  The autoencoder model uses an encoder to map data samples to a lower
dimensional latent space and then a decoder to map the latent space
representations back to the data space. Implicitly, it relies on the encoder to
approximate the inverse of the decoder network, so that samples can be mapped
to and back from the latent space faithfully. This approximation may lead to
sub-optimal latent space representations. In this work, we investigate a
decoder-only method that uses gradient flow to encode data samples in the
latent space. The gradient flow is defined based on a given decoder and aims to
find the optimal latent space representation for any given sample through
optimisation, eliminating the need of an approximate inversion through an
encoder. Implementing gradient flow through ordinary differential equations
(ODE), we leverage the adjoint method to train a given decoder. We further show
empirically that the costly integrals in the adjoint method may not be entirely
necessary. Additionally, we propose a $2^{nd}$ order ODE variant to the method,
which approximates Nesterov's accelerated gradient descent, with faster
convergence per iteration. Commonly used ODE solvers can be quite sensitive to
the integration step-size depending on the stiffness of the ODE. To overcome
the sensitivity for gradient flow encoding, we use an adaptive solver that
prioritises minimising loss at each integration step. We assess the proposed
method in comparison to the autoencoding model. In our experiments, GFE showed
a much higher data-efficiency than the autoencoding model, which can be crucial
for data scarce applications.
