Cavity flow PINNs example #8
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physics-informed-neural-networks-for-steady-cavity-flow/README.md
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| % Add anchor functions to strictly enforce boundary conditions on the | ||
| % streamfunction. | ||
| net = addLayers(net, [functionLayer(@(x)4.*x.*(1-x), Name="anchorX", Acceleratable=true); multiplicationLayer(3, Name="psi")]); |
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Similarly here, I think this needs some more explanation. This function is chosen to satisfy the boundary conditions? Can you give a short description about what the multiplicationLayer and connectLayers do?
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Yes I'll add some more explanation here. The main idea is to ensure that the boundary conditions for psi are always satisfied.
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| % Compute gradients. | ||
| psisum = sum(psi,1); | ||
| u = dlgradient(psisum, yeq, EnableHigherDerivatives=true); |
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Why not use dljacobian?
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I suppose it would mean you'd need at least R2024b. Otherwise I expect it should be ok - it'll do the sum(psi,1) line and the EnableHigherDerivatives=true by default, at the expense of the extra dimension argument dljacobian(psi,yeq,2).
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I do think dljacobian is cleaner here -- I'll switch to that.
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| u = dlgradient(psisum, y); | ||
| v = -1.*dlgradient(psisum, x); |
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It could be worth making this one dlgradient call, something like
[u,v] = dlgradient(psisum,y,x);
v = -v;
I had some feeling this was faster, although I'm not completely confident. I'm sure I found it faster when I did something like batching the interior points and boundary points together and performing one dlgradient, then splitting the output back into interior and boundary gradients. There could be a few places to try that out in the loss above.
Or maybe it was the forward pass, so something like:
allx = cat(1, xeq, xyBoundary(:,1));
ally = cat(1,yeq,xyBoundary(:,2));
psi = forward(net,allx,ally);
psibd = psi(size(xeq,1)+1:end,:);
psi = psi(1:size(xeq,1), :);
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Interesting -- I like both of these ideas. I think it is best practice to mimimize the number of network executions where possible, though it comes at the expense of readability.
I'll toy around with this, but I'd like to make the loss function nice and easy to read.
Thanks Mae! |
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| In order to automatically satisfy the continuity equation we use the stream function psi such that `u=psi_y` and `v=-psi_x`. The boundary conditions are `(u,v)=(1,0)` and the top boundary and `(u,v)=(0,0)` at the other boundaries. The Reynolds number `Re=100`. | ||
| In order to automatically satisfy the continuity equation we use the stream function $\psi$, such that $u=\partial \psi /\partial y$ and $v=-\partial \psi /\partial x$. The boundary conditions are $(u,v)=(1,0)$ at the top boundary and $(u,v)=(0,0)$ at the other boundaries. Additionally, $\psi =0$ is assumed on all the boundaries. The Reynolds number is $Re=100$. |
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I think we should say something like "the domain is
| psum = sum(p,1); | ||
| px = dlgradient(psum, xeq, EnableHigherDerivatives=true); | ||
| py = dlgradient(psum, yeq, EnableHigherDerivatives=true); | ||
| u = dljacobian(psi', yeq, 1); |
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If psi and yeq are both "batch x features" I think u = dljacobian(psi, yeq, 2) would work here. But then u is "hidden x batch x features" where hidden==size(psi,2), and since you end up wanting more derivatives along that "hidden" dimension, I guess you have to permute or swap to dljacobian(u,xeq,1) anyway. I think having stuff as "features x batch" works out more nicely for me typically, which means replacing featureInputLayer with inputLayer if you want to use unlabelled dlarray.
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