ISR: Invertible Symbolic Regression

TL;DR

This paper introduces ISR, a novel invertible symbolic regression method that combines invertible neural networks and symbolic architectures to learn interpretable, invertible relationships from data, applicable to density estimation and inverse problems.

Contribution

The paper presents a new invertible symbolic regression framework that integrates INNs and EQL, enabling efficient, interpretable, and invertible function learning with regularization for sparsity.

Findings

Effective in density estimation as a normalizing flow

Successfully applied to inverse kinematics

Demonstrated in geoacoustic inversion tasks

Abstract

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.