Discovering Scientific Formulae from Background Knowledge and Experimental Data Using Polynomial Optimization

Centrala begrepp

A new approach to scientific discovery that leverages polynomial optimization to derive scientific formulae consistent with background theory and experimental data.

Sammanfattning

The paper proposes a novel automated approach to scientific discovery, termed AI-Hilbert, that utilizes techniques from polynomial and sum-of-squares optimization to derive polynomial scientific laws that best explain a set of experimental data while maintaining consistency with a body of background knowledge.
Key highlights:

AI-Hilbert provides an axiomatic derivation of the correctness of the discovered scientific law, conditional on the correctness of the background theory. It can also identify inconsistencies in the background theory by performing best subset selection.

AI-Hilbert allows fine-grained control over the tractability of the scientific discovery process by bounding the degree of the polynomial certificates searched over. This differs from prior work which offers more limited control over time complexity.

Experiments show that AI-Hilbert can rediscover famous scientific laws like Kepler's Third Law, Einstein's Relativistic Time Dilation Law, and others, by combining background theory and experimental data. This is in contrast to state-of-the-art data-driven approaches which struggle to recover these laws, especially in limited data settings.

The paper argues that providing relevant background theory can decrease the amount of data required to recover a scientific law with high probability, by constraining the space of derivable laws.

Statistik

"The discovery of scientific formulae that parsimoniously explain natural phenomena and align with existing background theory is a key goal in science."
"Accordingly, recent works combine regression and reasoning to eliminate formulae inconsistent with background theory."
"We demonstrate that some famous scientific laws, including Kepler's Third Law of Planetary Motion, the Hagen-Poiseuille Equation, and the Radiated Gravitational Wave Power equation, can be derived in a principled manner from background axioms and experimental data."

Citat

"Remarkably, the optimization techniques leveraged in this paper allow our approach to run in polynomial time with fully correct background theory, or non-deterministic polynomial (NP) time with partially correct background theory."
"Accordingly, our approach discovers new scientific laws by solving an optimization problem to minimize a weighted sum of discrepancies between the proposed law and experimental data, plus the distance between the discovered law and its projection onto the set of symbolic laws derivable from background theory."

Viktiga insikter från

Evolving Scientific Discovery by Unifying Data and Background Knowledge with AI Hilbert

by Ryan Cory-Wr... på arxiv.org 04-30-2024

https://arxiv.org/pdf/2308.09474.pdf

Djupare frågor

How can the proposed approach be extended to discover scientific laws that are not expressible as polynomial equalities and inequalities

The proposed approach can be extended to discover scientific laws that are not expressible as polynomial equalities and inequalities by incorporating different basis functions or representations into the optimization framework. For example, trigonometric functions, exponential functions, or other non-polynomial functions can be used to represent the relationships between variables in the scientific laws. By expanding the set of basis functions or representations that the optimization algorithm can search over, the approach can be adapted to discover laws that have more complex or non-polynomial forms.
Additionally, techniques from computational algebraic geometry and optimization, such as sum-of-squares decomposition, can be applied to non-polynomial functions to represent them in a form that is amenable to optimization. By leveraging these mathematical tools, the approach can handle a broader range of functions and discover scientific laws that may not have a simple polynomial representation.

What are the limitations of the Positivestellensatz-based approach, and how can they be addressed to further improve the scalability and applicability of AI-Hilbert

The Positivestellensatz-based approach, while powerful in providing formal proofs of the correctness of derived scientific laws, has certain limitations that can impact its scalability and applicability. Some of these limitations include:

Computational Complexity: The approach may become computationally intensive as the degree of the polynomials and the complexity of the background theory increase. This can lead to longer optimization times and challenges in handling large-scale problems.

Assumptions and Constraints: The approach relies on certain assumptions, such as the Archimedean property, which may not always hold in practical scientific discovery settings. Relaxing these assumptions while maintaining the validity of the approach could enhance its applicability.

Data Efficiency: While the approach aims to reduce the amount of data needed for scientific discovery by incorporating background theory, there may still be scenarios where a significant amount of data is required to derive accurate scientific laws. Improving the data efficiency of the approach could make it more practical in data-scarce environments.

To address these limitations and improve the scalability and applicability of AI-Hilbert, researchers can explore optimizations in the algorithmic implementation, develop strategies to handle higher degrees of polynomials efficiently, relax restrictive assumptions without compromising the validity of the approach, and enhance the data efficiency by incorporating adaptive learning mechanisms or data augmentation techniques.

Can the ideas behind AI-Hilbert be applied to other domains beyond scientific discovery, such as machine learning model interpretability or knowledge integration

The ideas behind AI-Hilbert can indeed be applied to other domains beyond scientific discovery, such as machine learning model interpretability or knowledge integration. Here are some potential applications:

Machine Learning Model Interpretability: The principles of combining background knowledge with data-driven methods to derive interpretable models can be applied in the field of machine learning. By integrating domain expertise and constraints into the model discovery process, AI-Hilbert-like approaches can help in creating more transparent and understandable machine learning models.

Knowledge Integration in Decision Support Systems: AI-Hilbert's methodology of unifying data and background knowledge to derive new insights can be utilized in decision support systems. By incorporating existing rules, regulations, and expert knowledge into the decision-making process, these systems can provide more informed and reliable recommendations.

Natural Language Processing and Semantic Analysis: The approach can be adapted to extract meaningful patterns and relationships from textual data, enabling better semantic analysis and understanding of language. By combining linguistic rules and data-driven techniques, AI-Hilbert-like methods can enhance the accuracy and depth of natural language processing tasks.

By applying the core concepts of AI-Hilbert to these diverse domains, researchers can enhance the interpretability, reliability, and efficiency of various systems and processes.