Artificial Intelligence, Chaos, Prediction and Understanding in Science
aa r X i v : . [ c s . G L ] M a r Artificial Intelligence, Chaos, Prediction and Understanding inScience
Miguel A. F. Sanju´an
1, 2, ∗ Nonlinear Dynamics, Chaos and Complex Systems Group,Departamento de F´ısica, Universidad Rey Juan Carlos,M´ostoles, Madrid, Tulip´an s/n, 28933, Spain Department of Applied Informatics, Kaunas University of Technology,Studentu 50-415, Kaunas LT-51368, Lithuania (Dated: March 31, 2020)
Abstract
Machine learning and deep learning techniques are contributing much to the advancement ofscience. Their powerful predictive capabilities appear in numerous disciplines, including chaoticdynamics, but they miss understanding. The main thesis here is that prediction and understandingare two very different and important ideas that should guide us about the progress of science.Furthermore, it is emphasized the important role played by nonlinear dynamical systems for theprocess of understanding. The path of the future of science will be marked by a constructivedialogue between big data and big theory, without which we cannot understand. ∗ Corresponding Author : [email protected] . INTRODUCTION Techniques from artificial intelligence are contributing to transform every walk of life,enabling people to analyze data, integrating information and providing ways to improvedecision-making, and fields such as the biotechnology, health care, speech and voice recogni-tion, transport, finance, and climate change, among others. Computers learn from numerousexamples, after having been trained, and can recognize patterns in big data sets, and classifythem in different categories. As a consequence, they are able to carry out certain specifictasks with a much higher precision than humans. Many criticize that this does not signifythat the machine is intelligent, since intelligence is something far more complex, and fur-thermore, most of the serious scientists agree that we are extremely far away from a machinebeing more intelligent than a human being. No doubt, it would be fantastic that AI wouldbe more versatile, since right now in nearly all cases everything that has been achieved isrelated to pattern recognition, when practically all interesting problems are definitely muchmore complicated than that.One of the main threads of my article will be the analysis between artificial intelligenceand its relationship between prediction and understanding in science. Another key idea Iwish to emphasize throughout this article is the important role that ideas from nonlineardynamics and dynamical systems theory play in the process of understanding science, andthe evolutionary dynamics of physical and biological processes. As it will be commentedlater, even neuroscientists associate the very meaning of understanding to dynamical systemstheory.At the heart of the scientific endeavour lies the desire of understanding the universe,knowing what kind of reasons explain the past events, and acquiring the ability of forecastingthe future. Since the earliest times the main task of a scientist today is to observe nature,to build models from the observations, and to use them for predictions. Forecasting is theprocess of making predictions of the future based on past and present data. Thanks tothe scientific models, it is possible to understand nature and the mechanisms that explainthe observations, attempt to forecast extreme events and the weather, to prevent diseases,calculate the position of the celestial bodies, as well as develop the astronautic technologies,understand the behavior of the components of matter, fight against epidemics, etc.Nevertheless, the importance of forecasting goes beyond the practical purposes and points2o the essence of the scientific method itself. When we analyze a certain scientific problem,one of the goals is to catch the reality as faithful as possible from a model, with the ex-pectation to obtain an appropriate understanding of the involved physical processes. Thatis why making excellent predictions with our model and to test the predictions with newobservations is so important for the development of science by using the scientific method,as well as for our true understanding of the universe.Since the beginning of science, there has been an stimulating interaction between scienceand philosophy, though this relationship has not always received the same interest fromboth parts in the last decades. Actually, the English word scientist was first coined byWilliam Whewell in 1834. And is well known that before that, the name used was naturalphilosopher . I will begin by commenting on the need that science has of philosophy, asa group of scientists analyze in a very recent article published in the prestigious journalProceedings of the National Academy of Sciences of the United States of America, [1] as wellas other recent references that talk about the relationship between physics and philosophy[2], and physics and history [3].The authors of [1] argue that philosophy can have an important and productive impact onscience, and provide a series of recommendations to create a better atmosphere and dialoguebetween science and philosophy. After a persuasive discussion on the positive aspects of thisdialogue, they basically conclude that: “Modern science without philosophy will run upagainst a wall: the deluge of data within each field will make interpretation more and moredifficult”. Something that definitely is of the most importance considering our era of bigdata.On similar grounds the physicist Carlo Rovelli in his essay
