From d9362a591f85a7ac0b98ec1380ac26634d89578a Mon Sep 17 00:00:00 2001
From: Nathanael Bosch <nathanael.bosch@uni-tuebingen.de>
Date: Wed, 17 Jul 2024 15:38:26 +0200
Subject: [PATCH 1/9] Towards a JOSS paper

---
 paper/paper.bib |  71 +++++++++++++++++++++++++++
 paper/paper.md  | 125 ++++++++++++++++++++++++++++++++++++++++++++++++
 2 files changed, 196 insertions(+)
 create mode 100644 paper/paper.bib
 create mode 100644 paper/paper.md

diff --git a/paper/paper.bib b/paper/paper.bib
new file mode 100644
index 000000000..ca29f0ae6
--- /dev/null
+++ b/paper/paper.bib
@@ -0,0 +1,71 @@
+
+
+@InProceedings{krämer2021probabilistic,
+  title =        {Probabilistic {ODE} Solutions in Millions of Dimensions},
+  author =       {Kr{\"a}mer, Nicholas and Bosch, Nathanael and Schmidt,
+                  Jonathan and Hennig, Philipp},
+  booktitle =    {Proceedings of the 39th International Conference on Machine
+                  Learning},
+  pages =        {11634--11649},
+  year =         2022,
+  editor =       {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and
+                  Szepesvari, Csaba and Niu, Gang and Sabato, Sivan},
+  volume =       162,
+  series =       {Proceedings of Machine Learning Research},
+  month =        {17--23 Jul},
+  publisher =    {PMLR},
+  pdf =          {https://proceedings.mlr.press/v162/kramer22b/kramer22b.pdf},
+  url =          {https://proceedings.mlr.press/v162/kramer22b.html},
+}
+
+@InProceedings{tronarp2022fenrir,
+  title =        {Fenrir: Physics-Enhanced Regression for Initial Value
+                  Problems},
+  author =       {Tronarp, Filip and Bosch, Nathanael and Hennig, Philipp},
+  AUTHOR+an =    {2=highlight},
+  booktitle =    {Proceedings of the 39th International Conference on Machine
+                  Learning},
+  pages =        {21776--21794},
+  year =         2022,
+  editor =       {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and
+                  Szepesvari, Csaba and Niu, Gang and Sabato, Sivan},
+  volume =       162,
+  series =       {Proceedings of Machine Learning Research},
+  month =        {17--23 Jul},
+  publisher =    {PMLR},
+  pdf =          {https://proceedings.mlr.press/v162/tronarp22a/tronarp22a.pdf},
+  url =          {https://proceedings.mlr.press/v162/tronarp22a.html},
+}
+
+@inproceedings{bosch2023probabilistic,
+  title =        {Probabilistic Exponential Integrators},
+  author =       {Nathanael Bosch and Philipp Hennig and Filip Tronarp},
+  booktitle =    {Thirty-seventh Conference on Neural Information Processing
+                  Systems},
+  year =         2023,
+  url =          {https://openreview.net/forum?id=2dx5MNs2Ip}
+}
+
+@inproceedings{beck2024diffusion,
+  title =        {Diffusion Tempering Improves Parameter Estimation with
+                  Probabilistic Integrators for Ordinary Differential Equations},
+  author =       {Jonas Beck and Nathanael Bosch and Michael Deistler and Kyra
+                  L. Kadhim and Jakob H. Macke and Philipp Hennig and Philipp
+                  Berens},
+  booktitle =    {Forty-first International Conference on Machine Learning},
+  year =         2024,
+  url =          {https://openreview.net/forum?id=43HZG9zwaj}
+}
+
+
+@misc{wenger2021probnum,
+  title =        {ProbNum: Probabilistic Numerics in Python},
+  author =       {Jonathan Wenger and Nicholas Krämer and Marvin Pförtner and
+                  Jonathan Schmidt and Nathanael Bosch and Nina Effenberger and
+                  Johannes Zenn and Alexandra Gessner and Toni Karvonen and
+                  François-Xavier Briol and Maren Mahsereci and Philipp Hennig},
+  year =         2021,
+  eprint =       {2112.02100},
+  archivePrefix ={arXiv},
+  primaryClass = {cs.MS}
+}
\ No newline at end of file
diff --git a/paper/paper.md b/paper/paper.md
new file mode 100644
index 000000000..b02850549
--- /dev/null
+++ b/paper/paper.md
@@ -0,0 +1,125 @@
+---
+title: 'ProbNumDiffEq.jl: Probabilistic Numerical Solvers for Ordinary Differential Equations in Julia'
+tags:
+  - Julia
+  - probabilistic numerics
+  - differential equations
+  - Bayesian filtering and smoothing
+  - simulation
+authors:
+  - name: Nathanael Bosch
+    orcid: 0000-0003-0139-4622
+    affiliation: 1
+    corresponding: true
+affiliations:
+ - name: Tübingen AI Center, University of Tübingen, Germany
+   index: 1
+date: 17 July 2024
+bibliography: paper.bib
+---
+
+# Summary
+
+The forces on stars, galaxies, and dark matter under external gravitational
+fields lead to the dynamical evolution of structures in the universe. The orbits
+of these bodies are therefore key to understanding the formation, history, and
+future state of galaxies. The field of "galactic dynamics," which aims to model
+the gravitating components of galaxies to study their structure and evolution,
+is now well-established, commonly taught, and frequently used in astronomy.
+Aside from toy problems and demonstrations, the majority of problems require
+efficient numerical tools, many of which require the same base code (e.g., for
+performing numerical orbit integration).
+
+# Statement of need
+
+`Gala` is an Astropy-affiliated Python package for galactic dynamics. Python
+enables wrapping low-level languages (e.g., C) for speed without losing
+flexibility or ease-of-use in the user-interface. The API for `Gala` was
+designed to provide a class-based and user-friendly interface to fast (C or
+Cython-optimized) implementations of common operations such as gravitational
+potential and force evaluation, orbit integration, dynamical transformations,
+and chaos indicators for nonlinear dynamics. `Gala` also relies heavily on and
+interfaces well with the implementations of physical units and astronomical
+coordinate systems in the `Astropy` package [@astropy] (`astropy.units` and
+`astropy.coordinates`).
+
+`Gala` was designed to be used by both astronomical researchers and by
+students in courses on gravitational dynamics or astronomy. It has already been
+used in a number of scientific publications [@Pearson:2017] and has also been
+used in graduate courses on Galactic dynamics to, e.g., provide interactive
+visualizations of textbook material [@Binney:2008]. The combination of speed,
+design, and support for Astropy functionality in `Gala` will enable exciting
+scientific explorations of forthcoming data releases from the *Gaia* mission
+[@gaia] by students and experts alike.
+
+# Mathematics
+
+Single dollars ($) are required for inline mathematics e.g. $f(x) = e^{\pi/x}$
+
+Double dollars make self-standing equations:
+
+$$\Theta(x) = \left\{\begin{array}{l}
+0\textrm{ if } x < 0\cr
+1\textrm{ else}
+\end{array}\right.$$
+
+You can also use plain \LaTeX for equations
+\begin{equation}\label{eq:fourier}
+\hat f(\omega) = \int_{-\infty}^{\infty} f(x) e^{i\omega x} dx
+\end{equation}
+and refer to \autoref{eq:fourier} from text.
+
+# Citations
+
+Citations to entries in paper.bib should be in
+[rMarkdown](http://rmarkdown.rstudio.com/authoring_bibliographies_and_citations.html)
+format.
+
+If you want to cite a software repository URL (e.g. something on GitHub without a preferred
+citation) then you can do it with the example BibTeX entry below for @fidgit.
+
+For a quick reference, the following citation commands can be used:
+- `@author:2001`  ->  "Author et al. (2001)"
+- `[@author:2001]` -> "(Author et al., 2001)"
+- `[@author1:2001; @author2:2001]` -> "(Author1 et al., 2001; Author2 et al., 2002)"
+
+- `@bosch20_calib_adapt_probab_ode_solver`: developed + implemented
+- `@bosch21_pick_and_mix_infor_operat`: developed + implemented
+- `@krämer2021probabilistic`: implemented
+- `@tronarp2022fenrir`: developed + implemented
+- `@bosch2023probabilistic`: developed + implemented
+- `@beck2024diffusion`: developed
+- `@wenger2021probnum`: related software package
+- DALTON: did they compare against my code?
+- BlackBox PN? Probabilistic richardson extrapolation paper?
+- probdiffeq: related code
+
+# Figures
+
+Figures can be included like this:
+![Caption for example figure.\label{fig:example}](figure.png)
+and referenced from text using \autoref{fig:example}.
+
+Figure sizes can be customized by adding an optional second parameter:
+![Caption for example figure.](figure.png){ width=20% }
+
+# Acknowledgements
+
+The authors gratefully acknowledge co-funding by the European Union (ERC, ANUBIS, 101123955. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them). 
+The authors thank the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for supporting Nathanael Bosch.
+
+The authors are grateful to Nicholas Krämer for many valuable discussion and to Jonathan Schmidt for feedback on the manuscript.
+
+We acknowledge contributions from 
+Pietro Monticone (@pitmonticone),
+Vedant Puri (@vpuri3),
+Tim Holy (@timholy),
+Daniel González Arribas (@DaniGlez),
+David Widmann (@devmotion),
+Christopher Rackauckas (@ChrisRackauckas),
+Qingyu Qu (@ErikQQY),
+Cornelius Roemer (@corneliusroemer),
+and Jose Storopoli (@storopoli).
+
+
+# References

