payload: {"action":"created","comment":{"url":"https://api.github.com/repos/depthfirstlearning/depthfirstlearning.com/pulls/comments/297456610","pull_request_review_id":254347586,"id":297456610,"node_id":"MDI0OlB1bGxSZXF1ZXN0UmV2aWV3Q29tbWVudDI5NzQ1NjYxMA==","diff_hunk":"@@ -0,0 +1,289 @@\n+---\n+layout: post\n+title:  \"Neural ODEs\"\n+date:   2019-06-06 10:00:00 -0400\n+categories: neural-nets\n+author: luca\n+blurb: \"Neural ODEs are neural network models which generalize standard layer to layer propagation to continuous depth models. The forward propagation in many neural networks models can be seen as one step of discretation of an ODE; starting from this observation, one can construct and efficiently train models by numerically solving ODEs. On top of furnishing a new whole family of architectures, neural ODEs can be applied to gain memory efficiency in supervised learning tasks, construct a new class of invertible density models and model continuous time-series\"\n+feedback: true\n+---\n+\n+This guide would not have been possible without the help and feedback from many people. \n+\n+Special thanks to Prof. Joan Bruna and his class at NYU, [Mathematics of Deep Learning](https://github.com/joanbruna/MathsDL-spring19), and to Cinjon Resnick, who introduced me to DFL and helped me putting up this guide.\n+\n+Thank you to Avital Oliver, Matt Johnson, Dougal MacClaurin, David Duvenaud, and Ricky Chen for useful contributions to this guide.\n+\n+Thank you to Tinghao Li, Chandra Prakash Konkimalla, Manikanta Srikar Yellapragada, Shan-Conrad Wolf, Deshana Desai, Yi Tang, Zhonghui Hu for helping me prepare the notes.\n+\n+Finally, thank you to all my fellow students who attended the recitations and provided valuable feedback.\n+\n+\n+# Why\n+\n+Neural ODEs are neural network models which generalize standard layer to layer propagation to continuous depth models. The forward propagation in many neural networks models can be seen as one step of discretation of an ODE; starting from this observation, one can construct and efficiently train models by numerically solving ODEs. On top of furnishing a new whole family of architectures, neural ODEs can be applied to gain memory efficiency in supervised learning tasks, construct a new class of invertible density models and model continuous time-series.\n+\n+In this curriculum, we will go through all the background topics necessary to understand these models. At the end, you should be able to implement neural ODEs and apply them to different tasks.\n+\n+<br />\n+\n+# 1 Numerical solution of ODEs - Part 1\n+  **Motivation**: ODEs are used to mathematically model a number of natural processes and phenomena. The study of their numerical \n+    simulations is one of the main topics in numerical analysis and of fundamental importance in applied sciences. A first step to understand Neural ODEs is their definition and basic numerical techniques to solve them.\n+\n+  **Topics**:\n+\n+  1. Initial values problems\n+  2. One-step methods\n+  3. Consistency and convergence\n+\n+  **Required Reading**:\n+\n+  1. Sections 12.1-4 from [An Introduction to Numerical Analysis](https://www.cambridge.org/core/books/an-introduction-to-numerical-analysis/FD8BCAD7FE68002E2179DFF68B8B7237#) (S\\\"uli & Mayers)\n+  2. Sections 11.1-3 from [Numerical Mathematics](https://www.springer.com/us/book/9783540346586?token=holiday18&utm_campaign=3_fjp8312_us_dsa_springer_holiday18&gclid=Cj0KCQiAvebhBRD5ARIsAIQUmnlViB7VsUn-2tABSAhIvYaJgSEqmJXD7F4A7EgyDQtY9v_GeUsNif8aArGAEALw_wcB) (Quarteroni et al.)