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Parisi Equations Demo
Parisi Equations Demo
TODO
- Work out the 1-RSB estimate for the Parisi equations
- Plot the objective functions
- Create a second notebook that plots like RS, 1-RSB, 2-RSB, full-RSB estimates to demo the Parisi equations.
- Work out the 1-RSB estimate for the Parisi equations
- Plot the objective functions
- Create a second notebook that plots like RS, 1-RSB, 2-RSB, full-RSB estimates to demo the Parisi equations.
Let’s see how the Parisi equations breakdown into their constituent parts. The time is a bit backwards but I think that’s okay...
In[]:=
xMin=-10;
In[]:=
xMax=10;
In[]:=
initCond=u[0,x]Log[2*Cosh[x]];
Heat Equations
Heat Equations
We’ll start with the heat diffusion term in the Parisi equations.
In[]:=
pdeHeat=D[u[t,x],t]10*D[u[t,x],{x,2}];
Let’s solve it numerically
In[]:=
solHeat=NDSolve[{pdeHeat,initCond},u,{t,0,1},{x,xMin,xMax}];
And plot a 3-D plot of the solution
In[]:=
Plot3D[Evaluate[u[t,x]/.solHeat],{t,0,1},{x,xMin,xMax},PlotRangeAll]
Out[]=
Let’s also superimpose some level sets at different time-steps.
In[]:=
dataTableHeat=Evaluate[Table[u[t,x]/.solHeat,{t,0,1,0.1}]];
In[]:=
Plot[dataTableHeat,{x,xMin,xMax},PlotRangeAll,FillingAxis]
Out[]=
The heat equations alone also give the replica symmetric estimate. The value of this estimate is given by
In[]:=
u[1,0]/.solHeat
Out[]=
{3.6403}
Gradient Descent
Gradient Descent
Let’s try the other part of the PDE where you decay as the norm of your gradient. That is “u_t = - (u_x)^2” Note that this equation has a prefactor of 2, vs. the 10 in all the other sections.
In[]:=
pdeGrad=D[u[t,x],t]2(D[u[t,x],x])^2;
and solve the equations numerically again
solGrad=NDSolve[{pdeGrad,initCond},u,{t,0,1},{x,xMin,xMax},MaxStepSize0.0001,MaxSteps->10000];
Let’s plot the results
In[]:=
Plot3D[Evaluate[u[t,x]/.solGrad],{t,0,1},{x,xMin,xMax},PlotRangeAll]
Out[]=
And do a superimposted plot of levelsets
In[]:=
dataTableGrad=Evaluate[Table[u[t,x]/.solGrad,{t,0,1,0.1}]];
In[]:=
Plot[dataTableGrad,{x,0.5*xMin,0.5*xMax},PlotRangeAll,FillingAxis]
Out[]=
Putting it all together
Putting it all together
Okay now we put it all together to see what happens.
In[]:=
pde=D[u[t,x],t]10*(D[u[t,x],{x,2}]+(D[u[t,x],x])^2);
In[]:=
sol=NDSolve[{pde,initCond},u,{t,0,1},{x,xMin,xMax}];
Plotting things
In[]:=
Plot3D[Evaluate[u[t,x]/.sol],{t,0,1},{x,xMin,xMax},PlotRangeAll]
Out[]=
In[]:=
dataTable=Evaluate[Table[u[t,x]/.sol,{t,1,0,-0.1}]];
In[]:=
Plot[dataTable,{x,xMin,xMax},PlotRangeAll,FillingAxis]
Out[]=
Let’s spot check the value at the x=0 when t=1
In[]:=
u[1,0]/.sol
Out[]=
{10.6934}
Overlap given by the Gaussian CDF
Overlap given by the Gaussian CDF
What happens when we plot the Parisi equations with a non-trivial overlap CDF controlling the tradeoff between heat diffusion and gradient descent?
In[]:=
overlap1[t_]:=CDF[NormalDistribution[0.5,0.01],t];
In[]:=
Plot[overlap1[t],{t,-0.5,1.5}]
Out[]=
We then solve the PDE with the overlap distribution
Let’s plot this and see what happens...
What’s the value at x=0 when t=1?
Convex Combination of Step Functions
Convex Combination of Step Functions
Let’s try a convex combination of smooth step functions
The Parisi equations again
Plotting again
And finally the value of the solution achieved at x=0, t=1
The 1-RSB estimate
The 1-RSB estimate
So actually... I should use the replica calculations to figure out what the 1-RSB choice of mu should be and just plot that.