From 862804ce307ba80716bfe784ef55f3a4f295c498 Mon Sep 17 00:00:00 2001
From: Nathanael Bosch <nathanael.bosch@uni-tuebingen.de>
Date: Wed, 17 Jul 2024 18:25:55 +0200
Subject: [PATCH 2/9] Paper WIP, now with github action

---
 .github/workflows/draft-pdf.yml |  24 ++++
 paper/paper.bib                 | 238 +++++++++++++++++++++++++++++++-
 paper/paper.md                  | 158 +++++++++------------
 3 files changed, 320 insertions(+), 100 deletions(-)
 create mode 100644 .github/workflows/draft-pdf.yml

diff --git a/.github/workflows/draft-pdf.yml b/.github/workflows/draft-pdf.yml
new file mode 100644
index 000000000..b7053c0f4
--- /dev/null
+++ b/.github/workflows/draft-pdf.yml
@@ -0,0 +1,24 @@
+name: Draft PDF
+on: [push]
+
+jobs:
+  paper:
+    runs-on: ubuntu-latest
+    name: Paper Draft
+    steps:
+      - name: Checkout
+        uses: actions/checkout@v4
+      - name: Build draft PDF
+        uses: openjournals/openjournals-draft-action@master
+        with:
+          journal: joss
+          # This should be the path to the paper within your repo.
+          paper-path: paper/paper.md
+      - name: Upload
+        uses: actions/upload-artifact@v4
+        with:
+          name: paper
+          # This is the output path where Pandoc will write the compiled
+          # PDF. Note, this should be the same directory as the input
+          # paper.md
+          path: paper/paper.pdf
diff --git a/paper/paper.bib b/paper/paper.bib
index ca29f0ae6..b28310107 100644
--- a/paper/paper.bib
+++ b/paper/paper.bib
@@ -1,6 +1,4 @@
-
-
-@InProceedings{krämer2021probabilistic,
+@InProceedings{krämer2021highdim,
   title =        {Probabilistic {ODE} Solutions in Millions of Dimensions},
   author =       {Kr{\"a}mer, Nicholas and Bosch, Nathanael and Schmidt,
                   Jonathan and Hennig, Philipp},
@@ -57,7 +55,6 @@ @inproceedings{beck2024diffusion
   url =          {https://openreview.net/forum?id=43HZG9zwaj}
 }
 