Physics Needs Philosophy.Philosophy Needs Physics [2] argues in defense of the influence on physics that philosophyhas had, as well as the influence of physics in philosophy. The emphasis is mostly done onthe constructive role to conceptualize through theories after a simple recollection of data,also of much interest in our discussion here.Another thought-provoking article in this context has been written by Matthew Stanleywith the title
Why should physicists study history? [3], where he emphasizes the utility ofknowing the historical aspects and social interactions that affect the evolution of physics.Furthermore, it provides an intellectual flexibility exposing scientists to new ways of thinkingand forcing them to reexamine what is already known. Certainly, a knowledge of history3 igure 1. Artificial intelligence, machine learning and deep learning occupies almost the samespace nowadays, when in the recent past deep learning was simply a small part of machine learningand machine learning a small part of AI. can help us have an enriching reflection on how we know what we know and how it couldbe otherwise.Truly in recent times there have been spectacular developments made by machine learningand deep learning techniques in relation to numerous scientific predictions. These includechaotic systems as well, where is well known that they have prediction problems.Among them, we can highlight the tremendous impact of AlphaGo Zero [4] and Alp-haZero [5] that defeated the world’s best Go players and the best chess computer programs,respectively. The fascinating thing about these programs is that they are able to performvery specific and well defined tasks in a extraordinarily well manner. However, the programsdo not analyze the plays the way humans do. It happens that even the same programmerswho write the computer code do not understand why the programs make certain decisions.Interestingly, in the recently published book
Artificial Intelligence: A Guide for ThinkingHumans [6] Melanie Mitchell discusses, among other captivating issues, the evolution of theartificial intelligence methods during the last decades. And she describes that artificialintelligence, machine learning and deep learning occupies almost the same space nowadays,when in the recent past deep learning was simply a small part of machine learning andmachine learning a small part of AI (See Fig. 1).Enthusiasm is a necessary step to go ahead in any human enterprise, but it is also wise4o see whether some claims are real and true, since enthusiasm sometimes has replaced coolheads. Artificial intelligence has generated, needless to say, much enthusiasm, and this hasalso provoked certain reactions pointing out the flaws of some of the extreme claims, thatwill be discussed later on in this article. Naturally, in spite of all the enthusiasts on AI,there are certainly critics. In particular, Artificial intelligence owes a lot of its smarts toJudea Pearl, who won the Turing Award in 2011. In the 1980s he led efforts that allowedmachines to reason probabilistically. But, now he is one of the field’s sharpest critics. Inhis latest book,
The Book of Why: The New Science of Cause and Effect , [7] he arguesthat artificial intelligence has been handicapped by an incomplete understanding of whatintelligence really is. He has also declared recently that “All the impressive achievements ofdeep learning amount to just curve fitting,” [8]. He also defends the idea to teach machinesto understand why questions, by basically replacing reasoning by association with causalreasoning. This certainly goes to the core question of understanding.The paper is structured as follows. Section 2 is devoted to a general discussion on chaosand predictability, including recent developments on chaos and machine learning and thepredictability derived from the presence of fractal structures in phase space. A further dis-cussion on hetero-chaos, UDV and prediction, including shadowing will be discussed in Sect.3. Section 4 is focused in giving examples about the differences between the two differentnotions of prediction and understanding. In Sect. 5 different ways of understanding arecommented, as well as understanding by machines. Section 6 describes how recent devel-opments of machine learning has brought some authors to deny the value of the scientificmethod, and the reaction of many scientists to this situation. The paper ends emphasiz-ing the conclusions on the importance of keeping the prediction and understanding closetogether and claiming for a constructive dialogue between data-driven models and theoret-ical and conceptual models, as well as the important role that dynamical systems play forunderstanding. 5
I. CHAOS AND PREDICTABILITYA. Chaos and machine learning
Chaos theory has shown that long-term prediction is impossible. The slightest distur-bance of a chaotic system can lead us to be unable to specify the future state with sufficientprecision so that we cannot predict its evolution, what implies an intrinsic situation of un-certainty. In recent work by Ed Ott and collaborators [9, 10] having used machine learningtechniques, they have reported to be able to predict the future evolution of chaotic systemswith further precision than before, by extending the future horizon of the prediction furtherahead to what it could be done with current algorithms.They employed a machine-learning algorithm called reservoir computing to learn thedynamics of a well-known spacetime chaotic dynamical system used to study turbulenceand spatiotemporal chaos, called the Kuramoto-Sivashinsky equation. The important resultlies at the fact that after training the equation with past data, they were able to predict theevolution of the system out to eight Lyapunov times into the future, what basically meanseight