Distilling Free-Form Natural Laws from Experimental Data

Frequently Asked Questions (FAQ)

What exactly did you do in this project?

We developed a computing system that can observe a phenomenon in nature and then automatically identify various laws of nature and invariant equations that it obeys. For example, we had the system observe a double pendulum swinging chaotically using motion-tracking (see video). Without any knowledge about physics or geometry, the system identified the exact energy conservation and momentum relations that govern its dynamcis (see video).

Why is this difficult?

Identifying invariance laws is known to be a major challenge for human scientists; a large number of published invariant quantities have turned out to be coincidental. It turns out that identifying a meaningful invariance is very difficult computationally as well because so many trivial (coincidental) invariant equations exist. Our approach sifts out only the most meaningful relations.

Why is this important?

Mathematical symmetries and invariants underlie nearly all physical laws in nature. But, finding these invariants by hand for many natural phenomena is a laborious and arduous task. Allowing scientists to search for such relations more quickly and easily will one day help bring deeper understanding to the abundance of phenomena where the rules governing their behavior are currently unknown or incomplete.

Will this diminish the role of scientists?

While this advance will not replace scientists, it could greatly change how scientists approach investigating and modeling new phenomena. Rather than working directly with mathematical details, scientists could focus more on creative and conceptual issues, thinking of new conceptual frameworks and leaving machines to see if these new frameworks help generate more predictive and parsimonious explanations to observed phenomena.

How does it work?

The key insight into identifying nontrivial conservation laws computationally is that the invariant equations should be able to predict certain relationships between variables of the system. We used the partial derivatives between pairs of variables, which we can be derived directly from experimental data and also from an arbitrary equation using basic calculus.

What are your goals?

One of our main goals is to help enable mathematical modeling in domains that are exceedingly complex or difficult to decipher by hand, or where mathmatical modeling has not traditionally been employed due to lack of prior knowledge or availability of expertise.

What applications are there?

This project is applicable to many scientific domains where there exist theoretical gaps despite an abundance of data. In particular, a primary application is exploring data from different systems and to check if complex dynamics and phenomena can be explained by a simple underlying conservation or invariance.

What field(s) does this work fall under?

This work falls primarily under the field of artificial intelligence and machine learning. In particular, we employ techniques from evolutionary computation to search the space of equations.

What inspired this work?

While working on computational methods to model biological systems, we were surprised to realize that all of our methods for building explicit and differential equations seemed to break down and fail for modeling any type of implicit or invariant relationship. After several attempts to resolve the problem, it became obvious this was a much deeper and more challenging problem than it first seemed.

What are your next steps?

At the moment we are focused on important applications of this approach and welcome collaborations that could lead to high-impact publication.

Who are you?

This research was done at the Computational Synthesis Lab (CCSL) at Cornell University. Team members are Michael Schmidt and Hod Lipson. Michael Schmidt is a Ph.D. student and Hod Lipson is an Associate Professor.