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.

介绍了一种可逆符号回归（ISR）方法，结合了可逆神经网络（INNs）和方程学习者（EQL）的原理，通过可逆映射生成输入和输出之间的分析关系，提供了一种高效的基于梯度学习的可逆符号结构，可用于密度估计和解决反问题。

ISR：可逆符号回归

ISR: Invertible Symbolic Regression

In classical machine learning, regression is treated as a black box process
of identifying a suitable function from a hypothesis set without attempting to
gain insight into the mechanism connecting inputs and outputs. In the natural
sciences, however, finding an interpretable function for a phenomenon is the
prime goal as it allows to understand and generalize results. This paper
proposes a novel type of function learning network, called equation learner
(EQL), that can learn analytical expressions and is able to extrapolate to
unseen domains. It is implemented as an end-to-end differentiable feed-forward
network and allows for efficient gradient based training. Due to sparsity
regularization concise interpretable expressions can be obtained. Often the
true underlying source expression is identified.

该研究提出了一种名为等式学习器（EQL）的新型函数学习网络，可以学习解析表达式，并能够外推到未知的领域。通过稀疏正则化可以得到简洁明了的可解释表达式，可用于高效的基于梯度的训练。