Естественная индукция вместо естественного отбора
Интересная статья. Хотя и ожидаемая.
Физическая адаптация - без естественного отбора, в неорганических системах.
Подразумеваются вязкоупругие взимодействия. В таких системах под действием среды система обучается и постепенно адаптируется к воздействиям. Обучается - то есть учитывает прошлый опыт воздействий. Авторы называли это естественной индукцией
Buckley, C.L.; Lewens, T.; Levin, M.; Millidge, B.; Tschantz, A.; Watson, R.A. Natural Induction: Spontaneous Adaptive Organisation without Natural Selection. Entropy 2024, 26, 765. https://doi.org/ 10.3390/e26090765
Abstract: Evolution by natural selection is believed to be the only possible source of spontaneous adaptive organisation in the natural world. This places strict limits on the kinds of systems that can exhibit adaptation spontaneously, i.e., without design. Physical systems can show some properties relevant to adaptation without natural selection or design. (1) The relaxation, or local energy minimi-sation, of a physical system constitutes a natural form of optimisation insomuch as it f i nds locally optimal solutions to the frustrated forces acting on it or between its components. (2) When internal structure ‘gives way’ or accommodates a pattern of forcing on a system, this constitutes learning insomuch, as it can store, recall, and generalise past conf i gurations. Both these effects are quite natural and general, but in themselves insuff i cient to constitute non-trivial adaptation. However, here we show that the recurrent interaction of physical optimisation and physical learning together results in signif i cant spontaneous adaptive organisation. We call this adaptation by natural induction.
The effect occurs in dynamical systems described by a network of viscoelastic connections subject to occasional disturbances. When the internal structure of such a system accommodates slowly across many disturbances and relaxations, it spontaneously learns to preferentially visit solutions of increasingly greater quality (exceptionally low energy). We show that adaptation by natural induction thus produces network organisations that improve problem-solving competency with experience (without supervised training or system-level reward). We note that the conditions for adaptation by natural induction, and its adaptive competency, are different from those of natural selection. We therefore suggest that natural selection is not the only possible source of spontaneous adaptive organisation in the natural world
1.4. What Is Adaptation?
The quote from Darwin at the beginning of the Introduction Section draws our atten-tion to the exquisite and particular organisation of relationships between the parts of a system. (We would not use the word “perfected” in modern thinking, but there has to be something special about the arrangement of parts).
A system that is deformed or distorted by the forcing applied to it, or simply relaxes under the stresses it experiences, is not (normally) what we mean by adaptation (regardless of whether that deformation involves observable variables or internal structure). Even learning a function or a generalised class of patterns, though it might be analogous to simple forms of cognitive learning, is not the same sense of adaptation that we seem to refer to in biological evolution. Accordingly, the physical optimisation and physical learning examples above have not been claimed to be new sources of adaptation, although learning principles can deepen our understanding of adaptation by natural selection [34,49]. Thus, what do we really mean by adaptation and how does it relate to learning and optimisation?
Def i nitions of adaptation and adaptive organisation that are tied to natural selection, organismic reproduction, or Darwinian f i tness are not useful for answering this question.
The ‘appearance of design’ concept [4] is closer to what we need because it is not necessarily tied to Darwinian processes, but it is not easily quantif i able. In fact, def i ning exactly what we mean by adaptation is an open problem in biology. Note that Darwinian f i tness, the number of offspring produced, (or inclusive f i tness, for that matter), is part of an explanation of biological adaptation, not the explanandum itself (and besides, the relationship between natural selection and maximisation of these quantities is notoriously fraught [50,51]). It is not a high number of offspring that needs explaining, and believing that the ‘goodness of f i t’ between an organism and environment is def i ned as reproductive output fails to engage with the special properties of the biological organisations that facilitate this. To assess whether there can be any sources of adaptation other than natural selection, it is necessary to let go of the idea that survival and reproduction are the ultimate assessors of adaptation. Whilst these are undeniably important to biological organisms, they are part of the Darwinian explanation for how adaptation happens. In contrast, the goodness of f i t between an organism and its environment and the appearance of design are much more general notions and more conceptually apt to understanding what adaptation is. However, the former is too easily satisf i ed in a physical sense (e.g., an imprint in clay) and the latter is rather subjective.
