From d8333bc6b32c574d4f7ce0c9beefc1f1758f4327 Mon Sep 17 00:00:00 2001
From: amartyamukherjee <38194687+amartyamukherjee@users.noreply.github.com>
Date: Tue, 12 Oct 2021 19:18:04 -0400
Subject: [PATCH 1/3] Update hybrid-autoode.md

---
 .../physics-guided-deep-sequence-models/hybrid-autoode.md    | 5 +++++
 1 file changed, 5 insertions(+)

diff --git a/website/docs/physics-guided-deep-sequence-models/hybrid-autoode.md b/website/docs/physics-guided-deep-sequence-models/hybrid-autoode.md
index 58ce9a1a..d0d0cc72 100644
--- a/website/docs/physics-guided-deep-sequence-models/hybrid-autoode.md
+++ b/website/docs/physics-guided-deep-sequence-models/hybrid-autoode.md
@@ -2,3 +2,8 @@
 title: Hybrid AutoODE
 slug: /hybrid-autoode
 ---
+Assume the time series $x_t \in \mathbb{R}^d$ is governed by unknown differential equations and by other unknown factors that could affect its trajectory. The Hybrid AutoODE uses physics-guided models in conjunction with neural networks to improve the prediction of $x_t$. It is modelled by the following equations:
+
+$$ \begin{aligned} &\frac{dx}{dt} = f_\theta(t, x, u, F) \ &\frac{du}{dt} = g_\theta(t, x, u, F) \ &x(t_0) = x_0 \ &u(t_0) = u_0 \end{aligned} $$
+
+where $u \in \mathbb{R}^p$ are the unobserved variables and $F$ is a neural network. The Hybrid AutoODE uses auto-differentiation to estimate the parameters $\theta$ of the equations and the neural network.

From 91b508ed63f8fb45c18d2e1fa16c8add757a4247 Mon Sep 17 00:00:00 2001
From: "Amartya (Marty) Mukherjee"
 <38194687+amartyamukherjee@users.noreply.github.com>
Date: Sun, 14 Nov 2021 02:41:57 -0500
Subject: [PATCH 2/3] Update hybrid-autoode.md


From 619bfb007d5d213449b9c1bf8e68dcdfb1498a66 Mon Sep 17 00:00:00 2001
From: Kevin Lane <lane.kevin.a@gmail.com>
Date: Thu, 16 Dec 2021 19:07:30 -0800
Subject: [PATCH 3/3] Format documentation

---
 .../hybrid-autoode.md                                  | 10 +++++++++-
 1 file changed, 9 insertions(+), 1 deletion(-)

diff --git a/website/docs/physics-guided-deep-sequence-models/hybrid-autoode.md b/website/docs/physics-guided-deep-sequence-models/hybrid-autoode.md
index d0d0cc72..465574a0 100644
--- a/website/docs/physics-guided-deep-sequence-models/hybrid-autoode.md
+++ b/website/docs/physics-guided-deep-sequence-models/hybrid-autoode.md
@@ -2,8 +2,16 @@
 title: Hybrid AutoODE
 slug: /hybrid-autoode
 ---
+
 Assume the time series $x_t \in \mathbb{R}^d$ is governed by unknown differential equations and by other unknown factors that could affect its trajectory. The Hybrid AutoODE uses physics-guided models in conjunction with neural networks to improve the prediction of $x_t$. It is modelled by the following equations:
 
-$$ \begin{aligned} &\frac{dx}{dt} = f_\theta(t, x, u, F) \ &\frac{du}{dt} = g_\theta(t, x, u, F) \ &x(t_0) = x_0 \ &u(t_0) = u_0 \end{aligned} $$
+$$
+\begin{aligned}
+&\frac{dx}{dt} = f_\theta(t, x, u, F) \\
+&\frac{du}{dt} = g_\theta(t, x, u, F) \\
+&x(t_0) = x_0 \\
+&u(t_0) = u_0
+\end{aligned}
+$$
 
 where $u \in \mathbb{R}^p$ are the unobserved variables and $F$ is a neural network. The Hybrid AutoODE uses auto-differentiation to estimate the parameters $\theta$ of the equations and the neural network.