drsk0 / computational_donaldson_flow

Physics informed learning applied to Donaldson's geometric flow of symplectic structures on four-manifolds.

Computational solutions for the Donaldson geometric flow

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.

2d solutions

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 $$\partial_t u = d^*d\frac{1}{u}$$ This equation is closely related to the heat flow. In particular, there is a maximum-principle that guarantees convergence to a constant solution. To solve it, a finite-difference method is used to convert the problem to an ODE problem. Then the Rosenbrock23 solver is used to solve the ODE.

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.

Critical points

Looking for critical points of the flow means solving a (highly) non-linear elliptic PDE, $$\sum_{i = 1}^3 J_i X_{K_i} = 0,$$ where $X_{K_i}$ is defined by the equations $$dK_i = \rho(X_i, \cdot), \qquad K_i = \frac{\rho\wedge \omega_i}{{\rm dvol}_\rho}.$$

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 $X_i$ as free variables and couple them to the symplectic structure via the constraint $$dK_i = \rho(X_i, \cdot).$$ Using quasi-random point samples, we get an optimization problem that can be solved with stochastic gradient decent and the ADAM optimizer.

Unsurprisingly, this finds the lowest critical point of the problem with constant $K_i$ functions. To find higher critical points, additional constraints need to be imposed.