How can we make a general notion of adaptation more objective and quantif i able? Our approach is to def i ne adaptation as a process that provides an optimization or constraint-solving capability that is non-trivial, i.e., superior to a local hill-climber, and more specif-ically, one that improves its problem-solving capability with experience, i.e., learns to f i nd better solutions (without assuming that it f i nds optimal or perfect solutions). In evolutionary computation, the adaptive capability of natural selection is demonstrated through its ability to f i nd good solutions to optimisation problems [52]. Despite the con-troversy attached to viewing adaptation as a problem-solving process, we f i nd it useful to conceptualise adaptation in this way [53] A less value-laden conception of evolution as a dynamical process, without a problem to be solved and without a pre-existing niche to be occupied, is acknowledged [54,55]. However, if adaptation is construed as merely ‘whatever happens as a result of natural selection’, for example, then it remains tied to natural selection. Adaptation needs to address a design-like property (independent of natural selection), and problem-solving is one way to characterise this. At the least, a process that can provide a non-trivial problem-solving competency is a high bar for assess-ing adaptation. To formalise this, we begin with a notion of searching for points in some conf i guration space that optimises some quantity, i.e., a process of function optimisation def i ned over some given space of variables [50,53] (Figure 1A). An optimisation ability does not require that a process f i nds the globally optimal (best possible) conf i guration [50];
this would be too strict. Natural selection, for example, does not provide optimal solutions.
An adaptation built by natural selection need only be a good solution to a problem, not an optimal one. Conversely, neither do we want to def i ne adaptation in a manner that is trivially satisf i ed [47]. If we adopt a very simple notion of problem-solving, such as that provided by a hill-climbing or local gradient ascent process, that merely f i nds a local optimum of an objective function, it would be trivially satisf i ed by a physical system as discussed-and therefore not as satisfying as a concept of adaptation. The behaviour of any physical system that can be described by the local minimisation of an energy function can be interpreted as an optimisation process in the limited sense that it f i nds locally optimal solutions to its implicit energy-minimisation ’problem’ [42] (Figure 1A).
It is not very useful to adopt a def i nition of adaptation that is satisf i ed by a literal ball rolling down a literal hill (even if it is functionally equivalent to the process described by natural selection, see Discussion). Accordingly, the local optimisation behaviour of physical systems is not suff i cient to constitute adaptation. So, instead, we seek a non-trivial problem-solving competency-not necessarily optimal, but not trivial, either [47]. Physical systems can do better than the trivial case. Simulated annealing is a famous example of a computational optimisation process that f i nds solutions better than local optima [56], and actual annealing (e.g., in a cooling metal) occurs spontaneously. This shows that the spectrum of possibilities for natural adaptation is non-empty-but still, we aspire to more than cooling metal as a natural example of adaptation. In what follows, we show that the ability of a physical system to learn offers the possibility of a physical system that learns to solve problems better with experience. Put differently, we classify the ordinary (f i rst-order) local energy minimisation behaviour of physical systems as trivial, but optimisation that improves with experience or second-order optimisation is algorithmically interesting.
In the previous examples of physical optimisation, the quality of a solution improves over a given state trajectory, but it only reaches a local optimum and its ability to f i nd solutions of good quality does not change with experience. Lower-energy states, consti-tuting better solutions to the frustrated state variables in the system, may exist but are not obtained. The optimisation ability does not change over time-restarting the system results in the same locally optimal outcome if it restarts from the same initial position. In the examples of physical learning, a system incrementally improves the f i t of an (implicit) internal model to a conventional learning task such as classif i cation or representing a function, e.g., [29]. A problem-solving ability is not demonstrated (except in the same sense of f i nding a locally optimal f i t of model parameters to the data). Both these behaviours (learning and optimisation) are very natural and do not involve particularly limiting as-sumptions about the system. In both cases the behaviour is determined by local energy minimisation of the same energy function-the only difference is which variables give way under stress and which are held constant. In optimisation, the state gives way to the problem, and in learning, the model gives way to the data (Figure 1); however, so far we have considered these two processes in isolation: the outputs of the learning process do not affect the optimisation landscape, and the outputs of the optimisation process do not affect the learning data. This limiting assumption means that learning responds to data that is given by external conditions (f i xed training data), and optimisation responds to a landscape that is given by the problem def i nition (f i xed problem). These effects have been studied separately, but in general dynamical systems both these effects will happen at the same time. What happens to the structure and dynamics of the system in the general case where there is feedback between the two is much more interesting.
