  We introduce an Invertible Symbolic Regression (ISR) method. It is a machine
learning technique that generates analytical relationships between inputs and
outputs of a given dataset via invertible maps (or architectures). The proposed
ISR method naturally combines the principles of Invertible Neural Networks
(INNs) and Equation Learner (EQL), a neural network-based symbolic architecture
for function learning. In particular, we transform the affine coupling blocks
of INNs into a symbolic framework, resulting in an end-to-end differentiable
symbolic invertible architecture that allows for efficient gradient-based
learning. The proposed ISR framework also relies on sparsity promoting
regularization, allowing the discovery of concise and interpretable invertible
expressions. We show that ISR can serve as a (symbolic) normalizing flow for
density estimation tasks. Furthermore, we highlight its practical applicability
in solving inverse problems, including a benchmark inverse kinematics problem,
and notably, a geoacoustic inversion problem in oceanography aimed at inferring
posterior distributions of underlying seabed parameters from acoustic signals.
