American Scientist

What does it mean to understand the natural world? To a classically trained physicist, it means that one is able to construct a model that not only accounts for currently measured phenomena but is also capable of predicting future phenomena. The modern challenge posed by artificial intelligence is that if one has a sufficiently large dataset of a system, in principle one may train a machine learning model to predict future phenomena without having any understanding of the underlying physical mechanisms. It reignites the debate of what “understanding” really means.

In a mechanistic view of the world, any phenomenon may be understood as a large ensemble of interacting billiard balls. The reductionist asserts that as long as enough basic units (“billiard balls”) are present in such a system, one may model or simulate it exhaustively. For example, if one wishes to understand how waves behave in the ocean, one only has to simulate the interactions of each and every constituent molecule of water.

This reductionistic approach has been extremely successful at describing nature and making useful predictions, but it does not satisfy a more comprehensive meaning of “understanding.” Even if one could run such a simulation in one’s lifetime, it would be difficult to identify the underlying mechanisms responsible for producing different types of waves. Generally, examining the microscopic components of a complex system in order to understand how macroscopic behavior emerges falls short—think biology and economics. Implicitly, this brute-force approach suggests that one may study complex systems without having a scientific question in mind, and if enough computing power is deployed then insight naturally emerges. The billion-euro Human Brain Project is a spectacular example of the limitations of such an approach. Essentially, their simulations have failed to replace laboratory experiments.

As has been noted by the climate scientist Isaac Held, there is a tension between simulation and understanding. We simulate in order to mimic as much of an observed phenomenon as possible. But we achieve understanding by using idealized models to capture the essence of the phenomenon. By design, idealized models necessarily employ assumptions.

Models and Their Limitations

Models are approximations of reality, constructed by scientists to address specific questions of natural phenomena using the language of mathematics. Mathematical equations allow the practitioner to decide the level of abstraction needed to tackle a given problem, thus transcending the “billiard ball” approach. One example concerns the modeling of disease transmission. Human beings are complex entities whose individual behavior cannot be easily modeled by mathematical equations. However, the movement of ensembles of human beings within a city or country may be approximated by a set of fluid equations: the so-called compartmental models of disease transmission. Our incomplete understanding of the biological properties of a pathogen, as well as how it mutates and is transmitted across hosts, is encoded within a single parameter known as the reproduction number. In this manner, epidemiologists were able to study COVID-19 during the pandemic without having full knowledge of it, because the focus was on its macroscopic behavior across large spaces. (See “COVID-19 Models Demand an Abundance of Caution,” April 23, 2020.) In such an example, the theorist is employing the principle of separation of scale in order to isolate phenomena.

Ideally, one would like to construct a universal model that is able to answer every question that the scientist asks of it. In practice, mathematical solutions to equations describing nonlinear systems that can be written down on paper are exceedingly rare. (Nonlinear here means that changes between the inputs and outputs of the system do not obey a simple relationship.) Instead, one must solve these equations with a computer, which allows one to study how different components of a physical system interact and produce nonlinear outcomes. A universal simulation would be an emulation—a perfect replica of the actual system across all scales and at all times.

In practice, simulations are often limited by a dynamic range problem, meaning that one runs out of computing power to simulate behavior at all scales. The discretization of mathematical equations in order to program them into a computer introduces subtleties such as numerical dissipation—an artificial side effect of differential equation calculations that does not arise from physics. It is also likely that one needs different numerical methods to simulate behavior on different scales, and a single method is insufficient for all spatial scales. Perhaps the advent of AI will allow us to overcome some of these limitations, but for now emulations remain an aspiration.

Once AI technology is capable of designing models, writing programs to compute them, and interpreting their outcomes, it will forever change the way researchers use computers.

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If an astrophysicist wishes to understand how stars evolve over cosmic timescales (billions of years), then it is not unreasonable to model stars as spherical objects. Over such long timescales, the practitioner is less interested in transient phenomena and more interested in time-averaged effects. However, if an astrophysicist is interested in how sound waves propagate across a star so that they might understand its detailed structure (and therefore ultimately infer the age of the star, known as asteroseismology), then spherical symmetry becomes a questionable approximation. In these examples, the design of the model is not only tied to the specific scientific question being asked, but also the characteristic timescale being considered. Essentially, models often operate under idealized conditions in order to produce useful answers.

Once one accepts that models and simulations are necessarily limited in scope and may pragmatically be built only to address specific questions, then one has to accept that approximations are a feature and not a bug. Some practitioners even regard them as an art form. The intent and skill level of the modeler becomes relevant, because one needs to be able to ask a sharply defined scientific question and construct a model so that its output may be decisively confronted by data. Issues such as falsification and Occam’s razor become relevant—one wishes to construct models that are complex enough to include the relevant physics, chemistry, biology, and so on, but simple enough to be proven right or wrong by the best data available.

Understanding Mechanisms

Models and simulations have been used not only for prediction but also to seek insight into underlying mechanisms. Some practitioners find that incorporating their prior beliefs about a system as well as sources of uncertainty associated with the data—an approach called Bayesian statistics—facilitates this confrontation between models and data. In this process, the inferred values of the parameters of the model are probabilistic. Bayesian frameworks allow for empirical sources of uncertainty and partial theoretical ignorance, as well as degeneracies (different combinations of parameters that produce the same observable outcome), to be considered when using models to interpret data.

Even if it were possible to construct a true emulation, though, strictly speaking one would produce only correlations between phenomena. As scientists, we are interested in cause and effect. We want to understand why phenomena change in a particular way, which leads to better predictions, decision-making, and overall conceptualization about how a process works. To transform correlations into causal relationships, one needs to construct a simplified model to explain trends observed in the emulation. Such a model necessarily involves judiciously taken approximations in order to isolate the causal mechanism responsible for the observed trend. It is almost as if we are missing part of the language of models and simulations, a tool of intermediate complexity that goes beyond written linear models and full-blown numerical simulations.

Perhaps advances in AI will allow us to overcome this limitation of the human imagination. Perhaps scientists must ultimately let go of the ideal of explaining phenomena from first principles and simply accept that complex systems are not always amenable to the classical, reductionistic approach. It is plausible that an AI procedure, akin to an advanced version of the scientific computing software Mathematica, could produce approximate mathematical solutions of a governing equation much faster than any human being could. It is conceivable that this AI could visualize correlations and causations in multiple dimensions, beyond what a human brain could. Once AI technology is capable of taking over the tasks of designing models, writing programs to compute them, and interpreting their outcomes, it will forever change the way that researchers use computers to solve complex problems of the physical world, as well as how they train future generations of graduate students.

For example, a senior researcher may no longer need to be an expert computer programmer and may instead work with an AI that essentially acts as a super research assistant. Graduate students may spend less time on software engineering and more time asking deep, creative questions of the models or simulations they are studying. In other words, our computers would be only a small step away from telling us not only how to solve a problem, but also which problems to solve and what their solutions imply. AI would then acquire two human traits that are thought to be elusive to computers: physical intuition (or simply having a “feel” for how something works) and insight. Ironically, AI may end up emphasizing the most precious human trait we have as researchers, which is to ask deep, insightful questions led only by our curiosity.