On the Role of Fixed Points of Dynamical Systems in Training Physics-Informed Neural Networks

Franz M. Rohrhofer, Stefan Posch, Clemens Gößnitzer, Bernhard C Geiger

Abstract:

This paper empirically studies commonly observed training difficulties of Physics-Informed Neural Networks (PINNs) on dynamical systems.
Our results indicate that fixed points which are inherent to these systems play a key role in the optimization of the in PINNs embedded physics loss function.
We observe that the loss landscape exhibits local optima that are shaped by the presence of fixed points.
We find that these local optima contribute to the complexity of the physics loss optimization which can explain common training difficulties and resulting nonphysical predictions.
Under certain settings, e.g., initial conditions close to fixed points or long simulations times, we show that those optima can even become better than that of the desired solution.

Submission Length: Long submission (more than 12 pages of main content)

Changes Since Last Submission:

- Included "Validity Domain" in Discussion Section
- Code can now be found at https://github.com/frohrhofer/PINNs_fixed_points

Code: https://github.com/frohrhofer/PINNs_fixed_points

Assigned Action Editor: ~Nicolas_THOME2

License: Creative Commons Attribution 4.0 International (CC BY 4.0)

Submission Number: 490