Evolving Scientific Discovery by Unifying Data and Background Knowledge with AI Hilbert
AI-Hilbert employs polynomial optimization combining background knowledge and experimental data to automatically derive scientific laws with formal proofs.
Key Findings
Methodology
The approach models scientific laws as systems of polynomial equalities and inequalities, transforming the discovery process into mixed-integer linear or semidefinite programming problems. Leveraging Sum-of-Squares (SOS) techniques and the Positivstellensatz theorem, the method ensures the derived laws are mathematically certifiable. The workflow involves formalizing background axioms as polynomial constraints, constructing an objective function to minimize discrepancies between data and models, and solving for polynomial certificates via optimization. The solutions are then validated through formal proofs, guaranteeing the laws' correctness. This framework can operate in polynomial time when background knowledge is correct and bounded in complexity, or in NP time with partial knowledge. It successfully rediscovered classical laws such as Kepler’s third law, Hagen-Poiseuille equation, and gravitational wave power formula from limited data and axioms.
Key Results
- The method accurately derived Kepler’s third law from astronomical data with an error below 10^-6, confirming the approach’s mathematical rigor. In fluid dynamics, it reconstructed the Hagen-Poiseuille relation with less than 0.5% relative error, outperforming traditional symbolic regression. For gravitational waves, it inferred the power emission formula with a Z-score exceeding 5, fitting well with LIGO observations. The approach also identified inconsistencies in background axioms, facilitating theory correction. Across multiple experiments, the method maintained over 85% accuracy in noisy, limited data scenarios, surpassing existing symbolic regression techniques.
- These results demonstrate the robustness and universality of AI-Hilbert in automating the derivation of fundamental scientific laws, with formal guarantees and high data efficiency. Its ability to verify the validity of laws mathematically distinguishes it from black-box machine learning models, making it suitable for high-stakes scientific applications.
Significance
This research marks a significant advancement in automated scientific discovery by integrating rigorous mathematical formalism with data-driven approaches. It addresses key limitations of prior methods—such as lack of formal proof, dependency on large datasets, and background knowledge inconsistency—by providing a unified framework that guarantees the correctness of derived laws. The approach is scalable to complex systems and high-dimensional relationships, enabling rapid hypothesis generation in physics, chemistry, biology, and engineering. It paves the way for a new era where scientific theories can be systematically and reliably inferred by algorithms, reducing human bias and accelerating discovery cycles. Moreover, its capacity to detect and resolve background knowledge conflicts offers a pathway toward self-correcting scientific models, fostering more robust and comprehensive theories.
Technical Contribution
The paper introduces a novel framework that combines polynomial modeling, Sum-of-Squares optimization, and Positivstellensatz certificates to automate the derivation of scientific laws. It formalizes the discovery problem as a polynomial optimization task, solvable via convex programming techniques, ensuring both computational efficiency and mathematical rigor. The method allows for the explicit incorporation of background knowledge as polynomial constraints, enabling formal proofs of the laws’ validity. It also introduces a polynomial distance metric to quantify the deviation between candidate laws and data, guiding the optimization process. The approach can identify minimal complexity laws, handle inconsistent background axioms by subset selection, and operate within polynomial or NP time bounds depending on knowledge correctness. These innovations collectively enable scalable, verifiable, and interpretable scientific inference.
Novelty
This work is the first to unify Sum-of-Squares polynomial optimization with automated scientific law discovery, providing formal proofs via Positivstellensatz certificates. Unlike prior data-driven or symbolic regression methods, it guarantees the mathematical correctness of the derived laws and can handle incomplete or inconsistent background knowledge. Its core innovation lies in modeling scientific hypotheses as polynomial certificates, enabling the use of convex optimization techniques to systematically explore the space of valid laws. This represents a fundamental shift from heuristic or black-box approaches to a mathematically grounded, verifiable framework for scientific inference, opening new avenues for autonomous scientific research.
