May 9, 2019
As AI bulldozes its way through our everyday life, forcing a revolutionary change across the globe, it is vital to consider its impact in the field of science which doesn’t usually occur to the ordinary man.
Particularly in the fields of Physics and Maths, is AI playing a huge role in discovering new laws?
The field of physics almost appears antagonistic to the approach of Artificial Intelligence. Physics emphasizes on modeling real systems in order to enable interpretation. To this effect, it places a serious responsibility on the mathematical formalisms to negotiate with the fundamental physical principles (like conservation laws) and constitutive relations which transform into space-time equations (classical, quantum or relativistic) that can at least substantially quantify if not explain the reality.
For example, the fundamental motion of an oscillating pendulum can be explained by the principles of conservation of momentum and energy along with the definition of force. This simplifies to a second order partial differential equation in time and space which when solved numerically provides the time and position of the pendulum to a reasonable accuracy. How “good” the estimate is, depends on how “informed” the model is.
The “simple” model has several caveats: (i) the pendulum must not experience air drag (ii) the thread must be massless (iii) friction at the pivot is minimal etc. Under such idealized conditions, one would expect the pendulum to oscillate endlessly, or if some damping is accounted for, it could end up becoming static. However, the real world situations are much more complex and building an “explanation” model based on physical principles may become prohibitive. For example, the blob could have a non-uniform loading, the elasticity of the thread may change according to the stress-strain characteristics of the thread and there may be other attritional friction which could develop at the pivot. Can physics explain these theories? It certainly can. But can it predict the position of the pendulum with the accuracy that we seek using all these equations? This becomes a complicated exercise.
Now consider a scenario where they perform many experiments on this pendulum where space-time trajectory information is collected. The question is: is it possible to predict the motion of the pendulum given the data? To put in other words., do we really need a physics-based model to predict the coordinates of the pendulum if we are given sufficient data about its history of oscillations? Which method would produce a more accurate prediction of the coordinates? Or can AI at this stage be used to discover physics laws on its own without priorly assuming any model?
According to a published paper in 1987, Crutchfield and McNamara from U. Cal. Berkeley observed that temporal pattern learning, control and prediction, and chaotic data analysis shared a common problem: while deducing optimal equations of motion from observations of time-dependent behavior, each of them obtained models of the physical world with limited information. The authors went on to describe a method to reconstruct the deterministic portion of the equations of motion directly from a data series! They attempted to see if a data series could lead to a discovery of natural laws “automatically” without the assumption of any physics model in place.
Fast forward to 2009, a paper, published by Schmidt and Lipson in the Science journal, which introduced the idea that free-form natural laws” can be learned from experimental measurements in a physical system using symbolic regression algorithms. They addressed the problem of “what makes a correlation in observed data important and insightful” and proposed a framework for identifying non-trivial insights. Their algorithm, without any prior knowledge about physics, kinematics, or geometry, discovered Hamiltonians, Lagrangians, and other laws of geometric and momentum conservation which are fundamental. In other words, their algorithm recovered the laws of motion of simple mechanical systems, like a double pendulum, by searching over a space of mathematical expressions on the input variables.
In 2018, Iten et al published “Discovering physical concepts with neural networks” where they described the system of a one-dimensional damped pendulum using a neural-networks (Scinet) based on representation learning, and the results of the prediction were ridiculously accurate (See Figure 1)
Iten et al discovered the heliocentric model of the solar system ! SciNet mimics a physicist’s modeling process and applies it to study various physical scenarios. They explained the idea that the physicist (or SciNet) is exposed to some experimental observations pertaining to a physical setting (e.g., a time series of positions of the pendulum and will later be asked a question about this physical setting (e.g., where is the particle at time t10?”). Thus the representation should be as compact as possible while still fully characterizing the physics of the situation, so that the physicist may forget the original data and answer the question using only the representation.
So, is AI a possible solution for discovering laws of physics? Maybe. As Crutchfield lamented, the “intimate connection between dynamics, on the one hand, and modeling, prediction, and complexity, on the other, had been ignored” for our intellectual peril. AI seems to offer hope to bridge these. Research efforts are growing in this field and efforts are on to push the boundaries of discovering laws and theorems which currently although at infancy, could potentially lead to entirely new insights into our understanding of our universe! Who knows! Someday Schrodinger’s equation could jump out of an exploration of time-series data and how exciting would that be!