times further ahead than previous methods allowed. The Lyapunov time representshow long it takes for two almost-identical states of a chaotic system to exponentially diverge.It represents the inverse of the largest Lyapunov exponent of a dynamical system. Assuch, it typically sets the horizon of predictability. The algorithm knows nothing about thedynamical system itself; it only sees data recorded about its evolving solution. In essence,the results suggest that you can make the predictions with only data, without actuallyknowing the equations.In another research published in [11] by the same group, they showed that improvedpredictions of chaotic systems like the Kuramoto-Sivashinsky equation become possible byhybridizing the data-driven, machine-learning approach and traditional model-based predic-tion, so that accurate predictions have been extended out to twelve Lyapunov times, whatsuggests the importance of integrating both methods.A discussion on the relation on data science and dynamical systems theory is given in[12]. The authors combine ideas from dynamical systems theory and from learning theoryas a way to create a more effective framework to data-driven models for complex systems.They clearly comment that in spite of the tremendous successes of statistical models of6omplex systems, these models are treated as black-boxes with limited insights about thephysics involved and lacking understanding. They describe mathematical techniques forstatistical prediction phenomena usually studied in nonlinear dynamics. In spite of thenumerous mathematical techniques reviewed at the interface of dynamical systems theoryand data science for statistical modeling of dynamical systems, they do not discuss recentdevelopments in deep learning or reservoir computing.A notorious impact has received in a similar context a recent paper [13], where the maingoal has been to solve the chaotic three-body problem using deep neural networks. The mainidea is the use by the authors of an integrator for an n-body problem focusing in the three-body problem. The data they obtain with the integrations are used to train a neural networkso that they are able to obtain and predict trajectories much ahead the previous predictions,and in a very fast manner. The three-body problem is one of the classical unsolved problemsin physics that was formulated by Newton, which basically consists on solving the equationsof motion for three bodies under their own gravitational force. This constitutes also a clas-sical example of chaos in Physics after Poincar´e proved its non-integrability and its chaoticnature already at the end of the 19th century [14, 15]. The authors show that an ensembleof solutions obtained using an arbitrarily precise numerical integrator can be used to train adeep artificial neural network that, over a bounded time interval, provides accurate solutionsat fixed computational cost and up to 100 million times faster than a state-of-the-art solver.The main applications they have in mind are in astrophysics, black-hole systems or galacticdynamics. The success in accurately reproducing the results of the three-body problem, aclassical chaotic system, provides an stimulus for solving other chaotic problems of simi-lar complexity, by basically substituting classical solvers with machine learning algorithmstrained on the underlying physical processes [10, 16].
B. Predictability, attractors and basins
Issues related with chaos and prediction are very important in science and much discussedby many authors in the context not only of dynamical systems but in relation to differentscientific disciplines. In Physics we have laws that determine the time evolution of a givenphysical system commonly modeled by a dynamical system, depending on its parametersand its initial conditions. Precisely with these laws we can predict the future evolution of the7henomena we model. However, not always a good prediction can be done. Oftentimes andin particular when we have nonlinear or chaotic systems predictions are limited, what impliesan intrinsic unpredictability. There are different sources of uncertainty and unpredictabilityin dynamical systems. A small uncertainty in the initial conditions gives rise to a certainunpredictability of the final state as a consequence of the sensitive dependence on initialsconditions, which is a typical hallmark of chaos. Another source of uncertainty are thefractal structures commonly appearing in the basins of attraction in phase space. Chaoticsystems typically present fractal basins.Given a dynamical system possessing only one attractor in a certain region of phasespace, then the final state of the system is uniquely determined for any initial condition.Nevertheless, in most cases dynamical systems may possess more than one attractor in thesame region of phase space, that is, the system is multi-stable, so that to elucidate whichorbits tend to which attractor becomes a key issue. In multi-stable systems with manybasins of attraction, the dynamical system may possess fractal or even Wada boundaries sothat the prediction becomes harder, fundamentally based on the uncertainty associated tothe initial conditions. A thorough review of fractal basins and fractal structures in nonlineardynamics can be found in [17].Much work has been made in the past few years to clarify different aspects of unpre-dictability in dynamical systems [18] [19]. Among other efforts, the new notion of basinentropy [20] provides a new quantitative way to measure the unpredictability of the finalstates by analyzing basins of attraction. A detailed discussion of the issue of predictabilityand basins of attraction appears in the book [18].