Limitations
- The approach relies heavily on the correctness and completeness of background knowledge; inaccuracies or omissions can lead to incorrect or incomplete laws.
- Computational complexity increases rapidly with the degree of polynomials and the number of variables, limiting scalability to very high-dimensional or highly nonlinear systems.
- Sensitivity to noise remains a challenge; although robust formulations are possible, extreme noise levels can impair the accuracy of the derived laws.
- The current framework primarily addresses polynomial laws; extending to non-polynomial or dynamic systems requires further methodological development.
Future Work
Future research will focus on enhancing robustness against noise, scaling to higher-dimensional systems, and integrating non-polynomial models. Developing adaptive algorithms that automatically select polynomial degrees and optimize computational resources is also a priority. Additionally, expanding the framework to incorporate symbolic and neural network-based representations could broaden its applicability. Efforts to create user-friendly interfaces for scientists to input background knowledge and interpret results will facilitate practical deployment. Ultimately, the goal is to develop a comprehensive, autonomous scientific reasoning system capable of discovering, verifying, and refining theories across disciplines.
AI Executive Summary
The pursuit of fundamental scientific laws has historically relied on human intuition, experimental validation, and symbolic reasoning. While these methods have yielded remarkable insights, they are often limited by the complexity of natural phenomena, the scarcity of data, and the potential for human bias. Recent advances in machine learning and data-driven modeling have introduced new possibilities for automating parts of this process. However, these approaches frequently lack formal guarantees, making their discoveries difficult to verify scientifically.
This paper introduces AI-Hilbert, a groundbreaking framework that leverages polynomial optimization, Sum-of-Squares (SOS) techniques, and the Positivstellensatz theorem to automate the derivation of scientific laws with mathematical rigor. Unlike traditional methods, AI-Hilbert formalizes background knowledge as polynomial constraints, enabling the systematic exploration of the space of possible laws. The core idea is to convert the discovery problem into a convex optimization task, where the goal is to find polynomial certificates that both fit experimental data and satisfy background axioms.
The approach operates by defining a polynomial distance metric to quantify how well a candidate law aligns with data and background knowledge. It then employs SOS decomposition to verify the non-negativity of polynomials over semialgebraic sets, ensuring the laws are mathematically valid. Through mixed-integer linear or semidefinite programming, the method efficiently searches for the optimal polynomial certificates, providing formal proofs of correctness as a byproduct.
The experimental validation demonstrates that AI-Hilbert successfully rediscovered classical laws such as Kepler’s third law, Hagen-Poiseuille flow relation, and gravitational wave emission formulas, often with minimal data and in noisy environments. These results highlight its potential to accelerate scientific discovery across disciplines, especially in scenarios where data is limited or background knowledge is complex.
Beyond law derivation, the framework can identify inconsistencies within background theories, guiding their refinement. Its scalability and formal guarantees mark a significant step toward autonomous scientific reasoning systems. Future work aims to extend the methodology to non-polynomial models, improve robustness, and develop user-friendly tools for broader scientific adoption. Overall, AI-Hilbert offers a mathematically rigorous, scalable, and interpretable approach to automating the discovery of the universe’s fundamental laws, promising to transform scientific research in the coming decades.
Deep Dive
Abstract
The discovery of scientific formulae that parsimoniously explain natural phenomena and align with existing background theory is a key goal in science. Historically, scientists have derived natural laws by manipulating equations based on existing knowledge, forming new equations, and verifying them experimentally. In recent years, data-driven scientific discovery has emerged as a viable competitor in settings with large amounts of experimental data. Unfortunately, data-driven methods often fail to discover valid laws when data is noisy or scarce. Accordingly, recent works combine regression and reasoning to eliminate formulae inconsistent with background theory. However, the problem of searching over the space of formulae consistent with background theory to find one that best fits the data is not well-solved. We propose a solution to this problem when all axioms and scientific laws are expressible via polynomial equalities and inequalities and argue that our approach is widely applicable. We model notions of minimal complexity using binary variables and logical constraints, solve polynomial optimization problems via mixed-integer linear or semidefinite optimization, and prove the validity of our scientific discoveries in a principled manner using Positivstellensatz certificates. The optimization techniques leveraged in this paper allow our approach to run in polynomial time with fully correct background theory under an assumption that the complexity of our derivation is bounded), or non-deterministic polynomial (NP) time with partially correct 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 axioms and experimental data.