-
 @misc{wenger2021probnum,
   title =        {ProbNum: Probabilistic Numerics in Python},
   author =       {Jonathan Wenger and Nicholas Krämer and Marvin Pförtner and
@@ -68,4 +65,235 @@ @misc{wenger2021probnum
   eprint =       {2112.02100},
   archivePrefix ={arXiv},
   primaryClass = {cs.MS}
-}
\ No newline at end of file
+}
+
+@InProceedings{dalton2024,
+  title = 	 {Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equations},
+  author =       {Wu, Mohan and Lysy, Martin},
+  booktitle = 	 {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics},
+  pages = 	 {1018--1026},
+  year = 	 {2024},
+  editor = 	 {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen},
+  volume = 	 {238},
+  series = 	 {Proceedings of Machine Learning Research},
+  month = 	 {02--04 May},
+  publisher =    {PMLR},
+  pdf = 	 {https://proceedings.mlr.press/v238/wu24b/wu24b.pdf},
+  url = 	 {https://proceedings.mlr.press/v238/wu24b.html},
+}
+
+@article{rackauckas2017differentialequations,
+  title =        {{DifferentialEquations.jl} – A Performant and Feature-Rich
+                  Ecosystem for Solving Differential Equations in Julia},
+  author =       {Rackauckas, Christopher and Nie, Qing},
+  journal =      {Journal of Open Research Software},
+  volume =       5,
+  number =       1,
+  year =         2017,
+  publisher =    {Ubiquity Press}
+}
+
+@misc{probdiffeq,
+  author = {Kr{\"a}mer, Nicholas},
+  title = {probdiffeq: Probabilistic solvers for differential equations in JAX},
+  year = {2023},
+  publisher = {GitHub},
+  journal = {GitHub repository},
+  url = {https://github.com/pnkraemer/probdiffeq}
+}
+
+@misc{jax2018github,
+  author =       {James Bradbury and Roy Frostig and Peter Hawkins and Matthew
+                  James Johnson and Chris Leary and Dougal Maclaurin and George
+                  Necula and Adam Paszke and Jake Vander{P}las and Skye
+                  Wanderman-{M}ilne and Qiao Zhang},
+  title =        {{JAX}: composable transformations of {P}ython+{N}um{P}y
+                  programs},
+  version =      {0.2.5},
+  year =         2018,
+}
+
+@article{schober19,
+  author =       "Schober, Michael and S{\"a}rkk{\"a}, Simo and Hennig, Philipp",
+  title =        "A probabilistic model for the numerical solution of initial
+                  value problems",
+  journal =      "Statistics and Computing",
+  year =         2019,
+  month =        "Jan",
+  day =          01,
+  volume =       29,
+  number =       1,
+  pages =        "99--122",
+  issn =         "1573-1375",
+  doi =          "10.1007/s11222-017-9798-7",
+  url =          "https://doi.org/10.1007/s11222-017-9798-7"
+}
+
+@article{tronarp19,
+  author =       {Filip Tronarp and Hans Kersting and Simo S{\"{a}}rkk{\"{a}}
+                  and Philipp Hennig},
+  title =        {Probabilistic solutions to ordinary differential equations as
+                  nonlinear {B}ayesian filtering: a new perspective},
+  year =         2019,
+  volume =       29,
+  number =       6,
+  pages =        {1297-1315},
+  doi =          {10.1007/s11222-019-09900-1},
+  url =          {https://doi.org/10.1007/s11222-019-09900-1},
+  journal =      {Statistics and Computing},
+  timestamp =    {Wed, 25 Mar 2020 09:31:58 +0100},
+  biburl =       {https://dblp.org/rec/journals/sac/TronarpKSH19.bib},
+  bibsource =    {dblp computer science bibliography, https://dblp.org}
+}
+
+@article{kersting20,
+  author =       {Kersting, Hans and Sullivan, T. J. and Hennig, Philipp},
+  title =        {Convergence rates of Gaussian ODE filters},
+  journal =      {Statistics and Computing},
+  year =         2020,
+  month =        {Nov},
+  day =          01,
+  volume =       30,
+  number =       6,
+  pages =        {1791-1816},
+  issn =         {1573-1375},
+  doi =          {10.1007/s11222-020-09972-4},
+  url =          {https://doi.org/10.1007/s11222-020-09972-4}
+}
+
+@article{tronarp21,
+  author =       {Tronarp, Filip and S{\"a}rkk{\"a}, Simo and Hennig, Philipp},
+  title =        {Bayesian ODE solvers: the maximum a posteriori estimate},
+  journal =      {Statistics and Computing},
+  year =         2021,
+  month =        {Mar},
+  day =          03,
+  volume =       31,
+  number =       3,
+  pages =        23,
+  issn =         {1573-1375},
+  doi =          {10.1007/s11222-021-09993-7},
+  url =          {https://doi.org/10.1007/s11222-021-09993-7}
+}
+
+@misc{bosch2023parallelintime,
+  title =        {Parallel-in-Time Probabilistic Numerical {ODE} Solvers},
+  author =       {Nathanael Bosch and Adrien Corenflos and Fatemeh Yaghoobi and
+                  Filip Tronarp and Philipp Hennig and Simo Särkkä},
+  year =         2023,
+  eprint =       {2310.01145},
+  archivePrefix ={arXiv},
+  primaryClass = {math.NA}
+}
+
+@InProceedings{kersting20invprob,
+  title =        {Differentiable Likelihoods for Fast Inversion of
+                  ’{L}ikelihood-Free’ Dynamical Systems},
+  author =       {Kersting, Hans and Kr{\"a}mer, Nicholas and Schiegg, Martin
+                  and Daniel, Christian and Tiemann, Michael and Hennig,
+                  Philipp},
+  booktitle =    {Proceedings of the 37th International Conference on Machine
+                  Learning},
+  pages =        {5198--5208},
+  year =         2020,
+  editor =       {Hal Daumé III and Aarti Singh},
+  volume =       119,
+  series =       {Proceedings of Machine Learning Research},
+  month =        {13--18 Jul},
+  publisher =    {PMLR},
+  pdf =          {http://proceedings.mlr.press/v119/kersting20a/kersting20a.pdf},
+  url =          {http://proceedings.mlr.press/v119/kersting20a.html},
+}
+
+@inproceedings{schmidt21,
+  author =       {Schmidt, Jonathan and Kr\"{a}mer, Nicholas and Hennig,
+                  Philipp},
+  booktitle =    {Advances in Neural Information Processing Systems},
+  editor =       {M. Ranzato and A. Beygelzimer and Y. Dauphin and P.S. Liang
+                  and J. Wortman Vaughan},
+  pages =        {12374--12385},
+  publisher =    {Curran Associates, Inc.},
+  title =        {A Probabilistic State Space Model for Joint Inference from
+                  Differential Equations and Data},
+  url =
+                  {https://proceedings.neurips.cc/paper/2021/file/6734fa703f6633ab896eecbdfad8953a-Paper.pdf},
+  volume =       34,
+  year =         2021
+}
+
+@inproceedings{kraemer202bvp,
+  author =       {Kr\"{a}mer, Nicholas and Hennig, Philipp},
+  booktitle =    {Advances in Neural Information Processing Systems},
+  editor =       {M. Ranzato and A. Beygelzimer and Y. Dauphin and P.S. Liang
+                  and J. Wortman Vaughan},
+  pages =        {11160--11171},
+  publisher =    {Curran Associates, Inc.},
+  title =        {Linear-Time Probabilistic Solution of Boundary Value Problems},
+  url =
+                  {https://papers.nips.cc/paper/2021/hash/5ca3e9b122f61f8f06494c97b1afccf3-Abstract.html},
+  volume =       34,
+  year =         2021
+}
+
+@InProceedings{kraemer22mol,
+  title =        { Probabilistic Numerical Method of Lines for Time-Dependent
+                  Partial Differential Equations },
+  author =       {Kr\"amer, Nicholas and Schmidt, Jonathan and Hennig, Philipp},
+  booktitle =    {Proceedings of The 25th International Conference on Artificial
+                  Intelligence and Statistics},
+  pages =        {625--639},
+  year =         2022,
+  editor =       {Camps-Valls, Gustau and Ruiz, Francisco J. R. and Valera,
+                  Isabel},
+  volume =       151,
+  series =       {Proceedings of Machine Learning Research},
+  month =        {28--30 Mar},
+  publisher =    {PMLR},
+  pdf =          {https://proceedings.mlr.press/v151/kramer22a/kramer22a.pdf},
+  url =          {https://proceedings.mlr.press/v151/kramer22a.html},
+}
+
+@InProceedings{bosch22pick,
+  title =        { Pick-and-Mix Information Operators for Probabilistic {ODE}
+                  Solvers },
+  author =       {Bosch, Nathanael and Tronarp, Filip and Hennig, Philipp},
+  booktitle =    {Proceedings of The 25th International Conference on Artificial
+                  Intelligence and Statistics},
+  pages =        {10015--10027},
+  year =         2022,
+  editor =       {Camps-Valls, Gustau and Ruiz, Francisco J. R. and Valera,
+                  Isabel},
+  volume =       151,
+  series =       {Proceedings of Machine Learning Research},
+  month =        {28--30 Mar},
+  publisher =    {PMLR},
+  pdf =          {https://proceedings.mlr.press/v151/bosch22a/bosch22a.pdf},
+  url =          {https://proceedings.mlr.press/v151/bosch22a.html},
+}
+
+@InProceedings{bosch21capos,
+  title =        { Calibrated Adaptive Probabilistic {ODE} Solvers },
+  author =       {Bosch, Nathanael and Hennig, Philipp and Tronarp, Filip},
+  booktitle =    {Proceedings of The 24th International Conference on Artificial
+                  Intelligence and Statistics},
+  pages =        {3466--3474},
+  year =         2021,
+  editor =       {Banerjee, Arindam and Fukumizu, Kenji},
+  volume =       130,
+  series =       {Proceedings of Machine Learning Research},
+  month =        {13--15 Apr},
+  publisher =    {PMLR},
+  pdf =          {http://proceedings.mlr.press/v130/bosch21a/bosch21a.pdf},
+  url =          {http://proceedings.mlr.press/v130/bosch21a.html},
+}
+
+@article{kraemer24stableimp,
+  author =       {Nicholas Kr{{\"a}}mer and Philipp Hennig},
+  title =        {Stable Implementation of Probabilistic ODE Solvers},
+  journal =      {Journal of Machine Learning Research},
+  year =         2024,
+  volume =       25,
+  number =       111,
+  pages =        {1--29},
+  url =          {http://jmlr.org/papers/v25/20-1423.html}
+}
diff --git a/paper/paper.md b/paper/paper.md
index b02850549..fcad58c4b 100644
--- a/paper/paper.md
+++ b/paper/paper.md
@@ -10,7 +10,6 @@ authors:
   - name: Nathanael Bosch
     orcid: 0000-0003-0139-4622
     affiliation: 1
-    corresponding: true
 affiliations:
  - name: Tübingen AI Center, University of Tübingen, Germany
    index: 1
@@ -20,106 +19,75 @@ bibliography: paper.bib
 
