Evolving Scientific Discovery by Unifying Data and Background Knowledge with AI Hilbert
arxiv(2023)
摘要
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.
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