Physics-Informed Loss

Function¶

The learned function, $\boldsymbol{f_\theta}$, will map the spatial coordinates, $\mathbf{x}_\phi \in \mathbb{R}^{D\phi}$, and time coordinate, $t \in \mathbb{R}$, to sea surface height, $u \in \mathbb{R}$.

Loss¶

The standard loss term is data-driven

However, there is no penalization to make the field behave the way we would expect. We also want a regularization which makes the field, $u$, behave how we would expect. This can be achieved by adding a physics-informed loss regularization term to the total loss.

This loss term can be minimized by effectively minimizing a PDE function. For example:

where $\partial_t$ is the derivative of the field, $u$, wrt to time and $\mathcal{N}[\cdot]$ are some partial differential equations. We are interested in minimizing the full PDE, which we denote $\boldsymbol{f}_{phy}$, st it is 0. So the standard loss function applies.

Examples¶

Below are some examples of how I have used the PINNs loss/regularization formulation in my own research.

QG Equation¶

We have the following PDE for the QG dynamics:

where $q(x,t) \in \mathbb{R}^2 \times \mathbb{R} \rightarrow \mathbb{R}$ is the potential vorticity (PV), $\psi(x,t) \in \mathbb{R}^2 \times \mathbb{R} \rightarrow \mathbb{R}$ is the stream function, $\partial_t$ is the partial derivative wrt $t$, $\boldsymbol{J}$, is the Jacobian operator and $\det \boldsymbol{J}(\cdot,\cdot)$ is the determinant of the Jacobian.

Objective: We want to convert this PDE in terms of sea surface height (SSH) instead of PV and the stream function.

QG Equation 4 SSH (TLDR)¶

Note: For the ease of notation, let’s denote $u$ as the SSH. The above PDE can be written in terms of $u$

See the following page for more and how this equation was derived.

Example I: Divergence + Curl Free¶

Example II: Quasi-Geostrophic Equations¶