UniSymNet: A Unified Symbolic Network Guided by Transformer (2505.06091v1)

Abstract: Symbolic Regression (SR) is a powerful technique for automatically discovering mathematical expressions from input data. Mainstream SR algorithms search for the optimal symbolic tree in a vast function space, but the increasing complexity of the tree structure limits their performance. Inspired by neural networks, symbolic networks have emerged as a promising new paradigm. However, most existing symbolic networks still face certain challenges: binary nonlinear operators ${\times, \div}$ cannot be naturally extended to multivariate operators, and training with fixed architecture often leads to higher complexity and overfitting. In this work, we propose a Unified Symbolic Network that unifies nonlinear binary operators into nested unary operators and define the conditions under which UniSymNet can reduce complexity. Moreover, we pre-train a Transformer model with a novel label encoding method to guide structural selection, and adopt objective-specific optimization strategies to learn the parameters of the symbolic network. UniSymNet shows high fitting accuracy, excellent symbolic solution rate, and relatively low expression complexity, achieving competitive performance on low-dimensional Standard Benchmarks and high-dimensional SRBench.

Essay on "UniSymNet: A Unified Symbolic Network Guided by Transformer"

The paper "UniSymNet: A Unified Symbolic Network Guided by Transformer" proposes a novel approach to symbolic regression by introducing the Unified Symbolic Network (UniSymNet), which is guided by a Transformer model for structural optimization. Symbolic regression (SR) aims to discover mathematical expressions representing relationships in data. It is a challenging task due to the vast search space and lack of predefined structures. This paper addresses key limitations in existing symbolic networks, such as the representation of nonlinear operators and the issue of fixed architectural complexity leading to overfitting.

Key Contributions

1. Unified Symbolic Representation: UniSymNet brings forth a unified framework by transforming nonlinear binary operators $\{\times, \div, \mathrm{pow}\}$ into nested unary operators $\{\ln, \exp\}$. This transformation effectively converts binary interactions into multivariate ones. The authors provide conditions under which this unification reduces network complexity, improving both depth and node count efficiency. This adaptability is particularly beneficial for symbolic networks dealing with high-dimensional data.
2. Transformer Guided Structural Optimization: The paper leverages a pre-trained Transformer model to guide structural selection in the symbolic network. A novel label encoding method translates the structure of UniSymNet into sequences compatible with Transformer frameworks. Through encoding, the model can predict optimal network architectures tailored to specific tasks.
3. Objective-specific Optimization: The research introduces differentiated strategies for parameter estimation—Differentiable Network Optimization and Symbolic Function Optimization—each suited to distinct objectives. The former focuses on maintaining fitting accuracy, while the latter is tailored toward recovering underlying relationships in data.

Strong Results and Claims

The paper presents strong empirical results demonstrating UniSymNet's superior fitting accuracy, symbolic solution rate, and reduced expression complexity across various benchmarks. On the low-dimensional Standard Benchmarks and high-dimensional SRBench, UniSymNet consistently outperformed several baseline methods, showcasing its adaptability to dynamic SR tasks. Additionally, the use of nested unary operators proved effective in reducing architectural complexity when compared to existing paradigms using binary operators.

Implications and Speculation

The implications of this paper are profound for both theoretical and practical aspects of AI:

• Theoretical Advancement: By enabling multivariate operator interactions and leveraging learning-from-experience models, UniSymNet bridges theory and application in symbolic regression, providing a more comprehensive framework for interpreting scientific data.
• Practical Impact: The approach presents a compelling solution for automated equation discovery, offering potential deployment in physics, materials science, and engineering domains where discovering underlying laws and relationships from empirical data is crucial.

Future explorations could focus on refining encoding methods to improve automated discovery of mathematical equations. Another promising direction is the deployment of UniSymNet in specific domains, such as differential equation discovery, leveraging its reduced complexity and enhanced extrapolation capabilities.

In summary, the paper's approach to unifying symbolic networks with Transformer-guided optimization represents a significant stride in symbolic regression methodologies, aligning computational efficiency with interpretability and broadening the horizons for automated scientific discovery.