This example demonstrates physics informing of a data-driven model using to approaces - the DeepONet approach which computes the gradients using Autograd and an approach using Numerical derivatives (PINO).

This is an extension of the 2D darcy flow data-driven problem. In addition to the data loss, we will demonstrate the use of physics constranints, specifically the equation residual loss. Modulus Sym has utilities tailored for physics-informed machine learning. It also presents an abstracted APIs that allows users to think and model the problem from the lens of equations, constraints, etc. In this example, we will only levarage the physics-informed utilites 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 implictily, you may refer Modulus Sym

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

```
```pip install -r requirements.txt
python download_data.py

Do demonstrate the usefulness of the Physics loss, we will deliberately choose a smaller dataset size of 100 samples. In such regiemes, the effect of physics loss is more evident, as it regularizes the model in the absense of large data.

In this example we will use a Fourier Neural Operator (FNO). We will demonstrate two cases, in the first case, the FNO is used as the branch net and use a fully-connected network for the trunk net. The input to the branch network is the input permeability field and the input to the trunk network is the x, y coordinates. The output of the model is the pressure field. Having the mapping between the pressure field and the input x and y through a fully-differentiable network will allow us to compute the gradients of the pressure field w.r.t input x and y through automatic differentiation through Modulus sym utils.

In the second case, we will use just FNO and then compute the derivatives in a PINO style, using Numerical differentiation. Both approaches are viable ways to introduce physics in the loss function and the use of one over the other can change from case-to-case basis. With this example, we intend to demonstrate both such cases so that the users can compare and contrast the two approaches.

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.

To get started with the example, simply run,

```
```python
darcy_physics_informed_deeponet.py

or

```
```python
darcy_physics_informed_fno.py