Actually learning the solution? about harmonic-oscillator-pinn

[benmoseley]

Hi @pmorerio, some really great observations, thanks for trying the code!

Yes, you are exactly right, the PINN will only learn the solution over the range of its "physics loss training points". Basically there is no free lunch: we don't need many training examples of the solution (orange dots), but we do need lots of input points (green dots) sampled over the entire input domain we are interested in solving.

And then the question of whether PINNs are expressive enough to learn many oscillations of this system. It is very likely a small neural network will not be expressive enough, and a larger neural network is needed. Actually, we just wrote a paper on exactly this limitation. In the paper we propose "finite basis physics informed neural networks (FBPINNs)", which combine PINNs with domain decomposition to allow them to scale to large domains, see here: https://arxiv.org/pdf/2107.07871.pdf. In fact, I created an example comparing FBPINNs for PINNs using the harmonic oscillator problem, but for many more oscillations here: https://github.com/benmoseley/FBPINNs/blob/main/examples/Defining%20your%20own%20problem.ipynb.

The idea of using priors on the solution is a very good one. For example, others have used sinusoidal activation functions in the NN as a prior (https://vsitzmann.github.io/siren/). In our FBPINN paper, we show how the boundary conditions of the PDE can automatically be satisfied by using the PINN as part of a mathematical "ansatz" solution too.

[pmorerio]

Thank you very much for the comprehensive answer! I will definitely explore the follow-up paper and try the FBPINN code!
Best,
P.