 # Summary
 
-The forces on stars, galaxies, and dark matter under external gravitational
-fields lead to the dynamical evolution of structures in the universe. The orbits
-of these bodies are therefore key to understanding the formation, history, and
-future state of galaxies. The field of "galactic dynamics," which aims to model
-the gravitating components of galaxies to study their structure and evolution,
-is now well-established, commonly taught, and frequently used in astronomy.
-Aside from toy problems and demonstrations, the majority of problems require
-efficient numerical tools, many of which require the same base code (e.g., for
-performing numerical orbit integration).
+Probabilistic numerical solvers have emerged as an efficient framework for simulation, uncertainty quantification, and inference in dynamical systems.
+In comparison to traditional numerical methods which approximate the true trajectory of a system only by a single point estimate, probabilistic numerical solvers compute a _distribution_ over the true, unknown solution of the given differential equation
+and thereby provide information about the numerical error incurred during the computation.
+ProbNumDiffEq.jl implements such probabilistic numerical solvers for ordinary differential equations (ODEs) and differential algebraic equations (DAEs) in Julia, within the DifferentialEquations.jl ecosystem [@rackauckas2017differentialequations].
+
+More concretely, ProbNumDiffEq.jl provides a range of probabilistic numerical solvers for ordinary differential equations based on Bayesian filtering and smoothing,
+which have emerged as a particularly efficient and flexible class of methods for solving ODEs [@schober19; @kersting20; @tronarp19].
+These so-called "ODE filters" have known polynomial convergence rates 
+[@kersting20; @tronarp21]
+and numerical stability properties (such as A-stability or L-stability)
+[@tronarp19; @bosch2023probabilistic],
+their computational complexity is comparable to traditional numerical methods
+[@krämer2021highdim], 
+they are applicable to a range of numerical differential equation problems 
+[@kraemer202bvp; @kraemer22mol; @bosch22pick],
+and they can be formulated parallel-in-time 
+[@bosch2023parallelintime].
+ODE filters also provide a natural framework for ODE parameter inference 
+[@kersting20invprob; @tronarp2022fenrir; @schmidt21; @beck2024diffusion].
+ProbNumDiffEq.jl implements many of the methods referenced above and packages them in a software library with the aim to be easy-to-use, feature-rich, well-documented, and efficiently implemented.
 
 # Statement of need
 
-`Gala` is an Astropy-affiliated Python package for galactic dynamics. Python
-enables wrapping low-level languages (e.g., C) for speed without losing
-flexibility or ease-of-use in the user-interface. The API for `Gala` was
-designed to provide a class-based and user-friendly interface to fast (C or
-Cython-optimized) implementations of common operations such as gravitational
-potential and force evaluation, orbit integration, dynamical transformations,
-and chaos indicators for nonlinear dynamics. `Gala` also relies heavily on and
-interfaces well with the implementations of physical units and astronomical
-coordinate systems in the `Astropy` package [@astropy] (`astropy.units` and
-`astropy.coordinates`).
-
-`Gala` was designed to be used by both astronomical researchers and by
-students in courses on gravitational dynamics or astronomy. It has already been
-used in a number of scientific publications [@Pearson:2017] and has also been
-used in graduate courses on Galactic dynamics to, e.g., provide interactive
-visualizations of textbook material [@Binney:2008]. The combination of speed,
-design, and support for Astropy functionality in `Gala` will enable exciting
-scientific explorations of forthcoming data releases from the *Gaia* mission
-[@gaia] by students and experts alike.
-
-# Mathematics
-
-Single dollars ($) are required for inline mathematics e.g. $f(x) = e^{\pi/x}$
-
-Double dollars make self-standing equations:
-
-$$\Theta(x) = \left\{\begin{array}{l}
-0\textrm{ if } x < 0\cr
-1\textrm{ else}
-\end{array}\right.$$
-
-You can also use plain \LaTeX for equations
-\begin{equation}\label{eq:fourier}
-\hat f(\omega) = \int_{-\infty}^{\infty} f(x) e^{i\omega x} dx
-\end{equation}
-and refer to \autoref{eq:fourier} from text.
-
-# Citations
-
-Citations to entries in paper.bib should be in
-[rMarkdown](http://rmarkdown.rstudio.com/authoring_bibliographies_and_citations.html)
-format.
-
-If you want to cite a software repository URL (e.g. something on GitHub without a preferred
-citation) then you can do it with the example BibTeX entry below for @fidgit.
-
-For a quick reference, the following citation commands can be used:
-- `@author:2001`  ->  "Author et al. (2001)"
-- `[@author:2001]` -> "(Author et al., 2001)"
-- `[@author1:2001; @author2:2001]` -> "(Author1 et al., 2001; Author2 et al., 2002)"
-
-- `@bosch20_calib_adapt_probab_ode_solver`: developed + implemented
-- `@bosch21_pick_and_mix_infor_operat`: developed + implemented
-- `@krämer2021probabilistic`: implemented
-- `@tronarp2022fenrir`: developed + implemented
-- `@bosch2023probabilistic`: developed + implemented
-- `@beck2024diffusion`: developed
-- `@wenger2021probnum`: related software package
-- DALTON: did they compare against my code?
-- BlackBox PN? Probabilistic richardson extrapolation paper?
-- probdiffeq: related code
-
-# Figures
-
-Figures can be included like this:
-![Caption for example figure.\label{fig:example}](figure.png)
-and referenced from text using \autoref{fig:example}.
-
-Figure sizes can be customized by adding an optional second parameter:
-![Caption for example figure.](figure.png){ width=20% }
+Filtering-based probabilistic numerical ODE solvers have been an active field of research for the past decade, but their application in practical simulation and inference problems has been limited.
+ProbNumDiffEq.jl aims to help close this gap.
+ProbNumDiffEq.jl implements probabilistic numerical methods as performant, documented, and easy-to-use ODE solvers inside the well-established DifferentialEquations.jl ecosystem [@rackauckas2017differentialequations].
+Thereby, the package benefits from the extensive testing, documentation, performance optimization, and functionality that DifferentialEquations.jl provides.
+Users can easily find help and examples regarding many features that are not particular to ProbNumDiffEq.jl in the DifferentialEquations.jl documentation, 
+and we provide probabilistic numerics-specific examples and tutorials in the ProbNumDiffEq.jl documentation.
+We believe that this deep integration within DifferentialEquations.jl can be a key feature to attract users to probabilistic numerics by enabling users to use probabilistic ODE solvers as drop-in replacements for traditional ODE solvers.
+
+On the other hand, ProbNumDiffEq.jl also aims to accelerate the development of new probabilistic numerical ODE solvers by providing a solid foundation to both build on and compare against.
+Several publications have been developed with ProbNumDiffEq.jl, including contributions on
+step-size adaptation and calibration of these solvers [@bosch21capos],
+energy-preserving solvers and DAE solvers [@bosch22pick],
+probabilistic exponential integrators [@bosch2023probabilistic],
+and novel parameter inference algorithms [@tronarp2022fenrir; @beck2024diffusion].
+We also hope that by providing documented and performant implementations of published algorithms we facilitate researchers to use these methods as baselines when developing new numerical solvers.
+
+ProbNumDiffEq.jl is also the only software package in Julia, at the time of writing, that provides a comprehensive set of probabilistic numerical ODE solvers.
+Outside of Julia, two other software packages provide a similar functionality.
+ProbNum [@wenger2021probnum]
+is a Python package that provides probabilistic numerical solvers for ODEs, but also for other numerical problems such as linear systems and quadrature.
+It is thus broader in scope and provides functionality not covered by ProbNumDiffEq.jl, but it also lacks some of the specialized ODE solvers available in ProbNumDiffEq.jl.
+And with its reliance on Python and numpy, it is also generally less performant.
+ProbDiffEq [@probdiffeq]
+is a probabilistic numerical ODE solver package built on JAX.
+At the time of writing, it provides a very similar set of ODE solvers as ProbNumDiffEq.jl with the addition of certain filtering and smoothing methods and the lack of certain specialized ODE solvers, but as both ProbDiffEq and ProbNumDiffEq.jl are under active development this might change in the future.
+By building on JAX and leveraging its just-in-time compilation capabilities, ProbDiffEq provides ODE solvers with similar performance as those implemented in ProbNumDiffEq.jl.
+In summary, ProbNumDiffEq.jl provides one of the most feature-rich and performant probabilistic numerical ODE solver packages currently available and is the only one built on the Julia programming language.
 