4. Discussion
4.1. The Relationship between Natural Induction and Natural Selection Adaptation by natural induction and adaptation by natural selection share a number of features: Both involve the incremental accumulation of small changes over time, both can result in the increased f i t of an adaptive system to a system of constraints, and both involve simple gradient-following principles. They are also different in the mechanisms that they require (their necessary and suff i cient conditions), how they work (their algorithmic princi-ples) and, consequently, in their adaptive competencies. Whereas natural selection depends on the differential survival and/or reproduction of entities, natural induction operates by the differential easing of frustrated relationships between entities (The accommodation of internal connections to a state conf i guration has some similarity to the over-production of neural connections and their differential retention or reinforcement by ‘selective stabiliza-tion’, which may also result in increased response to or memory of activation patterns. The emphasis of natural induction on the differential easing of frustrated interactions agrees with an emphasis on network integration or ‘survival of the f i tted’, in contrast to ‘survival of the f i ttest’. However, natural induction is a physical model without any population or selection process, whereas selective stabilisation depends on an over-produced population of connections, and in terms of algorithmic competence, natural induction demonstrates an increase in adaptative capabilities). Moreover, the raison d’etre of selection as a theory of biological evolution is to avoid dependence on variation that is directed toward adaptive outcomes, whereas natural induction exploits the fact that variation directed toward easing frustrated interactions is normal in physical systems and adaptively signif i cant (Darwin suggested that some variation was developmentally environmentally directed but was not specif i c about its adaptive signif i cance. The nature of developmental bias and pheno-typic plasticity, and the potential of these and other factors to inf l uence genetic evolution adaptively, is an active topic [6,78]. These mechanistic differences are important, but it is perhaps the difference in their algorithmic principles and adaptive competencies that is more important. Even though both include gradient-following principles, they are different algorithms [79], and this is clear because they have different problem-solving capabilities.
A common interpretation of adaptation by natural selection, i.e., characterised as a pop-ulation following f i tness gradients to a local peak in a f i tness landscape [73,80], aligns well with local gradient-following principles of (f i rst-order) physical optimization. It is acknowl-edged that biological evolution is not necessarily a good optimiser [54], the idea of natural selection climbing gradients in a static f i tness landscape is a serious over-simplif i cation [55], identifying any quantity that natural selection maximises is problematic [50,51], and the conception of natural selection as a problem-solving process has been criticised [51,54,55,81].
However, these are mostly arguments that weaken the interpretation of natural selection as an optimisation process, i.e., natural selection is, at least sometimes, less effective at optimi-sation than a local gradient process. Accordingly, doing better than local optimisation is a conservative criterion for adaptation.
At a suitable level of abstraction, both natural selection and physical optimisation can be described as processes that follow local gradients to a local optimum. Clearly, though, they are mechanistically different ways of implementing this process. Specif i cally, natu-ral selection can be described as a process of random generation and selective retention, whereas in the Newtonian model of a physical system, a ball, for example, is deterministi-cally caused to move in a directional fashion by the reaction to the slope, i.e., to roll downhill, not uphill. No population of balls, random variation nor selection process need be involved.
It is also possible, however, to think about a statistical mechanics process where the position of the ball is represented by a probability distribution of possible future positions, which is updated to amplify positions that are lower in energy compared to those that are higher in energy. So, does it really matter whether it is a statistical-mechanics (variation and selection) process or a Newtonian (directed learning) process? If the gradient-following outcome is the same, does the mechanism matter?
In some contexts, the mechanism seems to matter a lot to biological thought; Evolu-tion by natural selection can only occur if there is a population, suitable variation, and selection-anything else is not natural selection. On the other hand, it is common to equate evolutionary adaptation with a hill-climbing process, and to the extent that the change in an evolving population is non-arbitrary, this level of functional equivalence seems to capture its adaptive competence. However, if that is true, it has a curious implication;
Although it is traditional for evolutionary thought to consider processes that go uphill (in a f i tness landscape), whereas physical models go downhill (in an energy function), this does not make either one cleverer, i.e., a more effective optimiser, than the other (regardless of whether they are implemented in a statistical manner or a Newtonian manner). That would suggest that a literal ball rolling down a literal hill would also constitute adaptation. If local hill-climbing is suff i cient to produce the biological adaptation we observe, this would mean that the genius of Darwin’s theory is just that it provides a hill-climbing process capable of operating in the appropriate organismic ‘design space’. Alternatively, the mechanistic details of evolution by natural selection, or the context in which it occurs, may matter to its adaptive competence-making it different from a hill climber [79]. It seems likely that the details do matter, but to the extent that evolution by natural selection is formally characterised as a simple (substrate-independent) hill-climbing process, it excludes these potentially important aspects of the actual biological process.
