Physics-Informed Loss

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, \(u \in \mathbb{R}\).

\[ u = \boldsymbol{f_\theta}(\mathbf{x}_\phi, t) \]

Loss#

The standard loss term is data-driven

\[ \mathcal{L}_{data} = \text{MSE}(u, \hat{u}) = \frac{1}{N} \sum_{n=1}^N \left(u - \boldsymbol{f_\phi}(\mathbf{x}_\phi, t) \right)^2 \]

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.

\[ \mathcal{L} = \mathcal{L}_{data} + \lambda \mathcal{L}_{phy} \]

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

\[ \boldsymbol{f}_{phy}(\mathbf{x},t):= \partial_t u(\mathbf{x},t) + \mathcal{N}[u(\mathbf{x},t)] = 0 \]

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:

\[ \partial_t q + \det J(\psi, q) = 0 \]

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 \(u\)

\[ \partial_t \nabla^2 u + c_2 \partial_t u + c_1\det \boldsymbol{J}(u, \nabla^2 u) + c_3 \det \boldsymbol{J}(u, u) = 0 \]

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