Solution Bundle

[tanmay2798]

The solution bundle method is used in this file - https://github.com/Isaac-Somerville/Neural-Networks-for-Solving-Differential-Equations/blob/main/ThreeBodyProblem/threeBodyOriginalMethod.py ?

[Isaac-Somerville]

The solution bundle method is used in this file - https://github.com/Isaac-Somerville/Neural-Networks-for-Solving-Differential-Equations/blob/main/ThreeBodyProblem/threeBodyOriginalMethod.py ?

Yes. The other files in the ThreeBodyProblem folder use this method as well, but also include curriculum learning

[tanmay2798]

Okay, also can you tell me if I use spring mass system where there are double derivatives, according to your naming system, would it be like. x = trialSolution(x0, xOut, t) dxTrial = dTrialSolution(xOut, dxOut, t) d2xTrial = dTrialSolution(, d2xOut, t) What should be in place of ?

should it be the value I get if I pass xOut through the network?

[Isaac-Somerville]

For second derivatives, you would require another function, d2TrialSolution, that uses the analytic expression for the second derivative of trialSolution. It would look something like this:

def d2TrialSolution(varOut, dVarOut, d2VarOut, t): """ Second derivative w.r.t. t of trial solution for a given variable varOut at times t with initial values varInitial

Arguments:
varOut (tensor of shape (batchSize,1)) -- network output for variable at times t
dVarOut (tensor of shape (batchSize,1)) -- derivative of network output for variable w.r.t. t at times t
d2VarOut (tensor of shape (batchSize,1)) -- second derivative of network output for variable w.r.t. t at times t
t (tensor of shape (batchSize,1)) -- times at which variable is evaluated

Returns:
d2trialSoln (tensor of shape (batchSize,1)) -- second derivative of trial solution for given variable at 
    times t with initial values varInitial"""
# d2TrialSoln = (torch.exp(-t)) * dVarOut  + ((1 - torch.exp(-t)) * d2VarOut) - torch.exp(-t) * varOut + torch.exp(-t) * dVarOut
d2TrialSoln = ((1 - torch.exp(-t)) * d2VarOut) + (torch.exp(-t) * (2 * d2VarOut  - varOut))
return d2TrialSoln

In the train function you would be required to evaluate d2xOut in the same was dxOut was calculated: d2xOut = grad(dxOut,t,torch.ones_like(dxOut),retain_graph=True, create_graph=True)[0]

Problems 5, 7 and 8 have more examples of doing this for second derivatives, as well as calculating mixed partial derivatives using autograd.

Hope this helps!

[tanmay2798]

@Isaac-Somerville https://github.com/Isaac-Somerville/Neural-Networks-for-Solving-Differential-Equations/blob/main/LagarisProblems/prob5.py

Is this and prob7 and prob8 in LagarisProblems folder are using solution bundle?

[Isaac-Somerville]

No but the methodology is the same regarding calculating derivatives of the trial solution

On Mon, 4 Mar 2024, 21:46 tanmay2798, @.***> wrote:

https://github.com/Isaac-Somerville/Neural-Networks-for-Solving-Differential-Equations/blob/main/LagarisProblems/prob5.py

Is this and prob7 and prob8 in LagarisProblems folder are using solution bundle?

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[tanmay2798]

Okay, and should it be d2TrialSoln = ((1 - torch.exp(-t)) d2VarOut) + (torch.exp(-t) (2 * dVarOut - varOut))

[Isaac-Somerville]

Yes that's right

[tanmay2798]

@Isaac-Somerville adam.txt I modified your code to work for spring mass dampener system (md2x/dt2 + bdx/dt + k(x-L)=0). But, for me it doesn't give promising results. Can you suggest what changes I need to make?