it looks like it starts deviates at the end. Also, would be interesting seeing a system that is not stable per say (although this system is not stable it has a limit cycle so that the NN would converge anyways)
That's due to the nature of the Lorenz equations - they are chaotics, meaning (among other things) that any two trajectories that are very close together will diverge at an exponential rate. If he had started 2 points that are extremely close to each other, say, 1e-5, and solved using *only* the ode solver, he would have seen divergence at the end of that as well. It's actually quite remarkable that the ode and NN trajectories stay as close to each other as they do, and shows the NN is doing about as well as is possible. The only way to really do better is to cheat and know what the functional form of the Lorenz equations are with certainty, which of course the NN doesn't have.
Is there an advantage of using a neural net instead of solving the equations numerically? What is the point of using neural nets if the system can be described by equations? I'm not trying to be rude, I just don't understand.
Nevermind, I get it. You can't always obtain an accurate mathematical expression for a system in real life. Also, non-linear systems are really difficult to solve, that's where neural networks come into play.
Can you please make a playlist that contains this video series so we can follow your lectures in ordrer ,it would be very helpful ! Thank you
Have equivalent code in tensorflow or pytorch?
thanks for sharing, great lecture
I used a similar technique to solve a two point boundary value problem a couple of years ago. Glad to know this approach is gaining traction.
When you say predict, we are still predicting points inside the training set? Is that correct
it looks like it starts deviates at the end. Also, would be interesting seeing a system that is not stable per say (although this system is not stable it has a limit cycle so that the NN would converge anyways)
That's due to the nature of the Lorenz equations - they are chaotics, meaning (among other things) that any two trajectories that are very close together will diverge at an exponential rate. If he had started 2 points that are extremely close to each other, say, 1e-5, and solved using *only* the ode solver, he would have seen divergence at the end of that as well. It's actually quite remarkable that the ode and NN trajectories stay as close to each other as they do, and shows the NN is doing about as well as is possible. The only way to really do better is to cheat and know what the functional form of the Lorenz equations are with certainty, which of course the NN doesn't have.
mind blown!
Is there an advantage of using a neural net instead of solving the equations numerically? What is the point of using neural nets if the system can be described by equations? I'm not trying to be rude, I just don't understand.
Nevermind, I get it. You can't always obtain an accurate mathematical expression for a system in real life. Also, non-linear systems are really difficult to solve, that's where neural networks come into play.
That color palette, though.