update readme

README.md

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 [![Dev](https://img.shields.io/badge/docs-dev-blue.svg)](https://vpuri3.github.io/NeuralROMs.jl/dev/)
 [![Build Status](https://github.com/vpuri3/NeuralROMs.jl/actions/workflows/CI.yml/badge.svg?branch=master)](https://github.com/vpuri3/NeuralROMs.jl/actions/workflows/CI.yml?query=branch%3Amaster)
 
+This repository implements machine learning (ML) based reduced order models (ROMs).
+Specifically, we introduce [smooth neural field ROM (SNF-ROM)](arxiv).
+
+## SNF-ROM: Projection-based nonlinear reduced order modeling with smooth neural fields
+
+### Abstract
+
+Reduced order modeling lowers the computational cost of solving PDEs by learning a low-order spatial representation from data and dynamically evolving these representations using manifold projections of the governing equations.
+While commonly used, linear subspace reduced-order models (ROMs) are often suboptimal for problems with a slow decay of Kolmogorov $n$-width, such as advection-dominated fluid flows at high Reynolds numbers.
+There has been a growing interest in nonlinear ROMs that use state-of-the-art representation learning techniques to accurately capture such phenomena with fewer degrees of freedom.
+We propose smooth neural field ROM (SNF-ROM), a nonlinear reduced modeling framework that combines grid-free reduced representations with Galerkin projection.
+The SNF-ROM architecture constrains the learned ROM trajectories to a smoothly varying path, which proves beneficial in the dynamics evaluation when the reduced manifold is traversed in accordance with the governing PDEs.
+Furthermore, we devise robust regularization schemes to ensure the learned neural fields are smooth and differentiable.
+This allows us to compute physics-based dynamics of the reduced system nonintrusively with automatic differentiation and evolve the reduced system with classical time-integrators.
+SNF-ROM leads to fast offline training as well as enhanced accuracy and stability during the online dynamics evaluation.
+We demonstrate the efficacy of SNF-ROM on a range of advection-dominated linear and nonlinear PDE problems where we consistently outperform state-of-the-art ROMs.
+
+### Setup and run
+
+Download the code and 
+
+```julia
+julia> import Pkg
+
+julia> Pkg.develop(url = "https://github.com/vpuri3/NeuralROMs.jl")
+
+julia> cd(joinpath(DEPOT_PATH[1], "dev", "NeuralROMs.jl"))
+
+julia> Pkg.activate() # activate the environment
+```
+
+We show how to run the 1D Advection test case corresponding to Section 6.1 of the paper.
+Each test case in Section 6 of the paper has a corresponding directory in `examples/`.
+
+```julia
+julia> include("examples/advect_fourier1D/datagen_advect1D.jl")
+```
+
+The script solves the 1D advection problem and stores the dataset as a JLD2 binary in
+`examples/advect_fourier1D/data_advect/`.
+To train SNF-ROM, run
+
+```julia
+julia> include("examples/advect_fourier1D/snf.jl")
+```
+
+### Results
+
+#### 1D Advection
+#### 2D Advection
+#### 1D Viscous Burgers (Re 10k)
+#### 2D Viscous Burgers (Re 1k)
+#### 1D Kuramoto-Sivashinsky
+
 ## Citing
 
 See [`CITATION.bib`](CITATION.bib) for the relevant reference(s).