Add a description for Hybrid AutoODE

website/docs/physics-guided-deep-sequence-models/hybrid-autoode.md

@@ -2,3 +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:
+
+$$
+\begin{aligned}
+&\frac{dx}{dt} = f_\theta(t, x, u, F) \\
+&\frac{du}{dt} = g_\theta(t, x, u, F) \\
+&x(t_0) = x_0 \\
+&u(t_0) = u_0
+\end{aligned}
+$$
+
+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.