III. HETERO-CHAOS, UDV AND PREDICTIONA. Predictability and shadowing
We are typically used to associate chaos with a lack of predictability. However, this isnot always the case. As a matter of fact, any scientist is commonly faced with the keyquestion of knowing how good a numerical prediction of a model is, and for how long isvalid. Quantitative answers to these questions were given by [21] by using the conceptsof shadowing distance, that measures the distance from the shadowing trajectory to the8 igure 2. The shadowing time is the time a numerical trajectory remains close to a true trajectorywhich is called a shadow . The distance to the shadow might be seen as an observational error, withinwhich the computer-generated orbits are considered reliable. At the glitch, the true trajectorydiverges from the computed-generated trajectory. computer-generated trajectory, and shadowing time, that measures the length of true shad-owing trajectories. This shadowing time will be the basis to assess the predictability of ourmodels. This idea is illustrated in Fig. 2.The shadowing property is frequently of great interest in practice. Often, one wants toknow whether there exists a true orbit that closely follows a given computer-generated orbit,where the noise results from roundoff error. A computer simulation of a chaotic system canbe distinct from the true trajectory. It may happen that after a certain time interval thedistance between the true orbit and the computed orbit increases greatly. Nevertheless, it ispossible that a true orbit might shadow the computed trajectory what means that it remainsclose to it for a long computing time.The shadowing time is directly linked to the hyperbolic or nonhyperbolic nature of theorbits. For hyperbolic chaotic systems, where the angle between the stable and unstablemanifolds is away from zero and the phase space is locally spanned by a fixed numberindependent stable and unstable directions, the shadowing is present during long timesand numerical trajectories stay close to the true ones. Even though there might be some9xceptions, it is rather common to accept that for most dynamical systems the shadowingproperty and hyperbolicity are indeed equivalent.Nevertheless, most chaotic attractors of physical interest are not hyperbolic. In mostcases, the attractors fail to be hyperbolic due to homoclinic tangencies, where stable andunstable manifolds intersect tangentially. Another mechanism for the appearance of nonhy-perbolicity is due to the presence of Unstable Dimension Variability (UDV) [22], where thedimension of the unstable and stable tangent spaces is not constant. In these cases, an orbitmay be shadowed, but only for a very short time, and the computed orbit behavior may becompletely different from the true one after this period of time.
B. Hetero-chaos
Some of these previous issues have been recently discussed in the context of the newconcept of hetero-chaos [19]. The presence of hetero-chaos has serious consequences forthe predictability of chaotic systems, that are common in science. As a matter of fact,predictability is more difficult when a chaotic attractor has different regions that are unstablein more directions than in others. This means that arbitrarily close to each point of theattractor there are different periodic points with different unstable dimensions. When thishappens, we say the chaos is heterogeneous, in contrast to homogeneous chaos occurringwhen there is only one unstable dimension, and the phenomenon receives the name of hetero-chaos.A relevant issue to our previous discussion on prediction and shadowing is also derivedfrom hetero-chaos. As it was mentioned earlier, knowing how good a numerical simulationworks and for how long the computed orbit is valid is of the most importance for a scientistusing numerical simulations of a model. The shadowing property makes that our simulationsare realistic, but they become unrealistic when shadowing fails, that may occur when thenumber of unstable directions increases for a trajectory in phase space. This transitionfrom a lower to a higher number of unstable directions has dynamical consequences thatare manifested through the fluctuations around zero of the finite time Lyapunov exponents,something typically happening for higher-dimensional dynamical systems. This is also acommon mechanism for the appearance of nonhyperbolicity, and as a consequence shadowingfails [23]. This poses a serious difficulty for predictability since hetero-chaotic systems cannot10ave the shadowing property while homogeneous chaotic systems usually do have.A short comment on Unstable Dimension Variability (UDV), that occurs when an at-tractor has two periodic orbits that are unstable in different numbers of dimensions. Asa consequence a typical trajectory on the chaotic attractor will visit small neighborhoodsof saddles and repellers in the attractor, so that a common good indicator of the UDV isprecisely the fluctuations about zero of the finite time Lyapunov exponents.As the authors of [19] express, hetero-chaos means that unstable periodic orbits embeddedin a chaotic set have distinct numbers of unstable directions. Accordingly, a trajectory willtypically move in regions with different unstable dimensions, leading to fluctuations aboutzero of some Lyapunov exponents, and affecting the shadowing property and its predictabil-ity. From this point of view, it can be understood as a unifying concept comprising differentphenomena observed in numerical simulations of chaotic dynamical systems and physicalexperiments, such as UDV, on-off intermittency, riddled basins, blowout and bubbling bifur-cations, where common patterns are present. Moreover, they conjectured that UDV almostalways implies hetero-chaos. Since shadowing fails for hetero-chaotic systems, ascertainingwhen a homogeneous chaotic system becomes heterogeneous is paramount when we are dis-cussing predictability. In addition, considering that models with high dimensional chaoticattractors are receiving much more attention by many researchers as models of numerousphysical phenomena, this indicates the relevance of this issue as what concerns predictionof physical systems. Hetero-chaos seems to be important for most physical systems withhigh-dimensional attractors, including weather prediction and climate modelling, what alsoshows a serious limitation to predictability either achieved with ordinary methods or meth-ods derived from artificial intelligence.