 # Acknowledgements
 
-The authors gratefully acknowledge co-funding by the European Union (ERC, ANUBIS, 101123955. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them). 
-The authors thank the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for supporting Nathanael Bosch.
-
-The authors are grateful to Nicholas Krämer for many valuable discussion and to Jonathan Schmidt for feedback on the manuscript.
-
-We acknowledge contributions from 
-Pietro Monticone (@pitmonticone),
-Vedant Puri (@vpuri3),
-Tim Holy (@timholy),
-Daniel González Arribas (@DaniGlez),
-David Widmann (@devmotion),
-Christopher Rackauckas (@ChrisRackauckas),
-Qingyu Qu (@ErikQQY),
-Cornelius Roemer (@corneliusroemer),
-and Jose Storopoli (@storopoli).
+The author gratefully acknowledges co-funding by the European Union (ERC, ANUBIS, 101123955. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them). 
+The author thanks the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for their support.
+
+I am grateful to Philipp Hennig for his support throughout the development of this package.
+I thank Nicholas Krämer, Filip Tronarp, and Jonathan Schmidt for many valuable discussions about probabilistic numerical ODE solvers and their correct, efficient, and elegant implementation.
+I also thank Christopher Rackauckas for support and feedback on how to integrate ProbNumDiffEq.jl with the DifferentialEquations.jl ecosystem and for including ProbNumDiffEq.jl into the testing pipeline of OrdinaryDiffEq.jl.
+I acknowledge contributions from 
+Pietro Monticone (\@pitmonticone),
+Vedant Puri (\@vpuri3),
+Tim Holy (\@timholy),
+Daniel González Arribas (\@DaniGlez),
+David Widmann (\@devmotion),
+Christopher Rackauckas (\@ChrisRackauckas),
+Qingyu Qu (\@ErikQQY),
+Cornelius Roemer (\@corneliusroemer),
+and Jose Storopoli (\@storopoli).
 
 
 # References

From 0bc0c7a7563c7e8c903f6e0dca9440135e8814f2 Mon Sep 17 00:00:00 2001
From: Nathanael Bosch <nathanael.bosch@uni-tuebingen.de>
Date: Fri, 19 Jul 2024 20:37:49 +0200
Subject: [PATCH 3/9] Add Julia citation

---
 paper/paper.bib | 14 ++++++++++++++
 paper/paper.md  |  2 +-
 2 files changed, 15 insertions(+), 1 deletion(-)

diff --git a/paper/paper.bib b/paper/paper.bib
index b28310107..16d318c84 100644
--- a/paper/paper.bib
+++ b/paper/paper.bib
@@ -297,3 +297,17 @@ @article{kraemer24stableimp
   pages =        {1--29},
   url =          {http://jmlr.org/papers/v25/20-1423.html}
 }
+
+@article{julia,
+  doi       = {10.1137/141000671},
+  url       = {https://doi.org/10.1137%2F141000671},
+  year      = 2017,
+  month     = {jan},
+  publisher = {Society for Industrial {\&} Applied Mathematics ({SIAM})},
+  volume    = {59},
+  number    = {1},
+  pages     = {65--98},
+  author    = {Jeff Bezanson and Alan Edelman and Stefan Karpinski and Viral B. Shah},
+  title     = {Julia: A Fresh Approach to Numerical Computing},
+  journal   = {{SIAM} Review}
+}
diff --git a/paper/paper.md b/paper/paper.md
index fcad58c4b..e9d2ff76f 100644
--- a/paper/paper.md
+++ b/paper/paper.md
@@ -22,7 +22,7 @@ bibliography: paper.bib
 Probabilistic numerical solvers have emerged as an efficient framework for simulation, uncertainty quantification, and inference in dynamical systems.
 In comparison to traditional numerical methods which approximate the true trajectory of a system only by a single point estimate, probabilistic numerical solvers compute a _distribution_ over the true, unknown solution of the given differential equation
 and thereby provide information about the numerical error incurred during the computation.
-ProbNumDiffEq.jl implements such probabilistic numerical solvers for ordinary differential equations (ODEs) and differential algebraic equations (DAEs) in Julia, within the DifferentialEquations.jl ecosystem [@rackauckas2017differentialequations].
+ProbNumDiffEq.jl implements such probabilistic numerical solvers for ordinary differential equations (ODEs) and differential algebraic equations (DAEs) in the Julia programming language [@julia], within the DifferentialEquations.jl ecosystem [@rackauckas2017differentialequations].
 
 More concretely, ProbNumDiffEq.jl provides a range of probabilistic numerical solvers for ordinary differential equations based on Bayesian filtering and smoothing,
 which have emerged as a particularly efficient and flexible class of methods for solving ODEs [@schober19; @kersting20; @tronarp19].

From cdaf90488e37162ad2d5ed9669e4c4f40d894907 Mon Sep 17 00:00:00 2001
From: Nathanael Bosch <nathanael.bosch@uni-tuebingen.de>
Date: Fri, 19 Jul 2024 20:39:50 +0200
Subject: [PATCH 4/9] Add DifferentialEquations.jl DOI

---
 paper/paper.bib | 4 +++-
 1 file changed, 3 insertions(+), 1 deletion(-)

diff --git a/paper/paper.bib b/paper/paper.bib
index 16d318c84..b48035718 100644
--- a/paper/paper.bib
+++ b/paper/paper.bib
@@ -90,7 +90,9 @@ @article{rackauckas2017differentialequations
   volume =       5,
   number =       1,
   year =         2017,
-  publisher =    {Ubiquity Press}
+  publisher =    {Ubiquity Press},
+  doi =          "10.5334/jors.151",
+  url =          "https://doi.org/10.5334/jors.151",
 }
 
 @misc{probdiffeq,

From 31e0a541a0f8f4d2afa25ab7cc068cc9f8c5c8f9 Mon Sep 17 00:00:00 2001
From: Nathanael Bosch <nathanael.bosch@uni-tuebingen.de>
Date: Fri, 19 Jul 2024 21:06:49 +0200
Subject: [PATCH 5/9] Add more DOIs and reformat bibtex

---
 paper/paper.bib | 78 +++++++++++++++++++++++++++++--------------------
 1 file changed, 46 insertions(+), 32 deletions(-)

diff --git a/paper/paper.bib b/paper/paper.bib
index b48035718..e89814c2f 100644
--- a/paper/paper.bib
+++ b/paper/paper.bib
@@ -41,7 +41,9 @@ @inproceedings{bosch2023probabilistic
   booktitle =    {Thirty-seventh Conference on Neural Information Processing
                   Systems},
   year =         2023,
-  url =          {https://openreview.net/forum?id=2dx5MNs2Ip}
+  url =          {https://openreview.net/forum?id=2dx5MNs2Ip},
+  doi =          "10.5555/3666122.3667881",
+  url =          "https://dl.acm.org/doi/10.5555/3666122.3667881",
 }
 