The models presented here can be interpreted as literal physical systems, but they can also be interpreted as models that stand in for other natural optimisation processes, includ-ing those involving the local gradient-following capacity of evolution by natural selection.
Adaptation by natural induction is instantiated as a Newtonian process in the models we have illustrated-with forces and (directional) reactions rather than a statistical mechanics process (random variation and selection) and because they are different mechanisms, it means that natural induction has different necessary and suff i cient conditions to natural selection, and may apply in cases where natural selection does not. However, in terms of adaptive competence, that is not the important difference between natural induction and natural selection. The problem-solving competence of natural induction is not the same as that of natural selection because they are different algorithms (not because they are different mechanisms for implementing the same algorithm). Adaptation by natural induction f i nds better solutions than a local gradient-following optimisation process, a.k.a. a hill-climber. If biological evolution is algorithmically equivalent to a hill-climber (f i rst-order optimisation only), then its adaptive competence is inferior to adaptation by natural induction. Conversely, if biological evolution is a more sophisticated optimiser than a hill climber, then the substrate-independent algorithm of random variation and selection does not describe it.
It might be appropriate to conceive biological evolution as a simple gradient process when natural selection acts on a simple vector of genes or vector of phenotypic traits individually determined by corresponding genes in a one-to-one fashion [66]. In this case, the action of evolution by natural selection is analogous to the bed of clay-a univariate model; that is, each selective coeff i cient is responsible for the change in frequencies of alleles at one locus. Other work has illustrated, however, that an evolutionary process operating on heritable variation in the connections of a dynamical network can exhibit the same kind of adaptive competence as natural induction. This includes gene-regulation networks [35,48,66]. In this case, the gene network constitutes a dynamical developmental process that generates phenotypes indirectly (and ‘disturbances’ are provided by a lifecycle that resets developmental states to a neutral, undifferentiated, epigenetic state). This gives it the possibility of representing relational interactions between traits (i.e., pleiotropic inter-actions in the genotype-phenotype map of such a developmental process may constitute an associative model of selected phenotypes [34]). Previous work shows that under these circumstances, the outcome is superior to local gradient optimisation [66]; that is, evolution by natural selection acting on the parameters of a developmental process (second-order) can solve problems that evolution by natural selection acting on a directly encoded phenotype (f i rst-order) cannot.
One interpretation of this is that natural selection can exhibit the same adaptive com-petence as natural induction, after all. However, adaptation by natural induction is not the algorithm that Darwin described (evolution by natural selection does not describe a process that provides model induction, a network of viscoelastic connections, an ability to learn to adapt better with experience, nor the signif i cance of ‘pulse’ or disturbances, amongst other things). So, even though the basic gradient-following process might, in biological cases, be provided by a variation and selection process, in cases where natural selection operates on the parameters of a developmental process constituted by a network of interactions [35,48,66], it might be more correct to attribute the adaptive competence to natural induction and not to natural selection. After all, whereas the theory of evolu-tion by natural selection focuses on the mechanism of variation and selection, this paper demonstrates (using purely Newtonian processes) that this superior adaptive competence arises from the dynamical feedback between model induction and optimisation, and not from random variation and selection. Whereas natural selection depends on the differential survival and reproduction of things, natural induction fundamentally depends on the differential easing of frustrated relationships between things. Although adaptation by natural induction is therefore fully compatible with a Darwinian model of evolutionary change (in suitable network contexts), these are different adaptive algorithms with different necessary conditions, different algorithmic principles, and different adaptive competencies.
We also note that the relationship between evolution and learning has been recognised and developed by many. At a suitable level of abstraction, evolution and some learning methods appear to be the same algorithm (i.e., ‘trial and error’ plus reinforcement equates to random variation and selection). Both can be understood as processes that optimise a function by, to a f i rst approximation, following local gradients. It has often been noted that reinforcement learning and evolution by natural selection are closely analogous [34,82], and indeed, the replicator equation (an abstraction of biological evolution under natural selection) and Bayesian updating (a learning optimisation process) have been shown to be formally equivalent [83,84]. See also [17,48,49,85-87] for the relationship between learning and evolution. These works expand and deepen our understanding of the adaptation provided by natural selection; however, note that adaptation by natural induction involves a two-way feedback between an optimisation process and an inductive learning process- the latter on its own is simply an optimisation process in model parameters, and not suff i cient to demonstrate an improvement in problem-solving competency.