\n+\n+  \n+  **Optional Reading**:\n+\n+  1. Runge-Kutta methods: Section 12.5 from (S\\\"uli & Mayers)\n+  2. [Prof. Trefethen's class ODEs and Nonlinear Dynamics 4.2](http://podcasts.ox.ac.uk/odes-and-nonlinear-dynamics-42)\n+\n+  **Questions**:\n+\n+  1. Exercise 1 in Section 11.12 of (Quarteroni et al.)\n+     <details><summary>Solutions</summary>\n+     <p>\n+     ...\n+     </p>\n+     </details>\n+\n+  2. Exercises 12.3,12.4, 12.7 in Section 12 of (S\\\"uli & Mayers)\n+     <details><summary>Solution</summary>\n+     <p>\n+     ...\n+     </p>\n+     </details>\n+\n+  3. Consider the following method for solving \\(y' = f(y)\\):\n+       \n+     $$y_{n+1} = y_n + h*(theta*f(y_n) + (1-theta)*f(y_{n+1}))$$\n+    \n+     Assuming sufficient smoothness of \\(y\\) and \\(f\\), for what value of \\(0 \\leq\\theta\\leq 1\\) is the truncation error the smallest? What does this mean about the accuracy of the method?\n+     <details><summary>Solution</summary>\n+     <p>\n+     ...\n+     </p>\n+     </details>\n+       \n+\n+  4. [Colab Notebook](https://colab.research.google.com/drive/1bNg-RzZoelB3w8AUQ6mefRQuN3AdrIqX)\n+     <details><summary>Solution</summary>\n+     <p>\n+     ...\n+     </p>\n+     </details>\n+\n+**Notes**: Here is a [link](/assets/nodes_notes/week1.pdf) to our notes for the lesson. \n+\n+<br />\n+\n+# 2 Numerical solution of ODEs - Part 2\n+  **Motivation**: In the previous class we introduced some simple schemes to numerically solve ODEs. In this class we go through some more involved schemes and their convergence analysis. ","path":"_posts/2019-06-06-NeuralODEs.markdown","position":null,"original_position":91,"commit_id":"8a1a2b639c9499d9b75b8c5d1ac40d70bb7d8986","original_commit_id":"9e6943e530bb117b0c7ebe05b521c22e08bee307","user":{"login":"luca-venturi","id":25848954,"node_id":"MDQ6VXNlcjI1ODQ4OTU0","avatar_url":"https://avatars0.githubusercontent.com/u/25848954?v=4","gravatar_id":"","url":"https://api.github.com/users/luca-venturi","html_url":"https://github.com/luca-venturi","followers_url":"https://api.github.com/users/luca-venturi/followers","following_url":"https://api.github.com/users/luca-venturi/following{/other_user}","gists_url":"https://api.github.com/users/luca-venturi/gists{/gist_id}","starred_url":"https://api.github.com/users/luca-venturi/starred{/owner}{/repo}","subscriptions_url":"https://api.github.com/users/luca-venturi/subscriptions","organizations_url":"https://api.github.com/users/luca-venturi/orgs","repos_url":"https://api.github.com/users/luca-venturi/repos","events_url":"https://api.github.com/users/luca-venturi/events{/privacy}","received_events_url":"https://api.github.com/users/luca-venturi/received_events","type":"User","site_admin":false},"body":"added this","created_at":"2019-06-26T01:48:43Z","updated_at":"2019-06-26T01:48:43Z","html_url":"https://github.com/depthfirstlearning/depthfirstlearning.com/pull/29#discussion_r297456610","pull_request_url":"https://api.github.com/repos/depthfirstlearning/depthfirstlearning.com/pulls/29","author_association":"NONE","_links":{"self":{"href":"https://api.github.com/repos/depthfirstlearning/depthfirstlearning.com/pulls/comments/297456610"},"html":{"href":"https://github.com/depthfirstlearning/depthfirstlearning.com/pull/29#discussion_r297456610"},"pull_request":{"href":"https://api.github.com/repos/depthfirstlearning/depthfirstlearning.com/pulls/29"}},"in_reply_to_id":294489091},"pull_request":{"url":"https://api.github.com/repos/depthfirstlearning/depthfirstlearning.com/pulls/29","id":287707493,"node_id":"MDExOlB1bGxSZXF1ZXN0Mjg3NzA3NDkz","html_url":"https://github.com/depthfirstlearning/depthfirstlearning.com/pull/29","diff_url":"https://github.com/depthfirstlearning/depthfirstlearning.com/pull/29.diff","patch_url":"https://github.com/depthfirstlearning/depthfirstlearning.com/pull/29.patch","issue_url":"https://api.github.com/repos/depthfirstlearning/depthfirstlearning.com/issues/29","number":29,"state":"open","locked":false,"title":"Neural