Misadventures in machine learning for mathematics (Otis Chodosh)
Misadventures in machine learning for mathematics (Otis Chodosh)
2021: Advancing mathematics by guiding human intuition with AI.
- No data?
E.g. prove that if $X$ has property $P$, then $X=X_ 0$.
Recall a,b,c method in PDE solving $\partial_ t^2 u-\partial^2_ xu =0$.
Consider $0 = \int M(\mathrm{PDE})$ where $M = a\partial_t u + b\partial_ x u + cu.$
Example 1 (real life example).
Consider $\triangle u =0$ for $\{u>0\}$ and $|\nabla u| = 1$ for $\partial\{u>0\}$.
For $\mathbb{R}^2,\mathbb{R}^3,\mathbb{R}^4$, $C^\infty$-solution; for $\mathbb{R}^7$, not $C^\infty$; for $\mathbb{R}^5$ and $\mathbb{R}^6$ unknown.
For $\mathbb{R}^3$, by substitue of variable, one can rewrite it as $w\triangle w\geq Q|\triangle w|^2$ for $\{u>0\}$ and $\partial_ v w + LHw\geq 0$ for $\partial\{u>0\}$ with $(1-Q)L\leq1$.
$w=|D^2 u|$, one has $Q = \frac{2}{3}$ and $L=\frac{7}{2}$.
Jerison-Savin: Take $w = (\sum_ {\lambda>0}\lambda^2 + a\sum_ {\lambda<0}\lambda^2)^{1/2}$, one has $Q=\frac{2}{3}$ but $L_ {a=4} = 3$.
Imagine we have a (numerical) solution $u$ and $w = F(D^2 u).$
Then $\nabla_ k w = F_ {ij}D_ {ijk}^3 u$. We use Taylor expansion and use 3-jet to approximate $u$. Then $$Q_ F = \textrm{min}_ {\textrm{3-jet}}\frac{w\triangle w}{|\nabla w|^2},$$ which is an easier optimization problem. The boundary term $L_ F$ is actually easier to compute and omitted for this talk.
Given a MLP $F$, find $Q_ F$, $L_ F$, then compute $(1-Q_ F)L_ F$. The goal is minimizing $(1- Q_ F)L_ F.$
Recall that for $f(x) = \textrm{min}_ s f(x,s)$, and $s_ x = \textrm{argmin}_ sf(x,s)$. Then $\nabla f(x) = D_ x f(x,s_ x).$
Assume $O(3)$ symmetry and homogeneous of degree 1, one can parametrize $F(D^2 u) = |D^2 u| G(k)$, where $k = \frac{\det D^u}{|Du|^{\frac{3}{2}}}.$