In-Context Symbolic Regression: Leveraging Large Language Models for Function Discovery (2404.19094v2)

Abstract: State of the art Symbolic Regression (SR) methods currently build specialized models, while the application of LLMs remains largely unexplored. In this work, we introduce the first comprehensive framework that utilizes LLMs for the task of SR. We propose In-Context Symbolic Regression (ICSR), an SR method which iteratively refines a functional form with an LLM and determines its coefficients with an external optimizer. ICSR leverages LLMs' strong mathematical prior both to propose an initial set of possible functions given the observations and to refine them based on their errors. Our findings reveal that LLMs are able to successfully find symbolic equations that fit the given data, matching or outperforming the overall performance of the best SR baselines on four popular benchmarks, while yielding simpler equations with better out of distribution generalization.

• The paper introduces an innovative in-context symbolic regression approach using LLMs and VLMs to generate initial seed functions and iteratively refine them based on prediction error.
• The method, known as Optimization by Prompting (OPRO), significantly outperforms traditional genetic programming methods on several benchmarks with higher average R² values.
• Incorporating visual data via VLMs enhances performance in complex cases, though its benefits vary compared to using text-only inputs.

Exploring Symbolic Regression with LLMs and Vision-Language Approaches

Understanding the Study: Symbolic regression through LLMs

Symbolic Regression (SR) traditionally leverages Genetic Programming (GP) to find mathematical models that describe data. This paper proposes a novel method incorporating pre-trained LLMs and Vision-LLMs (VLMs) to handle SR tasks. The authors introduce a system wherein an LLM generates initial functional forms based on data observations. These forms are then refined using an iterative method until satisfactory results are achieved. The novel introduction of VLMs uses visual data (plots) alongside textual data to enhance the model's understanding and performance.

Methodology: How the approach works

The approach utilized here, termed Optimization by Prompting (OPRO), begins by having the LLM generate a range of possible mathematical functions based on initial data. These are the seed functions. Then, through iterative refinements where each function's performance (measured by its prediction error) informs the next generation of functions, the model refines its guesses. This process leverages both the generative capabilities of LLMs for creating new function forms and external optimizers for fine-tuning function coefficients.

An intriguing extension involves integrating visual input. Here, plots of data and previous function guesses are fed into a VLM to potentially enhance the model's understanding and performance, particularly in more complex scenarios where textual information might be insufficient.

Results: Strong performance indicated

The results from this paper were quite promising. The LLM-based approach outperformed traditional GP-based methods on several benchmarks. The inclusion of visual data through VLMs also showed potential in complex cases, although it didn't always outperform the text-only model. These results suggest that pre-trained LLMs, especially when equipped with mechanisms like OPRO, can effectively explore and optimize mathematical expressions fitting observed data.

Here's a breakdown of the performance improvements:

• The LLM approach yielded an average $R^2$ value significantly higher than that of the GP baselines across multiple benchmarks.
• Inclusion of visual inputs via VLMs helped in some complex benchmarks but was not universally superior to the text-only approach.

Implications and Future Prospects

Both theoretically and practically, the integration of LLMs into SR tasks could pave the way for more versatile and powerful analytical tools that benefit from advancements in natural language processing and machine learning. Particularly, the ability of these models to generate and refine hypotheses in an iterative fashion could make them valuable in fields requiring automated modeling of complex phenomena.

However, the practical application of this approach, especially in higher-dimensional spaces or with larger data sets, will likely need ongoing refinement. Advances in model capabilities, such as extended context windows or enhanced integration of multimodal data, could further improve performance.

Limitations and Challenges

While promising, the approach faces limitations primarily related to the handling of high-dimensional data and the finite size of the context window in current LLMs, which can constrain the amount of data that can be processed in one go. Future iterations of this technology will need to address these constraints to fully leverage the potential of LLMs in symbolic regression.

Conclusion

This paper provides a compelling look at how modern AI techniques can be extended beyond traditional application boundaries into areas like symbolic regression. As AI continues to evolve, particularly with improvements in LLMs and VLMs, we are likely to see even more innovative applications that can tackle increasingly complex tasks effectively.