 @inproceedings{beck2024diffusion,
@@ -64,22 +66,26 @@ @misc{wenger2021probnum
   year =         2021,
   eprint =       {2112.02100},
   archivePrefix ={arXiv},
-  primaryClass = {cs.MS}
+  primaryClass = {cs.MS},
+  doi =          "10.48550/arXiv.2112.02100",
+  url =          "https://doi.org/10.48550/arXiv.2112.02100",
 }
 
 @InProceedings{dalton2024,
-  title = 	 {Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equations},
+  title =        {Data-Adaptive Probabilistic Likelihood Approximation for
+                  Ordinary Differential Equations},
   author =       {Wu, Mohan and Lysy, Martin},
-  booktitle = 	 {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics},
-  pages = 	 {1018--1026},
-  year = 	 {2024},
-  editor = 	 {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen},
-  volume = 	 {238},
-  series = 	 {Proceedings of Machine Learning Research},
-  month = 	 {02--04 May},
+  booktitle =    {Proceedings of The 27th International Conference on Artificial
+                  Intelligence and Statistics},
+  pages =        {1018--1026},
+  year =         2024,
+  editor =       {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen},
+  volume =       238,
+  series =       {Proceedings of Machine Learning Research},
+  month =        {02--04 May},
   publisher =    {PMLR},
-  pdf = 	 {https://proceedings.mlr.press/v238/wu24b/wu24b.pdf},
-  url = 	 {https://proceedings.mlr.press/v238/wu24b.html},
+  pdf =          {https://proceedings.mlr.press/v238/wu24b/wu24b.pdf},
+  url =          {https://proceedings.mlr.press/v238/wu24b.html},
 }
 
 @article{rackauckas2017differentialequations,
@@ -96,12 +102,13 @@ @article{rackauckas2017differentialequations
 }
 
 @misc{probdiffeq,
-  author = {Kr{\"a}mer, Nicholas},
-  title = {probdiffeq: Probabilistic solvers for differential equations in JAX},
-  year = {2023},
-  publisher = {GitHub},
-  journal = {GitHub repository},
-  url = {https://github.com/pnkraemer/probdiffeq}
+  author =       {Kr{\"a}mer, Nicholas},
+  title =        {probdiffeq: Probabilistic solvers for differential equations
+                  in JAX},
+  year =         2023,
+  publisher =    {GitHub},
+  journal =      {GitHub repository},
+  url =          {https://github.com/pnkraemer/probdiffeq}
 }
 
 @misc{jax2018github,
@@ -185,7 +192,9 @@ @misc{bosch2023parallelintime
   year =         2023,
   eprint =       {2310.01145},
   archivePrefix ={arXiv},
-  primaryClass = {math.NA}
+  primaryClass = {math.NA},
+  doi =          {10.48550/arXiv.2310.01145},
+  url =          {https://doi.org/10.48550/arXiv.2310.01145}
 }
 
 @InProceedings{kersting20invprob,
@@ -220,7 +229,9 @@ @inproceedings{schmidt21
   url =
                   {https://proceedings.neurips.cc/paper/2021/file/6734fa703f6633ab896eecbdfad8953a-Paper.pdf},
   volume =       34,
-  year =         2021
+  year =         2021,
+  doi =          {10.5555/3540261.3541208},
+  url =          {https://dl.acm.org/doi/10.5555/3540261.3541208},
 }
 
 @inproceedings{kraemer202bvp,
@@ -234,7 +245,9 @@ @inproceedings{kraemer202bvp
   url =
                   {https://papers.nips.cc/paper/2021/hash/5ca3e9b122f61f8f06494c97b1afccf3-Abstract.html},
   volume =       34,
-  year =         2021
+  year =         2021,
+  doi =          {10.5555/3540261.3541114},
+  url =          {https://dl.acm.org/doi/10.5555/3540261.3541114},
 }
 
 @InProceedings{kraemer22mol,
@@ -301,15 +314,16 @@ @article{kraemer24stableimp
 }
 
 @article{julia,
-  doi       = {10.1137/141000671},
-  url       = {https://doi.org/10.1137%2F141000671},
-  year      = 2017,
-  month     = {jan},
-  publisher = {Society for Industrial {\&} Applied Mathematics ({SIAM})},
-  volume    = {59},
-  number    = {1},
-  pages     = {65--98},
-  author    = {Jeff Bezanson and Alan Edelman and Stefan Karpinski and Viral B. Shah},
-  title     = {Julia: A Fresh Approach to Numerical Computing},
-  journal   = {{SIAM} Review}
+  doi =          {10.1137/141000671},
+  url =          {https://doi.org/10.1137%2F141000671},
+  year =         2017,
+  month =        {jan},
+  publisher =    {Society for Industrial {\&} Applied Mathematics ({SIAM})},
+  volume =       59,
+  number =       1,
+  pages =        {65--98},
+  author =       {Jeff Bezanson and Alan Edelman and Stefan Karpinski and Viral
+                  B. Shah},
+  title =        {Julia: A Fresh Approach to Numerical Computing},
+  journal =      {{SIAM} Review}
 }

From 6a6f95b9be9488ad5e09a4628776f89aedfad79a Mon Sep 17 00:00:00 2001
From: Nathanael Bosch <nathanael.bosch@uni-tuebingen.de>
Date: Sat, 20 Jul 2024 12:19:21 +0200
Subject: [PATCH 6/9] Remove invalid DOIs

---
 paper/paper.bib | 6 ------
 1 file changed, 6 deletions(-)

diff --git a/paper/paper.bib b/paper/paper.bib
index e89814c2f..ef6df1f04 100644
--- a/paper/paper.bib
+++ b/paper/paper.bib
@@ -42,8 +42,6 @@ @inproceedings{bosch2023probabilistic
                   Systems},
   year =         2023,
   url =          {https://openreview.net/forum?id=2dx5MNs2Ip},
-  doi =          "10.5555/3666122.3667881",
-  url =          "https://dl.acm.org/doi/10.5555/3666122.3667881",
 }
 
 @inproceedings{beck2024diffusion,
@@ -230,8 +228,6 @@ @inproceedings{schmidt21
                   {https://proceedings.neurips.cc/paper/2021/file/6734fa703f6633ab896eecbdfad8953a-Paper.pdf},
   volume =       34,
   year =         2021,
-  doi =          {10.5555/3540261.3541208},
-  url =          {https://dl.acm.org/doi/10.5555/3540261.3541208},
 }
 
 @inproceedings{kraemer202bvp,
@@ -246,8 +242,6 @@ @inproceedings{kraemer202bvp
                   {https://papers.nips.cc/paper/2021/hash/5ca3e9b122f61f8f06494c97b1afccf3-Abstract.html},
   volume =       34,
   year =         2021,
-  doi =          {10.5555/3540261.3541114},
-  url =          {https://dl.acm.org/doi/10.5555/3540261.3541114},
 }
 
 @InProceedings{kraemer22mol,

From d6155580e0bcf815f29e10fa63132cafdc82cbd2 Mon Sep 17 00:00:00 2001
From: Nathanael Bosch <nathanael.bosch@uni-tuebingen.de>
Date: Tue, 24 Sep 2024 16:30:16 +0200
Subject: [PATCH 7/9] More details on the performance comparison to ProbNum and
 ProbDiffEq