IV. PREDICTION AND UNDERSTANDING
When we approach the issue of machine learning and understanding in science, importantquestions arise. As a matter of fact, an excellent ability in prediction could not imply acorrect understanding of the physical processes involved. Prediction and understanding arecertainly two different concepts. Actually, to illustrate this idea we can draw inspirationfrom the history of the planetary motion. We can start with Ptolemy’s methods and hisgeocentric method to predict how planets move in the sky. As is well known, Ptolemy did11 igure 3. The figure shows the trajectories of the planets of the solar system by using the geocentricand the heliocentric model. (Taken from [24]) not know the theory of gravity, not even that the sun occupied the center of the solar system.While it was possible to predict the motions of the planets, it was not known why thesemethods worked. This theory lasted for a quite long time and it was followed by the workof several brilliant scientists. Years later the heliocentric system of Nicolaus Copernicuschanged everything. This is illustrated in the Fig. 3. Later at the dawn of the modern timescame the astronomical observations of Galileo Galilei, that were continued by the work ofJohannes Kepler and his famous laws.And at the end, Isaac Newton found the differential equations that governed the motionof the planets. This was a highly important step, since that contributed to understand whythe planets move. The Universal Law of Gravitation formulated in 1687 [25], allowed tosuccessfully explain the motion of the planets, from Mercury up to Saturn, already knownfrom ancient times. The same idea, that of finding the differential equation, is the key tounderstanding, and as a consequence to predict even the existence of other planets.That was the case of the planet Uranus that was discovered by the British astronomerFrederick William Herschel in 1781. Once it was realised that it was a genuine planet,12urther observations continued the following decades that revealed substantial deviationsfrom the tables based on predictions done by the Newton’s law of universal gravitation.So confident was the scientific community in the goodness of the Newton’s laws, that itwas hypothesised that an unknown body was perturbing the orbit through gravitationalinteraction. The position of this body was predicted in 1846 by the French mathematicianUrbain Le Verrier and finally the planet Neptune was found. Some years later, and aftercareful analyses of its orbit the existence of another new planet was predicted, leading to thediscovery of Pluto by the American astronomer Clyde Tombaugh in 1930. All these successesgave strong confidence in the infallibility of the Newton’s law of universal gravitation. Evenit was postulated the existence of the planet Vulcan [26] between the Sun and Mercury, thatwould explain the perihelion precession of Mercury, but in this case the problem was solvedby changing the Newton gravitational law by the Einstein General Relativity theory.In science the notion of understanding leads us to a similar pattern. Reducing a com-plicated phenomenon to a simple set of rules or principles, implies an understanding of theconsidered phenomenon. Machines make their predictions much better than us. But theyare not able to explain why. Certainly, artificial intelligence techniques are contributingand will contribute much in science. The predictions can be excellent, but the key issue iswhether we can understand what is happening. Prediction without understanding affects thevery notion and sense of scientific knowledge as we know it today. Needless to say, there areinnumerable unknowns, and all this discussion is not simple at all. However, the importantissue is to elucidate the authentic meaning of science. We understand science as the abilityto know, understand and predict. Keeping only the predictive capacity is not enough. Ifwe forget understanding, then we could conclude that machines could successfully developscientific work by themselves.This tension between prediction and understanding has been permanent in the history ofscience, as is the case in fields where there exist a large amount of data such as genomics,computational biology, economy and finance. What is usually missing is understanding.But not always this tension has been derived by data. As an example, I will mention adiscussion made by Alex Broadbent in his article
Prediction, Understanding, and Medicine [27], where he argues that understanding is the core intellectual competence of medicine andas a practical consequence comes the ability to make predictions about health and disease.There are different ways of doing science, or characteristics and aspects of science that are13ore relevant in some disciplines than others. The following characteristics might help toclassify different scientific disciplines, though in some sense all of them might be necessary. • UnderstandingThis attempts mainly to the formulation of important questions in science. Physicsis one of them, where after relevant questions we expect to have the answers to thewhy of things. But of course, the same pattern affects to natural sciences wheneverwe ask deep questions that we want to answer. Do neutrinos have mass? And if so,why? Why we sleep? Why stars