The results in this paper thus demonstrate that a dynamical system described by a network of viscoelastic connections, and subject to occasional disturbances, exhibits adaptation in the more stringent sense of learning to optimise better with experience or improving its problem-solving competency over time-and this is not the same mechanism, algorithm, or competency as natural selection.
Where might networks of suitable connections occur naturally? By far the most fa-miliar examples of viscoelastic networks are in fact biological ones. Ecological networks, protein networks, cytoskeletal networks, metabolic networks, bio-electrical networks, social networks, and the biosphere as a whole are all networks at least partially characterised by linkages that are likely to give way under stress and are subject to at least occasional perturbations. Although these networks all involve biological individuals and materials, most of them are not (always) evolutionary units, so natural selection does not straightfor-wardly apply. This suggests that the interaction of natural selection and natural induction may be complex and possibly widespread. Outside of systems that we already recognise as biological, another obvious candidate where natural induction may be important is the origins of life and origins of evolution [88,89]. To the extent that a pre-biotic chemical network has an internal conformation structure that gives way under stress, we speculate that it has the potential to induce a model of its past experience that can anticipate and generalise without having properties suff i cient to be a bonaf i de evolutionary unit.
Whilst we wish to make the case that the conditions for this are quite natural and not onerous (i.e., do not require selection or design), we do not claim that these conditions are ubiquitous or even frequently or commonly met. Bear in mind that the conditions for evolution by natural selection, namely self-replicating systems with heritable variation in reproductive success, are hardly ubiquitous in the physical world (even though they may be realised in various substrates [90,91]) and their origin is not known. Instead, we claim that the contrary assumption, that natural selection is the only possible naturally occurring mechanism of spontaneous adaptation, is not correct. Who would make such a claim? Actually, this assumption is quite widely adopted, usually implicitly, with very wide-reaching and important implications [4].
Reasoning in many biological domains often depends on this assumption implicitly or explicitly. For example, the assertion that loose ecological communities (or the biosphere as a whole) cannot possibly be adapted because they are not evolutionary units (or members of a population) is an argument that presupposes natural selection to be the only possible source of adaptation. Biological systems are replete with dynamical systems described by networks in many different substrates (e.g., chemical reaction networks, metabolic networks, protein networks, gene-regulatory networks, bio-electric networks, ecological networks, and social networks). Linkages in many networks give way under stress (perfect elasticity is an idealisation) and biological systems are rarely isolated from shocks or disturbances. To the extent that these come together in natural systems, natural induction has the potential to provide a mechanism of spontaneous adaptation that is relevant to biological systems in many domains. This includes those where natural selection does not apply and those where natural selection also applies. How natural induction interacts with natural selection, and the broader implications for biological thought and the origins of adaptive complexity, will be analysed in future work.
Conclusions It has been argued strongly, and widely assumed across biological thinking, that natu-ral selection is the only possible mechanism capable of producing spontaneous adaptation in natural systems. Here we show that this assumption is false; adaptive organisation occurs spontaneously in physical systems with suitable natural properties, through an effect we call adaptation by natural induction. This occurs in dynamical systems described by a network of viscoelastic connections subject to disturbances. A viscoelastic connection is simply one that ‘gives way’ slightly under stress, which is a natural property of many physical materials, biological networks, and complex systems more generally. When dis-turbances cause the system to visit a distribution of locally optimal solutions, the changes to the connections in the network learn a generalised associative model of the local so-lutions visited, which causes the system to adapt in the rigorous sense of improving its problem-solving competency; that is, it provides optimisation ability that is superior to local optimisation and improves with experience. The simplicity of natural induction, and its necessary and suff i cient conditions, offer a solution to both the chicken and the egg problems-i.e., natural selection is not involved at run time nor in the construction or setup of the system. This has important implications for our understanding of biological evolution and adaptation in other complex systems-not least that adaptation can occur spontaneously in systems that are not units of selection.
Физическая адаптация - без естественного отбора, в неорганических системах.