---
 paper/paper.bib | 47 +++++++++++++++++++++++++++++++++++++++++++++++
 paper/paper.md  | 11 ++++++-----
 2 files changed, 53 insertions(+), 5 deletions(-)

diff --git a/paper/paper.bib b/paper/paper.bib
index ef6df1f04..823ed1505 100644
--- a/paper/paper.bib
+++ b/paper/paper.bib
@@ -321,3 +321,50 @@ @article{julia
   title =        {Julia: A Fresh Approach to Numerical Computing},
   journal =      {{SIAM} Review}
 }
+
+@Article{numpy,
+ title         = {Array programming with {NumPy}},
+ author        = {Charles R. Harris and K. Jarrod Millman and St{\'{e}}fan J.
+                 van der Walt and Ralf Gommers and Pauli Virtanen and David
+                 Cournapeau and Eric Wieser and Julian Taylor and Sebastian
+                 Berg and Nathaniel J. Smith and Robert Kern and Matti Picus
+                 and Stephan Hoyer and Marten H. van Kerkwijk and Matthew
+                 Brett and Allan Haldane and Jaime Fern{\'{a}}ndez del
+                 R{\'{i}}o and Mark Wiebe and Pearu Peterson and Pierre
+                 G{\'{e}}rard-Marchant and Kevin Sheppard and Tyler Reddy and
+                 Warren Weckesser and Hameer Abbasi and Christoph Gohlke and
+                 Travis E. Oliphant},
+ year          = {2020},
+ month         = sep,
+ journal       = {Nature},
+ volume        = {585},
+ number        = {7825},
+ pages         = {357--362},
+ doi           = {10.1038/s41586-020-2649-2},
+ publisher     = {Springer Science and Business Media {LLC}},
+ url           = {https://doi.org/10.1038/s41586-020-2649-2}
+}
+
+@ARTICLE{scipy,
+  author  = {Virtanen, Pauli and Gommers, Ralf and Oliphant, Travis E. and
+            Haberland, Matt and Reddy, Tyler and Cournapeau, David and
+            Burovski, Evgeni and Peterson, Pearu and Weckesser, Warren and
+            Bright, Jonathan and {van der Walt}, St{\'e}fan J. and
+            Brett, Matthew and Wilson, Joshua and Millman, K. Jarrod and
+            Mayorov, Nikolay and Nelson, Andrew R. J. and Jones, Eric and
+            Kern, Robert and Larson, Eric and Carey, C J and
+            Polat, {\.I}lhan and Feng, Yu and Moore, Eric W. and
+            {VanderPlas}, Jake and Laxalde, Denis and Perktold, Josef and
+            Cimrman, Robert and Henriksen, Ian and Quintero, E. A. and
+            Harris, Charles R. and Archibald, Anne M. and
+            Ribeiro, Ant{\^o}nio H. and Pedregosa, Fabian and
+            {van Mulbregt}, Paul and {SciPy 1.0 Contributors}},
+  title   = {{{SciPy} 1.0: Fundamental Algorithms for Scientific
+            Computing in Python}},
+  journal = {Nature Methods},
+  year    = {2020},
+  volume  = {17},
+  pages   = {261--272},
+  adsurl  = {https://rdcu.be/b08Wh},
+  doi     = {10.1038/s41592-019-0686-2},
+}
\ No newline at end of file
diff --git a/paper/paper.md b/paper/paper.md
index e9d2ff76f..feb4f9fc5 100644
--- a/paper/paper.md
+++ b/paper/paper.md
@@ -61,13 +61,14 @@ We also hope that by providing documented and performant implementations of publ
 ProbNumDiffEq.jl is also the only software package in Julia, at the time of writing, that provides a comprehensive set of probabilistic numerical ODE solvers.
 Outside of Julia, two other software packages provide a similar functionality.
 ProbNum [@wenger2021probnum]
-is a Python package that provides probabilistic numerical solvers for ODEs, but also for other numerical problems such as linear systems and quadrature.
-It is thus broader in scope and provides functionality not covered by ProbNumDiffEq.jl, but it also lacks some of the specialized ODE solvers available in ProbNumDiffEq.jl.
-And with its reliance on Python and numpy, it is also generally less performant.
+is a Python package that implements probabilistic numerical for various numerical problems, including linear systems, quadrature, and ODEs.
+ProbNum particularly aims to facilitate rapid experimentation and accelerate the development of new methods [@wenger2021probnum].
+It is therefore broader in scope and provides functionality not covered by ProbNumDiffEq.jl.
+But, it also lacks some of the specialized ODE solvers available in ProbNumDiffEq.jl, and with its reliance on Python and NumPy [@numpy] and the lack of just-in-time compilation, it is also generally less performant.
 ProbDiffEq [@probdiffeq]
 is a probabilistic numerical ODE solver package built on JAX.
-At the time of writing, it provides a very similar set of ODE solvers as ProbNumDiffEq.jl with the addition of certain filtering and smoothing methods and the lack of certain specialized ODE solvers, but as both ProbDiffEq and ProbNumDiffEq.jl are under active development this might change in the future.
-By building on JAX and leveraging its just-in-time compilation capabilities, ProbDiffEq provides ODE solvers with similar performance as those implemented in ProbNumDiffEq.jl.
+At the time of writing, it provides a very similar set of ODE solvers as ProbNumDiffEq.jl with the addition of certain filtering and smoothing methods and the lack of certain specialized ODE solvers---but as both ProbDiffEq and ProbNumDiffEq.jl are under active development, this might change in the future.
+By building on JAX and leveraging its just-in-time compilation capabilities, ProbDiffEq provides ODE solvers with similar performance as those implemented in ProbNumDiffEq.jl (shown through benchmarks in both packages comparing to SciPy [@scipy]).
 In summary, ProbNumDiffEq.jl provides one of the most feature-rich and performant probabilistic numerical ODE solver packages currently available and is the only one built on the Julia programming language.
 
 # Acknowledgements

From 02805e2082f28688df1b69710e050ef8b1aaacda Mon Sep 17 00:00:00 2001
From: Mehmet Hakan Satman <mhsatman@gmail.com>
Date: Fri, 27 Sep 2024 21:16:08 +0300
Subject: [PATCH 8/9] Minor fix on paper and bibtex (#330)

* update bibtex

* update paper.md

* Update paper/paper.md

---------

Co-authored-by: Nathanael Bosch <nathanael.bosch@uni-tuebingen.de>
---
 paper/paper.bib | 8 ++++----
 paper/paper.md  | 4 ++--
 2 files changed, 6 insertions(+), 6 deletions(-)

diff --git a/paper/paper.bib b/paper/paper.bib
index 823ed1505..b7f7ff5b8 100644
--- a/paper/paper.bib
+++ b/paper/paper.bib
@@ -56,7 +56,7 @@ @inproceedings{beck2024diffusion
 }
 
 @misc{wenger2021probnum,
-  title =        {ProbNum: Probabilistic Numerics in Python},
+  title =        {ProbNum: Probabilistic Numerics in {P}ython},
   author =       {Jonathan Wenger and Nicholas Krämer and Marvin Pförtner and
                   Jonathan Schmidt and Nathanael Bosch and Nina Effenberger and
                   Johannes Zenn and Alexandra Gessner and Toni Karvonen and
@@ -87,8 +87,8 @@ @InProceedings{dalton2024
 }
 
 @article{rackauckas2017differentialequations,
-  title =        {{DifferentialEquations.jl} – A Performant and Feature-Rich
-                  Ecosystem for Solving Differential Equations in Julia},
+  title =        {{DifferentialEquations.jl} – {A} Performant and Feature-Rich
+                  Ecosystem for Solving Differential Equations in {J}ulia},
   author =       {Rackauckas, Christopher and Nie, Qing},
   journal =      {Journal of Open Research Software},
   volume =       5,
@@ -155,7 +155,7 @@ @article{tronarp19
 