shine? As an answer of these questions we genuinelylook for a clear understanding of how things work the way they do. • PredictionThis is another key aspect of the scientific endeavour. We want to know what willhappen. According to what we know we want to predict something unknown. Thischaracteristic is so fundamental, that even it could be argued that if you cannottruly predict a phenomenon you cannot consider it under a scientific discipline. Andconsequently if you cannot predict you cannot understand. We can predict solar cycles,failures in engineering designs, and natural disasters. The predictive power of scienceis one of the driving forces of progress and development. • DescriptionClearly, this is another important aspect of science that not necessarily needs logicaldeductions of the same nature as the why questions. It concerns mainly with an-swering what and how questions. There are certain scientific disciplines where thischaracteristic is more common than others. What is consciousness? How did life be-gin? How a Lyapunov exponent evolve with time? Or merely consider a taxonomy ofsome concepts or natural objects, a mere description of natural phenomena withoutgoing any further.We can learn physics and predict in physics or other sciences through machine learning,but we still do not know if machines can actually understand. Actually, according to somephilosophers and neuroscientists we do not even know what it means to understand. Thereis another issue that we should discuss here. AI is not able to make interpretations. Unques-tionably, they are highly sophisticated optimization algorithms that constantly feed on data14ntil they find enough patterns to make their own predictions. Nevertheless, these patternsare purely empirical laws; they have no theoretical basis or physical interpretation, such asKepler’s or Maxwell’s laws.Precisely in this context is worth to mention again the critical view of certain develop-ments of AI that are mainly based on data done by Judea Pearl in [7], where he affirms: “Incertain circles there is an almost religious faith that we can find the answers to these ques-tions in the data itself, if only we are sufficiently clever at data mining. However, readers ofthis book will know that this hype is likely to be misguided. The questions I have just askedare all causal, and causal questions can never be answered from data alone. They requireus to formulate a model of the process that generates the data, or at least some aspects ofthat process. Anytime you see a paper or a study that analyzes the data in a model-freeway, you can be certain that the output of the study will merely summarize, and perhapstransform, but not interpret the data”.
V. DIFFERENT FORMS OF UNDERSTANDING
We can learn and predict in science through machine learning, but we still do not know ifit can be understood. Then, a key question arises: What does understanding mean?. Thisis precisely the question that the neuroscientist Gilles Laurent [28] raises himself in a shortessay, where he highlights the importance of the power of explanation of theory, since in orderto understand the brain it is necessary to understand a system of interacting elements, theneurons, and how their interactions and structure generate functions. Actually, he stronglyemphasizes the role played by the theory of dynamical systems that definitely contribute tohelp us to have a mental and mathematical conceptualization, and ultimately to understand.As a matter of fact, even though philosophers have been worried about the meaning ofunderstanding, it seems that it is not something very clear according to the philosopher R.L. Franklin in his article
On Understanding [29], when he dares to write: ““Understand” isa word we understand as well as any, but we do not understand philosophically what it isto understand.”In any case, in his discussion on the subject, he points out that the notion of understand-ing is linked to the capacity to explain something, and the explanation is often linked to thecausal law that relates what we observe. 15 igure 4.
The elephant and the six blind men. (Cartoon originally copyrighted by theauthors of [30]; G. Renee Guzlas, artist). Taken from [30].
By analogy to the story of the blind men and the elephant (Fig. 4), each scientist hasa strong knowledge and supposedly lots of data on a particular area of the elephant, butno one has the knowledge that in reality what they are observing is an elephant. No oneof these observations can provide a global view of the unifying concept. The story is welldescribed in the poem
The Blind Men and the Elephant of the American poet John GodfreySaxe (1816-1887) that can be found in [30].Another interesting question related to understanding and prediction is the issue of un-derstanding machines, which has been investigated by some researchers [31, 32], thoughapparently not so many. They consider that for the term ”Understand” to be useful in thefield of AI, it must refer to something measurable. Among the criteria to consider, theymention: (1) to predict the behavior of the phenomenon, (2) to achieve the objectives re-garding the phenomenon, (3) to explain the phenomenon and (4) to create or recreate thephenomenon. In any case the notion of understanding by machines is by no means a simple16roblem.
