly seen at work.
- Due to the inversion
$\frac{1}{u}$ the convergence is much faster where$u$ is small, and can become extremely slow when$u$ is big. - Hence, the flow favours points where
$u$ is big, while points where$u$ is small quickly make$u$ grow.
Looking for critical points of the flow means solving a (highly) non-linear elliptic PDE,
We are using NeuralPDE.jl to define a loss function on the jet-bundle of
the four-torus corresponding to the critical point equation. The symplectic
structure is then approximated by a fully connected neural network with
several hidden layers. To make the optimization more flexible, we also treat
the Hamiltonian vector fields
Unsurprisingly, this finds the lowest critical point of the problem with
constant