Task:

PINNs original paper [2019] can be accessed here.

The remaining of this document is structured as follows:

1. Equations: Navier-Stokes and Schrodinger
2. PINNs : How they work and which gap they fill in the literature.
3. PINNs : Limitations.
4. Reproducing paper experiments: Pinns-torch
5. Discussion and Conclusion

Some people love it, other people hate it. However, it is undeniable that solving Navier-Stokes and fully understanding it remains a million dollar question (or should I say equation? 😄).

In Raissi et al., they experiment PINNs on the 2D version of Navier Stokes equation. The equation is described explicitly as follows:

$$ u_t \lambda_{1}(uu_x vu_y) = -p_x \lambda_{2}(u_{xx} u_{yy}) (1),

$$

$v_t \lambda_{1}(uv_x vv_y) = -p_y \lambda_{2}(v_{xx} v_{yy}) (2)$

$$

$$

$$

$$

$$

$$

where $u(t,x,y)$ denotes the x-component of the velocity field, $v(t,x,y)$ the y-component, whereas the pressure is described by $p(t,x,y)$. I highlight that in this setup, we have no knowledge about the $\lambda$ values ($\lambda_1 , \lambda_2$).

For the purposes of the paper and this assignment, the experiments consider the pseudo problem of a "incompressible flow past a circular cylinder" (Raissi et al., 693) using the Navier Stokes 2D formulation shown above.