ODEs - added .markdown and notes","user":{"login":"luca-venturi","id":25848954,"node_id":"MDQ6VXNlcjI1ODQ4OTU0","avatar_url":"https://avatars0.githubusercontent.com/u/25848954?v=4","gravatar_id":"","url":"https://api.github.com/users/luca-venturi","html_url":"https://github.com/luca-venturi","followers_url":"https://api.github.com/users/luca-venturi/followers","following_url":"https://api.github.com/users/luca-venturi/following{/other_user}","gists_url":"https://api.github.com/users/luca-venturi/gists{/gist_id}","starred_url":"https://api.github.com/users/luca-venturi/starred{/owner}{/repo}","subscriptions_url":"https://api.github.com/users/luca-venturi/subscriptions","organizations_url":"https://api.github.com/users/luca-venturi/orgs","repos_url":"https://api.github.com/users/luca-venturi/repos","events_url":"https://api.github.com/users/luca-venturi/events{/privacy}","received_events_url":"https://api.github.com/users/luca-venturi/received_events","type":"User","site_admin":false},"body":"Hi all and sorry for the wait!\r\nThis is a first draft and I need to finish some stuff, but it would be great to have your feedback in the meantime. I need to:\r\n\r\n- add solutions to the exercises\r\n- do a pass on the notes and check for typos, etc.\r\n- add the concept maps\r\n\r\nPlease let me know any thoughts that you have on the guide, if you think that there is some more stuff missing, or anything else. I will keep update the PR as I go over the missing things / make changes. \r\n","created_at":"2019-06-12T23:13:58Z","updated_at":"2019-06-26T01:48:43Z","closed_at":null,"merged_at":null,"merge_commit_sha":"355b9b93a3b6a8e9d3f3cd18cb234da1833bc93c","assignee":null,"assignees":[],"requested_reviewers":[],"requested_teams":[],"labels":[],"milestone":null,"commits_url":"https://api.github.com/repos/depthfirstlearning/depthfirstlearning.com/pulls/29/commits","review_comments_url":"https://api.github.com/repos/depthfirstlearning/depthfirstlearning.com/pulls/29/comments","review_comment_url":"https://api.github.com/repos/depthfirstlearning/depthfirstlearning.com/pulls/comments{/number}","comments_url":"https://api.github.com/repos/depthfirstlearning/depthfirstlearning.com/issues/29/comments","statuses_url":"https://api.github.com/repos/depthfirstlearning/depthfirstlearning.com/statuses/8a1a2b639c9499d9b75b8c5d1ac40d70bb7d8986","head":{"label":"luca-venturi:master","ref":"master","sha":"8a1a2b639c9499d9b75b8c5d1ac40d70bb7d8986","user":{"login":"luca-venturi","id":25848954,"node_id":"MDQ6VXNlcjI1ODQ4OTU0","avatar_url":"https://avatars0.githubusercontent.com/u/25848954?v=4","gravatar_id":"","url":"https://api.github.com/users/luca-venturi","html_url":"https://github.com/luca-venturi","followers_url":"https://api.github.com/users/luca-venturi/followers","following_url":"https://api.github.com/users/luca-venturi/following{/other_user}","gists_url":"https://api.github.com/users/luca-venturi/gists{/gist_id}","starred_url":"https://api.github.com/users/luca-venturi/starred{/owner}{/repo}","subscriptions_url":"https://api.github.com/users/luca-venturi/subscriptions","organizations_url":"https://api.github.com/users/luca-venturi/orgs","repos_url":"https://api.github.com/users/luca-venturi/repos","events_url":"https://api.github.com/users/luca-venturi/events{/privacy}","received_events_url":"https://api.github.com/users/luca-venturi/received_events","type":"User","site_admin":false},"repo":{"id":190387470,"node_id":"MDEwOlJlcG9zaXRvcnkxOTAzODc0NzA=","name":"depthfirstlearning.com","full_name":"luca-venturi/depthfirstlearning.com","private":false,"owner":{"login":"luca-venturi","id":25848954,"node_id":"MDQ6VXNlcjI1ODQ4OTU0","avatar_url":"https://avatars0.githubusercontent.com/u/25848954?v=4","gravatar_id":"","url":"https://api.github.com/users/luca-venturi","html_url":"https://github.com/luca-venturi","followers_url":"https://api.github.com/users/luca-venturi/followers","following_url":"https://api.github.com/users/luca-venturi/following{/other_user}","gists_url":"https://api.github.com/users/luca-venturi/gists{/gist_id}","starred_url":"https://api.github.com/users/luca-venturi/starred{/owner}{/repo}","subscriptions_url":"https://api.github.com/users/luca-venturi/subscriptions","organizations_url":"https://api.github.com/users/luca-venturi/orgs","repos_url":"https://api.github.com/users/luca-venturi/repos","events_url":"https://api.github.com/users/luca-venturi/events{/privacy}","received_events_url":"https://api.github.com/users/luca-venturi/received_events","type":"User","site_admin":false},"html_url":"https://github.com/luca-venturi/depthfirstlearning.com","description":null,"fork":true,"url":"https://api.github.com/repos/luca-venturi/depthfirstlearning.com","forks_url":"https://api.github.com/repos/luca-venturi/depthfirstlearning.com/forks","keys_url":"https://api.github.com/repos/luca-venturi/depthfirstlearning.com/keys{/key_id}","collaborators_url":"https://api.github.com/repos/luca-venturi/depthfirstlearning.com/collaborators{/collaborator}","teams_url":"https://api.github.com/repos/luca-venturi/depthfirstlearning.com/teams","hooks_url":"https://api.github.com/repos/luca-venturi/depthfirstlearning.com/hooks","issue_events_url":"https://api.github.com/repos/luca-venturi/depthfirstlearning.com/issues/events{/number}","events_url":"https://api.github.com/repos/luca-venturi/depthfirstlearning.com/events","assignees_url":"https://api.github.com/repos/luca-venturi/depthfirstlearning.com/assignees{/user}","branches_url":"https://api.github.com/repos/luca-venturi/depthfirstlearning.com/branches{/branch}","tags_url":"https://api.github.com/repos/luca-venturi/depthfirstlearning.com/tags","blobs_url":"https://api.github.com/repos/luca-venturi/depthfirstlearning.com/git/blobs{/sha}","git_tags_url":"https://api.github.com/repos/luca-venturi/depthfirstlearning.com/git/tags{/sha}","git_refs_url":"https://api.github.com/repos/luca-venturi/depthfirstlearning.com/git/refs{/sha}","trees_url":"https://api.github.com/repos/luca-venturi/depthfirstlearning.com/git/trees{/sha}","statuses_url":"https://api.github.com/repos/luca-venturi/depthfirstlearning.com/statuses/{sha}","languages_url":"https://api.github.com/repos/luca-venturi/depthfirstlearning.com/languages","stargazers_url":"https://api.github.com/repos/luca-venturi/depthfirstlearning.com/stargazers","contributors_url":"https://api.github.com/repos/luca-venturi/depthfirstlearning.com/contributors","subscribers_url":"https://api.github.com/repos/luca-venturi/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