Neural Chronos ODE: Unveiling Temporal Patterns and Forecasting Future
  and Past Trends in Time Series Data
C. Coelho M. Fernanda P. Costa L. L. Ferrás
This work introduces Neural Chronos Ordinary Differential Equations (Neural
CODE), a deep neural network architecture that fits a continuous-time ODE
dynamics for predicting the chronology of a system both forward and backward in
time. To train the model, we solve the ODE as an initial value problem and a
final value problem, similar to Neural ODEs. We also explore two approaches to
combining Neural CODE with Recurrent Neural Networks by replacing Neural ODE
with Neural CODE (CODE-RNN), and incorporating a bidirectional RNN for full
information flow in both time directions (CODE-BiRNN), and variants with other
update cells namely GRU and LSTM: CODE-GRU, CODE-BiGRU, CODE-LSTM, CODE-BiLSTM.
  Experimental results demonstrate that Neural CODE outperforms Neural ODE in
learning the dynamics of a spiral forward and backward in time, even with
sparser data. We also compare the performance of CODE-RNN/-GRU/-LSTM and
CODE-BiRNN/-BiGRU/-BiLSTM against ODE-RNN/-GRU/-LSTM on three real-life time
series data tasks: imputation of missing data for lower and higher dimensional
data, and forward and backward extrapolation with shorter and longer time
horizons. Our findings show that the proposed architectures converge faster,
with CODE-BiRNN/-BiGRU/-BiLSTM consistently outperforming the other
architectures on all tasks.