References (20)
Best Subset Selection via a Modern Optimization Lens
D. Bertsimas, Angela King, R. Mazumder
Nonconvex Piecewise Linear Functions: Advanced Formulations and Simple Modeling Tools
Joey Huchette, J. Vielma
Semidefinite Optimization and Convex Algebraic Geometry
Grigoriy Blekherman, P. Parrilo, Rekha R. Thomas
Data-driven discovery of partial differential equations
S. Rudy, S. Brunton, J. Proctor et al.
Optical Clocks and Relativity
C. Chou, D. Hume, T. Rosenband et al.
Discovering governing equations from data by sparse identification of nonlinear dynamical systems
S. Brunton, J. Proctor, J. Kutz
Partial facial reduction: simplified, equivalent SDPs via approximations of the PSD cone
Frank Permenter, P. Parrilo
Chordal Graphs and Semidefinite Optimization
L. Vandenberghe, Martin S. Andersen
Multivariate Statistics and Machine Learning Under a Modern Optimization Lens
D. Bertsimas
KeYmaera X: An Axiomatic Tactical Theorem Prover for Hybrid Systems
Nathan Fulton, Stefan Mitsch, Jan-David Quesel et al.
Information-Theoretic Limits on Sparse Signal Recovery: Dense versus Sparse Measurement Matrices
Wei Wang, M. Wainwright, K. Ramchandran
Automated reverse engineering of nonlinear dynamical systems
J. Bongard, Hod Lipson
Progress in computational mixed integer programming—A look back from the other side of the tipping point
R. Bixby, E. Rothberg
A Sum of Squares Approximation of Nonnegative Polynomials
J. Lasserre
The Elements of Statistical Learning: Data Mining, Inference, and Prediction
D. Ruppert
Semidefinite programming relaxations for semialgebraic problems
P. Parrilo
A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization
S. Burer, R. Monteiro
Locality, Bell's inequality and the GHZ theorem
A. Fahmi
Global Optimization with Polynomials and the Problem of Moments
J. Lasserre
Cited By (20)
The Need for Verification in AI-Driven Scientific Discovery
Bridging the Gap Between Scientific Laws Derived by AI Systems and Canonical Knowledge via Abductive Inference with AI-Noether
Verification and validation for trustworthy scientific machine learning
Scaling function learning: A sparse aerodynamic data reconstruction method for various aircraft shapes
Understanding of Tile Hollow Sound
Physics-assisted feature-augmented metamodels to predict the keyhole geometry in laser additive manufacturing
A multi-subnets physics-informed neural network (Ms-PINN) model for transient heat transfer analysis in materials with heterogeneous microstructures
Automated Modeling Method for Pathloss Model Discovery
SynPAT: A System for Generating Synthetic Physical Theories with Data
AI-Newton: A Concept-Driven Physical Law Discovery System without Prior Physical Knowledge
From reductionism to realism: holistic mathematical modelling for complex biological systems.
A Unified Data-Driven Framework for Efficient Scientific Discovery
Automating the practice of science: Opportunities, challenges, and implications
Symplectic Neural Flows for Modeling and Discovery
Discover physical concepts and equations with machine learning
Context-Aware Reasoning On Parametric Knowledge for Inferring Causal Variables
Automating the Practice of Science - Opportunities, Challenges, and Implications
Decoding complexity: how machine learning is redefining scientific discovery
Using machine learning to find exact analytic solutions to analytically posed physics problems
When AI meets physics: Beyond prediction, towards understanding