Dynamic Bayesian Optimization Framework for Instruction Tuning in Partial Differential Equation Discovery

2025-12-31

arXiv.org

AI Summary

This work addresses the instability of large language models (LLMs) in symbolic discovery of partial differential equations (PDEs), which arises from their reliance on static prompts and inability to adapt to the dynamic demands of multi-step reasoning. To overcome this limitation, the authors formulate prompt engineering as a sequential decision-making problem and propose an adaptive prompt selection mechanism. By maintaining a discrete library of reasoning strategies and integrating Bayesian optimization with numerical feedback, the method dynamically selects the optimal instruction at each reasoning step to guide symbolic generation. This approach represents the first application of dynamic Bayesian optimization to instruction tuning in PDE discovery, significantly outperforming fixed-prompt baselines by achieving higher equation recovery rates and yielding more concise analytical expressions on standard benchmarks.

Technology Category

• Natural Language Processing: Prompt Engineering / Prompting
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation
• Search and Retrieval-Augmented AI: Search Tool Learning with LLM: Teaching LLMs to invoke search and make use of retrieved information
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Large Language Models (LLMs) show promise for equation discovery, yet their outputs are highly sensitive to prompt phrasing, a phenomenon we term instruction brittleness. Static prompts cannot adapt to the evolving state of a multi-step generation process, causing models to plateau at suboptimal solutions. To address this, we propose NeuroSymBO, which reframes prompt engineering as a sequential decision problem. Our method maintains a discrete library of reasoning strategies and uses Bayesian Optimization to select the optimal instruction at each step based on numerical feedback. Experiments on PDE discovery benchmarks show that adaptive instruction selection significantly outperforms fixed prompts, achieving higher recovery rates with more parsimonious solutions.

Problem

• instruction brittleness
• prompt engineering
• PDE discovery
• Large Language Models
• adaptive instruction

Innovation

• Dynamic Bayesian Optimization
• Instruction Tuning
• Large Language Models
• PDE Discovery
• Prompt Engineering