Подразумеваются вязкоупругие взимодействия. В таких системах под действием среды система обучается и постепенно адаптируется к воздействиям. Обучается - то есть учитывает прошлый опыт воздействий. Авторы называли это естественной индукцией
Buckley, C.L.; Lewens, T.; Levin, M.; Millidge, B.; Tschantz, A.; Watson, R.A. Natural Induction: Spontaneous Adaptive Organisation without Natural Selection. Entropy 2024, 26, 765. https://doi.org/ 10.3390/e26090765
Abstract: Evolution by natural selection is believed to be the only possible source of spontaneous adaptive organisation in the natural world. This places strict limits on the kinds of systems that can exhibit adaptation spontaneously, i.e., without design. Physical systems can show some properties relevant to adaptation without natural selection or design. (1) The relaxation, or local energy minimi-sation, of a physical system constitutes a natural form of optimisation insomuch as it f i nds locally optimal solutions to the frustrated forces acting on it or between its components. (2) When internal structure ‘gives way’ or accommodates a pattern of forcing on a system, this constitutes learning insomuch, as it can store, recall, and generalise past conf i gurations. Both these effects are quite natural and general, but in themselves insuff i cient to constitute non-trivial adaptation. However, here we show that the recurrent interaction of physical optimisation and physical learning together results in signif i cant spontaneous adaptive organisation. We call this adaptation by natural induction.
The effect occurs in dynamical systems described by a network of viscoelastic connections subject to occasional disturbances. When the internal structure of such a system accommodates slowly across many disturbances and relaxations, it spontaneously learns to preferentially visit solutions of increasingly greater quality (exceptionally low energy). We show that adaptation by natural induction thus produces network organisations that improve problem-solving competency with experience (without supervised training or system-level reward). We note that the conditions for adaptation by natural induction, and its adaptive competency, are different from those of natural selection. We therefore suggest that natural selection is not the only possible source of spontaneous adaptive organisation in the natural world
1.4. What Is Adaptation?
The quote from Darwin at the beginning of the Introduction Section draws our atten-tion to the exquisite and particular organisation of relationships between the parts of a system. (We would not use the word “perfected” in modern thinking, but there has to be something special about the arrangement of parts).
A system that is deformed or distorted by the forcing applied to it, or simply relaxes under the stresses it experiences, is not (normally) what we mean by adaptation (regardless of whether that deformation involves observable variables or internal structure). Even learning a function or a generalised class of patterns, though it might be analogous to simple forms of cognitive learning, is not the same sense of adaptation that we seem to refer to in biological evolution. Accordingly, the physical optimisation and physical learning examples above have not been claimed to be new sources of adaptation, although learning principles can deepen our understanding of adaptation by natural selection [34,49]. Thus, what do we really mean by adaptation and how does it relate to learning and optimisation?
Def i nitions of adaptation and adaptive organisation that are tied to natural selection, organismic reproduction, or Darwinian f i tness are not useful for answering this question.
The ‘appearance of design’ concept [4] is closer to what we need because it is not necessarily tied to Darwinian processes, but it is not easily quantif i able. In fact, def i ning exactly what we mean by adaptation is an open problem in biology. Note that Darwinian f i tness, the number of offspring produced, (or inclusive f i tness, for that matter), is part of an explanation of biological adaptation, not the explanandum itself (and besides, the relationship between natural selection and maximisation of these quantities is notoriously fraught [50,51]). It is not a high number of offspring that needs explaining, and believing that the ‘goodness of f i t’ between an organism and environment is def i ned as reproductive output fails to engage with the special properties of the biological organisations that facilitate this. To assess whether there can be any sources of adaptation other than natural selection, it is necessary to let go of the idea that survival and reproduction are the ultimate assessors of adaptation. Whilst these are undeniably important to biological organisms, they are part of the Darwinian explanation for how adaptation happens. In contrast, the goodness of f i t between an organism and its environment and the appearance of design are much more general notions and more conceptually apt to understanding what adaptation is. However, the former is too easily satisf i ed in a physical sense (e.g., an imprint in clay) and the latter is rather subjective.