 @article{kersting20,
   author =       {Kersting, Hans and Sullivan, T. J. and Hennig, Philipp},
-  title =        {Convergence rates of Gaussian ODE filters},
+  title =        {Convergence rates of {G}aussian ODE filters},
   journal =      {Statistics and Computing},
   year =         2020,
   month =        {Nov},
diff --git a/paper/paper.md b/paper/paper.md
index feb4f9fc5..c77da7ed0 100644
--- a/paper/paper.md
+++ b/paper/paper.md
@@ -43,7 +43,7 @@ ProbNumDiffEq.jl implements many of the methods referenced above and packages th
 # Statement of need
 
 Filtering-based probabilistic numerical ODE solvers have been an active field of research for the past decade, but their application in practical simulation and inference problems has been limited.
-ProbNumDiffEq.jl aims to help close this gap.
+ProbNumDiffEq.jl aims to bridge this gap.
 ProbNumDiffEq.jl implements probabilistic numerical methods as performant, documented, and easy-to-use ODE solvers inside the well-established DifferentialEquations.jl ecosystem [@rackauckas2017differentialequations].
 Thereby, the package benefits from the extensive testing, documentation, performance optimization, and functionality that DifferentialEquations.jl provides.
 Users can easily find help and examples regarding many features that are not particular to ProbNumDiffEq.jl in the DifferentialEquations.jl documentation, 
@@ -61,7 +61,7 @@ We also hope that by providing documented and performant implementations of publ
 ProbNumDiffEq.jl is also the only software package in Julia, at the time of writing, that provides a comprehensive set of probabilistic numerical ODE solvers.
 Outside of Julia, two other software packages provide a similar functionality.
 ProbNum [@wenger2021probnum]
-is a Python package that implements probabilistic numerical for various numerical problems, including linear systems, quadrature, and ODEs.
+is a Python package that implements probabilistic numerical methods for various numerical problems, including linear systems, quadrature, and ODEs.
 ProbNum particularly aims to facilitate rapid experimentation and accelerate the development of new methods [@wenger2021probnum].
 It is therefore broader in scope and provides functionality not covered by ProbNumDiffEq.jl.
 But, it also lacks some of the specialized ODE solvers available in ProbNumDiffEq.jl, and with its reliance on Python and NumPy [@numpy] and the lack of just-in-time compilation, it is also generally less performant.

From 0bc7944a7ffe042974bdfb8423318788d615a5d4 Mon Sep 17 00:00:00 2001
From: Nathanael Bosch <nathanael.bosch@uni-tuebingen.de>
Date: Sat, 28 Sep 2024 14:25:50 +0200
Subject: [PATCH 9/9] Minor improvements to wording and commas in the paper

---
 paper/paper.md | 14 +++++++-------
 1 file changed, 7 insertions(+), 7 deletions(-)

diff --git a/paper/paper.md b/paper/paper.md
index c77da7ed0..c8b801a7c 100644
--- a/paper/paper.md
+++ b/paper/paper.md
@@ -20,9 +20,9 @@ bibliography: paper.bib
 # Summary
 
 Probabilistic numerical solvers have emerged as an efficient framework for simulation, uncertainty quantification, and inference in dynamical systems.
-In comparison to traditional numerical methods which approximate the true trajectory of a system only by a single point estimate, probabilistic numerical solvers compute a _distribution_ over the true, unknown solution of the given differential equation
+In comparison to traditional numerical methods, which approximate the true trajectory of a system only by a single point estimate, probabilistic numerical solvers compute a _distribution_ over the true unknown solution of the given differential equation
 and thereby provide information about the numerical error incurred during the computation.
-ProbNumDiffEq.jl implements such probabilistic numerical solvers for ordinary differential equations (ODEs) and differential algebraic equations (DAEs) in the Julia programming language [@julia], within the DifferentialEquations.jl ecosystem [@rackauckas2017differentialequations].
+ProbNumDiffEq.jl implements such probabilistic numerical solvers for ordinary differential equations (ODEs) and differential-algebraic equations (DAEs) in the Julia programming language [@julia] within the DifferentialEquations.jl ecosystem [@rackauckas2017differentialequations].
 
 More concretely, ProbNumDiffEq.jl provides a range of probabilistic numerical solvers for ordinary differential equations based on Bayesian filtering and smoothing,
 which have emerged as a particularly efficient and flexible class of methods for solving ODEs [@schober19; @kersting20; @tronarp19].
@@ -47,8 +47,8 @@ ProbNumDiffEq.jl aims to bridge this gap.
 ProbNumDiffEq.jl implements probabilistic numerical methods as performant, documented, and easy-to-use ODE solvers inside the well-established DifferentialEquations.jl ecosystem [@rackauckas2017differentialequations].
 Thereby, the package benefits from the extensive testing, documentation, performance optimization, and functionality that DifferentialEquations.jl provides.
 Users can easily find help and examples regarding many features that are not particular to ProbNumDiffEq.jl in the DifferentialEquations.jl documentation, 
-and we provide probabilistic numerics-specific examples and tutorials in the ProbNumDiffEq.jl documentation.
-We believe that this deep integration within DifferentialEquations.jl can be a key feature to attract users to probabilistic numerics by enabling users to use probabilistic ODE solvers as drop-in replacements for traditional ODE solvers.
+and we provide additional examples and tutorials specific to the probabilistic solvers in the ProbNumDiffEq.jl documentation.
+We believe that this deep integration within DifferentialEquations.jl is a key feature to attract users to probabilistic numerics by enabling the use of probabilistic ODE solvers as drop-in replacements for traditional ODE solvers.
 
 On the other hand, ProbNumDiffEq.jl also aims to accelerate the development of new probabilistic numerical ODE solvers by providing a solid foundation to both build on and compare against.
 Several publications have been developed with ProbNumDiffEq.jl, including contributions on
@@ -56,15 +56,15 @@ step-size adaptation and calibration of these solvers [@bosch21capos],
 energy-preserving solvers and DAE solvers [@bosch22pick],
 probabilistic exponential integrators [@bosch2023probabilistic],
 and novel parameter inference algorithms [@tronarp2022fenrir; @beck2024diffusion].
-We also hope that by providing documented and performant implementations of published algorithms we facilitate researchers to use these methods as baselines when developing new numerical solvers.
+We also hope that by providing documented and performant implementations of published algorithms, we facilitate researchers to use these methods as baselines when developing new numerical solvers.
 
 ProbNumDiffEq.jl is also the only software package in Julia, at the time of writing, that provides a comprehensive set of probabilistic numerical ODE solvers.
 Outside of Julia, two other software packages provide a similar functionality.
 ProbNum [@wenger2021probnum]
 is a Python package that implements probabilistic numerical methods for various numerical problems, including linear systems, quadrature, and ODEs.
 ProbNum particularly aims to facilitate rapid experimentation and accelerate the development of new methods [@wenger2021probnum].
-It is therefore broader in scope and provides functionality not covered by ProbNumDiffEq.jl.
-But, it also lacks some of the specialized ODE solvers available in ProbNumDiffEq.jl, and with its reliance on Python and NumPy [@numpy] and the lack of just-in-time compilation, it is also generally less performant.
+It is therefore broader in scope and provides functionality not covered by ProbNumDiffEq.jl, but it also lacks some of the specialized ODE solvers available in ProbNumDiffEq.jl.
+In addition, with its reliance on Python and NumPy [@numpy] and the lack of just-in-time compilation, it is also generally less performant.
 ProbDiffEq [@probdiffeq]
 is a probabilistic numerical ODE solver package built on JAX.
 At the time of writing, it provides a very similar set of ODE solvers as ProbNumDiffEq.jl with the addition of certain filtering and smoothing methods and the lack of certain specialized ODE solvers---but as both ProbDiffEq and ProbNumDiffEq.jl are under active development, this might change in the future.