How to solve equations with a second or higher order derivative in time, like the wave equation? about pinns HOT 2 OPEN

You can refer fractional PINN for higher order time derivatives.

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In the paper "Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations" to which this repository refers, it is said that:

In this work, we consider parametrized and nonlinear partial differential equations of the general form
u t + N [ u ; λ] = 0 , x ∈ Ω, t ∈ [ 0 , T ]
where u ( t , x ) denotes the latent (hidden) solution, N [ · ; λ] is a nonlinear operator parametrized by λ , and Ω is a subset
of R^D

Thus it's not considered the case with higher time derivatives.
Nevertheless, I guess that some trials can be done. The discrete time approach employs the Runge Kutta method that's tailored to a form like the one shown above, so I would forget it.
Let's focus on the continuous time approach, i.e. the one employed in the example with the Schrodinger equation. The only adaptation I would do to the script by Raissi would be the definition of the part of the loss computed in the collocation points (the N_f points extracted from the interior of the spatio-temporal domine).
In the Raissi's repo, in the file main/continuous_time_inference (Schrodinger)/Schrodinger.py, at line 148 it's defined net_f_uv. I would change its content computing a higher time derivative and substituting f_u, f_v with the residual that you need.
I actually do not know if that's a good approach since I'm totally not an expert, but I hope my opinion will help you somehow.

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