How can we make a general notion of adaptation more objective and quantif i able? Our approach is to def i ne adaptation as a process that provides an optimization or constraint-solving capability that is non-trivial, i.e., superior to a local hill-climber, and more specif-ically, one that improves its problem-solving capability with experience, i.e., learns to f i nd better solutions (without assuming that it f i nds optimal or perfect solutions). In evolutionary computation, the adaptive capability of natural selection is demonstrated through its ability to f i nd good solutions to optimisation problems [52]. Despite the con-troversy attached to viewing adaptation as a problem-solving process, we f i nd it useful to conceptualise adaptation in this way [53] A less value-laden conception of evolution as a dynamical process, without a problem to be solved and without a pre-existing niche to be occupied, is acknowledged [54,55]. However, if adaptation is construed as merely ‘whatever happens as a result of natural selection’, for example, then it remains tied to natural selection. Adaptation needs to address a design-like property (independent of natural selection), and problem-solving is one way to characterise this. At the least, a process that can provide a non-trivial problem-solving competency is a high bar for assess-ing adaptation. To formalise this, we begin with a notion of searching for points in some conf i guration space that optimises some quantity, i.e., a process of function optimisation def i ned over some given space of variables [50,53] (Figure 1A). An optimisation ability does not require that a process f i nds the globally optimal (best possible) conf i guration [50];
this would be too strict. Natural selection, for example, does not provide optimal solutions.
An adaptation built by natural selection need only be a good solution to a problem, not an optimal one. Conversely, neither do we want to def i ne adaptation in a manner that is trivially satisf i ed [47]. If we adopt a very simple notion of problem-solving, such as that provided by a hill-climbing or local gradient ascent process, that merely f i nds a local optimum of an objective function, it would be trivially satisf i ed by a physical system as discussed-and therefore not as satisfying as a concept of adaptation. The behaviour of any physical system that can be described by the local minimisation of an energy function can be interpreted as an optimisation process in the limited sense that it f i nds locally optimal solutions to its implicit energy-minimisation ’problem’ [42] (Figure 1A).
It is not very useful to adopt a def i nition of adaptation that is satisf i ed by a literal ball rolling down a literal hill (even if it is functionally equivalent to the process described by natural selection, see Discussion). Accordingly, the local optimisation behaviour of physical systems is not suff i cient to constitute adaptation. So, instead, we seek a non-trivial problem-solving competency-not necessarily optimal, but not trivial, either [47]. Physical systems can do better than the trivial case. Simulated annealing is a famous example of a computational optimisation process that f i nds solutions better than local optima [56], and actual annealing (e.g., in a cooling metal) occurs spontaneously. This shows that the spectrum of possibilities for natural adaptation is non-empty-but still, we aspire to more than cooling metal as a natural example of adaptation. In what follows, we show that the ability of a physical system to learn offers the possibility of a physical system that learns to solve problems better with experience. Put differently, we classify the ordinary (f i rst-order) local energy minimisation behaviour of physical systems as trivial, but optimisation that improves with experience or second-order optimisation is algorithmically interesting.
In the previous examples of physical optimisation, the quality of a solution improves over a given state trajectory, but it only reaches a local optimum and its ability to f i nd solutions of good quality does not change with experience. Lower-energy states, consti-tuting better solutions to the frustrated state variables in the system, may exist but are not obtained. The optimisation ability does not change over time-restarting the system results in the same locally optimal outcome if it restarts from the same initial position. In the examples of physical learning, a system incrementally improves the f i t of an (implicit) internal model to a conventional learning task such as classif i cation or representing a function, e.g., [29]. A problem-solving ability is not demonstrated (except in the same sense of f i nding a locally optimal f i t of model parameters to the data). Both these behaviours (learning and optimisation) are very natural and do not involve particularly limiting as-sumptions about the system. In both cases the behaviour is determined by local energy minimisation of the same energy function-the only difference is which variables give way under stress and which are held constant. In optimisation, the state gives way to the problem, and in learning, the model gives way to the data (Figure 1); however, so far we have considered these two processes in isolation: the outputs of the learning process do not affect the optimisation landscape, and the outputs of the optimisation process do not affect the learning data. This limiting assumption means that learning responds to data that is given by external conditions (f i xed training data), and optimisation responds to a landscape that is given by the problem def i nition (f i xed problem). These effects have been studied separately, but in general dynamical systems both these effects will happen at the same time. What happens to the structure and dynamics of the system in the general case where there is feedback between the two is much more interesting.
