This example demonstrates physics informing of a data-driven model using numerical derivatives (PINO).

To examine the properties of PINOs with 3 coupled nonlinear equations, we examined the ability of the networks to reproduce the nonlinear shallow water equations. These equations are applicable in several physical scenarios including tsunami modeling. We assumed that the total fluid column height $\eta(x,y,t)$ was composed of a mean height plus some perturbation, but the initial velocity fields $u(x,y,t)$ and $v(x,y,t)$ were initially zero. These equations are given by

$$\begin{align} \frac{\partial(\eta)}{\partial t}+\frac{\partial(\eta u)}{\partial x}+ \frac{\partial(\eta v)}{\partial y}&=0, \\ \frac{\partial(\eta u)}{\partial t}+ \frac{\partial}{\partial x}\left(\eta u^{2}+\frac{1}{2} g \eta^{2}\right)+ \frac{\partial(\eta u v)}{\partial y}&=\nu\left(u_{xx} + u_{yy}\right), \\ \frac{\partial(\eta v)}{\partial t}+\frac{\partial(\eta u v)}{\partial x}+ \frac{\partial}{\partial y}\left(\eta v^{2}+\frac{1}{2} g \eta^{2}\right)&=\nu\left(v_{xx} + v_{yy}\right), \end{align}$$

$$\begin{align} \textrm{with} \quad \eta(x,y,0) = \eta_{0}(x,y),\ u(x,y,0)=0, v(x,y,0)=0,\ \quad x,y \in[0,1), \ t \in[0,1], \end{align}$$

where the gravitational coefficient $g=1$ and the viscosity coefficient $\nu=0.002$ to prevent the formation of shocks. Below we plot how each of these fields evolves in space and time according to the PINO predictions and the simulated data. We observe that the error in each of these cases is relatively small.

We will demonstrate the use of data loss and physics constraints, specifically the equation residual loss, to create accurate predictions. Modulus Sym has utilities tailored for physics-informed machine learning. It also presents abstracted APIs that allow users to think and model the problem from the lens of equations, constraints, etc. In this example, we will only leverage the physics-informed utilities to see how we can add physics to an existing data-driven model with ease while still maintaining the flexibility to define our own training loop and other details. For a more abstracted definition of these type of problems, where the training loop definition and other things is taken care of implicitly, you may refer Modulus Sym

The training and validation datasets for this example can be found on the PINO Applications Github page. The downloading and pre-processing of the data can also be done by running the below set of commands:

python download_data.py

To demonstrate the usefulness of the Physics loss, we will deliberately choose a smaller dataset size of 45 samples. In such regiemes, the effect of physics loss is more evident, as it regularizes the model in the absence of large data.

In this example, we will use a Fourier Neural Operator (FNO). and then compute the derivatives in a PINO style, using Numerical differentiation with Fourier derivatives. With this example, we intend to demonstrate how to implement multiple equations into the loss function.

In this example, we will also use the PDE class from Modulus-Sym to symbolically define the PDEs. This is very convinient and most natural way to define these PDEs and allows us to print the equations to check for correctness. This also abstracts out the complexity of converting the equation into a pytorch representation. Modulus Sym also provides several complex, well-tested PDEs like 3D Navier-Stokes, Linear elasticity, Electromagnetics, etc. pre-defined which can be used directly in physics-informing applications. We will also give you the option to choose between the derivative functions from Modulus-Sym or from the original paper.

To get started with the example, simply run,

python train_swe_nl_pino.py

• Applications of physics informed neural operators
• Fourier Neural Operator for Parametric Partial Differential Equations
• Physics-Informed Neural Operator for Learning Partial Differential Equations