Enhance landing page

docs/index.md

@@ -1,10 +1,76 @@
-# Exponax
+# Getting Started
 
-A fast and differentiable PDE solver framework in JAX.
+## Installation
 
-Hello World
+```bash
+pip install git+ssh://git@github.com/Ceyron/exponax@main
+```
 
-Below is some math
+## Quickstart
 
-$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
+1d Kuramoto-Sivashinsky Equation.
 
+```python
+import jax
+import exponax as ex
+import matplotlib.pyplot as plt
+
+ks_stepper = ex.stepper.KuramotoSivashinskyConservative(
+    num_spatial_dims=1, domain_extent=100.0,
+    num_points=200, dt=0.1,
+)
+
+u_0 = ex.ic.RandomTruncatedFourierSeries(
+    num_spatial_dims=1, cutoff=5
+)(num_points=200, key=jax.random.PRNGKey(0))
+
+trajectory = ex.rollout(ks_stepper, 500, include_init=True)(u_0)
+
+plt.imshow(trajectory[:, 0, :].T, aspect='auto', cmap='RdBu', vmin=-2, vmax=2, origin="lower")
+plt.xlabel("Time"); plt.ylabel("Space"); plt.show()
+```
+
+![](imgs/ks_rollout.png)
+
+For a next step, check out the [simple_advection_example_1d.ipynb](examples/simple_advection_example_1d.ipynb) notebook in the `examples` folder, and check out the <a href="#documentation">Documentation</a>.
+
+## Features
+
+
+1. **JAX** as the computational backend:
+    1. **Backend agnotistic code** - run on CPU, GPU, or TPU, in both single and
+        double precision.
+    2. **Automatic differentiation** over the timesteppers - compute gradients
+        of solutions with respect to initial conditions, parameters, etc.
+    3. Also helpful for **tight integration with Deep Learning** since each
+        timestepper is just an
+        [Equinox](https://github.com/patrick-kidger/equinox) Module.
+    4. **Automatic Vectorization** using `jax.vmap` (or `equinox.filter_vmap`)
+        allowing to advance multiple states in time or instantiate multiple
+        solvers at a time that operate efficiently in batch.
+2. **Lightweight Design** without custom types. There is no `grid` or `state`
+    object. Everything is based on `jax.numpy` arrays. Timesteppers are callable
+    PyTrees.
+3. More than 35 pre-built dynamics:
+    1. Linear PDEs in 1d, 2d, and 3d (advection, diffusion, dispersion, etc.)
+    2. Nonlinear PDEs in 1d, 2d, and 3d (Burgers, Kuramoto-Sivashinsky,
+        Korteweg-de Vries, Navier-Stokes, etc.)
+    3. Reaction-Diffusion (Gray-Scott, Swift-Hohenberg, etc.)
+4. Collection of initial condition distributions (truncated Fourier series,
+   Gaussian Random Fields, etc.)
+5. **Utilities** for spectral derivatives, grid creation, autogressive rollout,
+   etc.
+6. Easily extendable to new PDEs by subclassing from the `BaseStepper` module.
+7. Normalized interface for reduced number of parameters to uniquely define any
+   dynamics.
+
+
+## License
+
+MIT, see [here](https://github.com/Ceyron/apebench/blob/main/LICENSE.txt)
+
+---
+
+> [fkoehler.site](https://fkoehler.site/) &nbsp;&middot;&nbsp;
+> GitHub [@ceyron](https://github.com/ceyron) &nbsp;&middot;&nbsp;
+> X [@felix_m_koehler](https://twitter.com/felix_m_koehler)