VI. MACHINE LEARNING AND SCIENTIFIC METHOD
Recent advances in machine learning and the associated hype behind has provoked theappearance of AI enthusiasts and skeptics. There are some enthusiasts of the AI thathave dared to announce even the end of the scientific method as we know it today [33].Other enthusiasts pretend to extract predictions and even natural laws by simply usingexperimental data [34] or even creating machines for scientific discovery able to win a Nobelprize [35]. Not to mention the recent book by Max Tegmark on being human in the age ofArtificial Intelligence [36].A few years ago, a provocative article published by Chris Anderson, editor in chief of themagazine Wired, with the title
The End of Theory: The Data Deluge Makes the ScientificMethod Obsolete [33] provoked a large discussion among scientists. In his article Andersonargued that it was enough to establish correlations by using enough data that eventuallycould be analyzed without any need for models or hypothesis. Basically, by throwing thedata into the huge computers was enough letting only the algorithms to find statisticalpatterns.Others argue that in some occasions there is a trade-off where we can renounce under-standing, since obviously is more complicated than simply compute something and makesome quick predictions.Gary Smith in his recent book
The AI Delusion [37] encourages scepticism about artificialintelligence and the blind trust we put in it. In a certain sense, his book represents a responseto the philosophy represented by the article of Anderson, because unfortunately too manypeople have been attracted by these claims. He expresses it by explicitly writing: “Far toomany intelligent and well-meaning people believe that number-crunching is enough. We donot need to understand the world. We do not need theories. It is enough to find patternsin data. Computers are really good at that, so we should turn our decision-making over tocomputers.”In reality, he explains with numerous examples why we should not be intimidated intothinking that computers are infallible, that data-mining is knowledge discovery, and thatblack boxes should be trusted, emphasizing the importance of human reasoning as funda-17entally different from artificial intelligence, which is why is needed more than ever.In spite of the enthusiasts denying the scientific method, there are voices that oppose thisviewpoint and mark the limits of machine prediction. Among them we can cite [38, 43–45].In particular in [38] the authors critically assess the claim that bigger data leads tobigger predictions. They use analogies and ideas from atmospheric sciences and essentiallyconclude that a compromise between modelling and quantitative analysis is the best strategyfor forecasting, as already anticipated long ago by Lewis Fry Richardson and John vonNeumann, as pioneers in numerical weather prediction. They highlight that too many datado not make necessarily more accurate predictions, as is well known in weather forecasts.They also emphasize the important role played by the high dimension of systems with a highenough number of degrees of freedom versus the intrinsic role of chaos as a limiting factor topredictability in low dimensional systems. All this is nicely described in great detail in [39]and in other recent and enlightening papers by Angelo Vulpiani and collaborators [40–42].Similar ideas have been also recently defended by Mark Buchanan [43] as well, arguing thatthe limits on the predictive accuracy of big data is derived from the theory of dynamicalsystems in the context of high-dimensional systems, the case in many typically complexproblems like the weather and other real-world applications. This same idea was alreadycommented when the new notion of hetero-chaos was discussed in Sect. 3.Analogously Jim Cruthfield [44] argues in a fantastic manner on the importance of com-bining data, theory and computations, and intuition.A defense of the scientific method versus the mere analysis of data is well documentedin [45] in the context of biological and medical sciences. The authors clearly point outthe weaknesses of pure big data approaches that cannot provide a true understanding andconceptual vision of the physical processes involved and subsequent applications. Theymake a strong defense of the theory as a guide to experimental design and to producereliable predictive models and conceptual knowledge and understanding. They also remarkthe importance for biology and bioinformatics students to be trained to understand thetheory of dynamical systems that are needed to describe and model biological dynamicalprocesses.Their skepticism brings them to affirm “More attention needs to be given to theory if themany attempts at integrating computation, big data and experiment are to provide usefulknowledge. A substantial portion of funding used to gather and process data should be18 igure 5. The figure illustrates one of the dreams of AI, a robot attempting to do maths. We arevery far from that. diverted towards efforts to discern the laws of biology.” And one of the authors rhetoricallyhad expressed it with the following sentence: “Does anyone really believe that data miningcould produce the general theory of relativity?” [47]. In another recent article [46] SauroSucci and Peter Coveney argue that some extravagant claims of big data need to be revisedin view of some obstacles such as nonlinearity, non-locality, and high dimensionality derivedfrom the science of complex systems, defending a hybrid method where data and theorycome together, and somehow improving the scientific method as we know it.In an interesting document published by edge.org and edited by John Brockman in 2015,a key question was asked to numerous scientists and artists including Nobel Prize winnersabout
