Vernon Austel

Cristina Cornelio, Sanjeeb Dash, Vernon Austel et al.

Scientists have long aimed to discover meaningful formulae which accurately describe experimental data. A common approach is to manually create mathematical models of natural phenomena using domain knowledge, and then fit these models to data. In contrast, machine-learning algorithms automate the construction of accurate data-driven models while consuming large amounts of data. The problem of incorporating prior knowledge in the form of constraints on the functional form of a learned model (e.g., nonnegativity) has been explored in the literature. However, finding models that are consistent with prior knowledge expressed in the form of general logical axioms (e.g., conservation of energy) is an open problem. We develop a method to enable principled derivations of models of natural phenomena from axiomatic knowledge and experimental data by combining logical reasoning with symbolic regression. We demonstrate these concepts for Kepler's third law of planetary motion, Einstein's relativistic time-dilation law, and Langmuir's theory of adsorption, automatically connecting experimental data with background theory in each case. We show that laws can be discovered from few data points when using formal logical reasoning to distinguish the correct formula from a set of plausible formulas that have similar error on the data. The combination of reasoning with machine learning provides generalizeable insights into key aspects of natural phenomena. We envision that this combination will enable derivable discovery of fundamental laws of science and believe that our work is an important step towards automating the scientific method.

Vernon Austel, Cristina Cornelio, Sanjeeb Dash et al.

The Symbolic Regression (SR) problem, where the goal is to find a regression function that does not have a pre-specified form but is any function that can be composed of a list of operators, is a hard problem in machine learning, both theoretically and computationally. Genetic programming based methods, that heuristically search over a very large space of functions, are the most commonly used methods to tackle SR problems. An alternative mathematical programming approach, proposed in the last decade, is to express the optimal symbolic expression as the solution of a system of nonlinear equations over continuous and discrete variables that minimizes a certain objective, and to solve this system via a global solver for mixed-integer nonlinear programming problems. Algorithms based on the latter approach are often very slow. We propose a hybrid algorithm that combines mixed-integer nonlinear optimization with explicit enumeration and incorporates constraints from dimensional analysis. We show that our algorithm is competitive, for some synthetic data sets, with a state-of-the-art SR software and a recent physics-inspired method called AI Feynman.

Vernon Austel, Sanjeeb Dash, Oktay Gunluk et al.

In this study we introduce a new technique for symbolic regression that guarantees global optimality. This is achieved by formulating a mixed integer non-linear program (MINLP) whose solution is a symbolic mathematical expression of minimum complexity that explains the observations. We demonstrate our approach by rediscovering Kepler's law on planetary motion using exoplanet data and Galileo's pendulum periodicity equation using experimental data.