Motion Prediction Problem
I would try to train a Recurrent Neural Network for this problem, the LSTM architecture is particularly interesting for time series prediction.

Sources:


karpathy.github.io/2015/05/21/rnn-effectiveness/
colah.github.io/posts/2015-08-Understanding-LSTMs/


You would of course need a training dataset consisting of a sequence of events. Let's say $x$ and $y$ coordinates for every timestamp $t$, (possibly $z$ as well to predict a jumping puppy).

Both the network input layer and the output layer would consist of 2 units.
Then you would train your model to predict $(x_{t+1}, y_{t+1})$ given $(x_{t}, y_{t})$. Once the network is trained, you can predict the position of the dog at any point in the future.

Let $M$ be our trained model and let's say you want to predict the position of the dog at time $k$ and you know the current position of the dog at time $t$.

$M(x_{t}, y_{t}) = (x_{t+1}, y_{t+1})$

$t = t+1$

$M(x_{t}, y_{t}) = (x_{t+1}, y_{t+1})$

$...$ increment $t$ and keep predicting until $t+1 = k-1$

$M(x_{t+1}, y_{t+1}) = (x_{k}, y_{k})$

Put in pictures this corresponds to:
 (picture from Udacity lecture about Deep Learning)