What do you think about machines that think? [48]. Nearly two hundred responses areincluded, where one can see all kind of responses, from enthusiasts to skeptics and between.I have selected here the response of a well-known physicist, Freeman Dyson, that is concise,surprising and with a bit of humour: ”I could be wrong: I do not believe that machinesthat think exist, or that they are likely to exist in the foreseeable future. If I am wrong, asI often am, any thoughts I might have about the question are irrelevant. If I am right, thenthe whole question is irrelevant.” Figure 5 shows a machine doing mathematics.In the context of geosciences and weather prediction is worth to mention here a fascinating19ecent book written by an atmospheric physicist, Shaun Lovejoy [49], who besides leads thenew discipline of nonlinear geophysics [50]. He strongly emphasizes the idea that concepts ofnonlinear geophysics, mainly derived from nonlinear dynamics, fractal geometry and complexsystems theory, can provide a rational basis for the statistics and models of natural systems,making our understanding of the world more complete. Furthermore, he makes a detaileddiscussion on the limits of predictability either by using the standard deterministic chaoticmodels or the lesser known stochastic models in weather predictions.Of great interest on our discussion on prediction and understanding are the insightfulcomments on the current role played by theory and quick numerical results in atmosphericscience, and how this is affecting understanding. He writes: ”Theory of any kind wasincreasingly seen as superfluous; it was either irrelevant or a luxury that could no longer beafforded. Any and all atmospheric questions were answered using the now- standard tools:NWPs and GCMs. Unfortunately, these models are massive constructs built by teams ofscientists spanning generations. They were already “black boxes,” and even when theyanswered questions, they did not deliver understanding. Atmospheric science was graduallybeing transformed from an effort at comprehending the atmosphere to one of imitating itnumerically (i.e., into a purely applied field). New areas— such as the climate— were beingtotally driven by applications and technology: climate change and computers. In this bravenew world, few felt the need or had the resources to tackle basic scientific problems.”Likewise in the context of geosciences, a nice perspective article was recently publishedin Nature [51] where the authors defend similar ideas, focusing mainly in geoscientific data,and analysing the relationship between deep learning and process understanding in data-driven Earth system science. They review in a superb manner the developments of machinelearning in geosciences, and discussed that there are certain predictive problems related toforecasting extreme events such as floods or fires or predicting in the biosphere, where notsubstantial advances have been seen in the past few years, in spite of the deluge of datathat we are accumulating nowadays. In few words, there has not been much progress inprediction even though the capacity to accumulate more data has tremendously increased.They unreservedly defend that the most promising and challenging future would be togain understanding in addition to optimizing prediction, so that they propose an integra-tion of machine learning with physical modelling. The idea is that data-driven machinelearning approaches will successfully complement and enrich the physical modelling, featur-20ng a conceptualized and interpretable understanding. Precisely one of the challenges theyestablish for deep learning methods is the need for understanding and for what they callinterpretability, and causal discovery from observational data. Definitely, machine learningmethods provide an excellent improvement of classification and prediction, but it does nothelp much to scientific understanding.
VII. CONCLUSIONS
We are witnessing an era in which big data and machine learning and deep learning tech-niques will contribute, as they are already doing, in a very important way to the advancementof science, whether applied or basic. Numerous examples in many different disciplines haveillustrated the powerful predictive capabilities of these techniques, including examples inchaotic dynamics. All this has created an enormous hype on the new possibilities, and fur-ther creating very high expectations for the future. Likewise, in the face of some perhapsexaggerated positions about the potential of these techniques, a reaction has been provokedin the scientific community by pointing out the flaws of these positions, as well as somelimits, sometimes affecting the core of the scientific method.As a result of these efforts, it can be concluded that we cannot do without the role ofmodeling, conceptualization and other tools provided by theoretical science and scientificmethod, when one of the important goals is understanding. Prediction and understandingare two fundamental ideas that should guide us about the progress of science.I want to emphasize again here the importance of the dynamical systems for the processof understanding and to get insights about the physical and biological processes involved inour observations and describe and model them.There is no doubt that the path of the future of science will be marked by a constructivedialogue between big data and big theory. Data science has much to contribute, but withouttheoretical and conceptual models we cannot understand.Despite all the above, there are some who think that one day the machines will be ableto carry out all the activities that the human brain is capable of doing. If we extend itto scientific creation, as well as to the possibility of finding new laws of physics and to thesame elaboration of scientific theories, we could conclude that man’s contribution to sciencewould have ended. No matter how much excitement the machine learning techniques are21reating, it seems that we are very far from that and, therefore, we have as humans muchfuture ahead to discover, understand and predict.
ACKNOWLEDGMENTS
I acknowledge an interesting encounter and a further discussion with Mark Barthelemy,after which he encouraged me to write this article. Some of these ideas were exposed ina meeting around the question ”Is it possible that Artificial Intelligence generate scientifictheories?” that took place at the Spanish Royal Academy of Sciences for what I wish toacknowledge Jos´e Mar´ıa Fuster and Jes´us M. Sanz-Serna. In addition, I thank AngeloVulpiani and Peter Coveney for providing me some new interesting references. This workwas supported by the Spanish State Research Agency (AEI) and the European RegionalDevelopment Fund (ERDF) under Project No. FIS2016-76883-P.
CONFLICT OF INTEREST
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