4. Discussion
4.1. The Relationship between Natural Induction and Natural Selection Adaptation by natural induction and adaptation by natural selection share a number of features: Both involve the incremental accumulation of small changes over time, both can result in the increased f i t of an adaptive system to a system of constraints, and both involve simple gradient-following principles. They are also different in the mechanisms that they require (their necessary and suff i cient conditions), how they work (their algorithmic princi-ples) and, consequently, in their adaptive competencies. Whereas natural selection depends on the differential survival and/or reproduction of entities, natural induction operates by the differential easing of frustrated relationships between entities (The accommodation of internal connections to a state conf i guration has some similarity to the over-production of neural connections and their differential retention or reinforcement by ‘selective stabiliza-tion’, which may also result in increased response to or memory of activation patterns. The emphasis of natural induction on the differential easing of frustrated interactions agrees with an emphasis on network integration or ‘survival of the f i tted’, in contrast to ‘survival of the f i ttest’. However, natural induction is a physical model without any population or selection process, whereas selective stabilisation depends on an over-produced population of connections, and in terms of algorithmic competence, natural induction demonstrates an increase in adaptative capabilities). Moreover, the raison d’etre of selection as a theory of biological evolution is to avoid dependence on variation that is directed toward adaptive outcomes, whereas natural induction exploits the fact that variation directed toward easing frustrated interactions is normal in physical systems and adaptively signif i cant (Darwin suggested that some variation was developmentally environmentally directed but was not specif i c about its adaptive signif i cance. The nature of developmental bias and pheno-typic plasticity, and the potential of these and other factors to inf l uence genetic evolution adaptively, is an active topic [6,78]. These mechanistic differences are important, but it is perhaps the difference in their algorithmic principles and adaptive competencies that is more important. Even though both include gradient-following principles, they are different algorithms [79], and this is clear because they have different problem-solving capabilities.
A common interpretation of adaptation by natural selection, i.e., characterised as a pop-ulation following f i tness gradients to a local peak in a f i tness landscape [73,80], aligns well with local gradient-following principles of (f i rst-order) physical optimization. It is acknowl-edged that biological evolution is not necessarily a good optimiser [54], the idea of natural selection climbing gradients in a static f i tness landscape is a serious over-simplif i cation [55], identifying any quantity that natural selection maximises is problematic [50,51], and the conception of natural selection as a problem-solving process has been criticised [51,54,55,81].
However, these are mostly arguments that weaken the interpretation of natural selection as an optimisation process, i.e., natural selection is, at least sometimes, less effective at optimi-sation than a local gradient process. Accordingly, doing better than local optimisation is a conservative criterion for adaptation.
At a suitable level of abstraction, both natural selection and physical optimisation can be described as processes that follow local gradients to a local optimum. Clearly, though, they are mechanistically different ways of implementing this process. Specif i cally, natu-ral selection can be described as a process of random generation and selective retention, whereas in the Newtonian model of a physical system, a ball, for example, is deterministi-cally caused to move in a directional fashion by the reaction to the slope, i.e., to roll downhill, not uphill. No population of balls, random variation nor selection process need be involved.
It is also possible, however, to think about a statistical mechanics process where the position of the ball is represented by a probability distribution of possible future positions, which is updated to amplify positions that are lower in energy compared to those that are higher in energy. So, does it really matter whether it is a statistical-mechanics (variation and selection) process or a Newtonian (directed learning) process? If the gradient-following outcome is the same, does the mechanism matter?
In some contexts, the mechanism seems to matter a lot to biological thought; Evolu-tion by natural selection can only occur if there is a population, suitable variation, and selection-anything else is not natural selection. On the other hand, it is common to equate evolutionary adaptation with a hill-climbing process, and to the extent that the change in an evolving population is non-arbitrary, this level of functional equivalence seems to capture its adaptive competence. However, if that is true, it has a curious implication;
Although it is traditional for evolutionary thought to consider processes that go uphill (in a f i tness landscape), whereas physical models go downhill (in an energy function), this does not make either one cleverer, i.e., a more effective optimiser, than the other (regardless of whether they are implemented in a statistical manner or a Newtonian manner). That would suggest that a literal ball rolling down a literal hill would also constitute adaptation. If local hill-climbing is suff i cient to produce the biological adaptation we observe, this would mean that the genius of Darwin’s theory is just that it provides a hill-climbing process capable of operating in the appropriate organismic ‘design space’. Alternatively, the mechanistic details of evolution by natural selection, or the context in which it occurs, may matter to its adaptive competence-making it different from a hill climber [79]. It seems likely that the details do matter, but to the extent that evolution by natural selection is formally characterised as a simple (substrate-independent) hill-climbing process, it excludes these potentially important aspects of the actual biological process.
The models presented here can be interpreted as literal physical systems, but they can also be interpreted as models that stand in for other natural optimisation processes, includ-ing those involving the local gradient-following capacity of evolution by natural selection.
Adaptation by natural induction is instantiated as a Newtonian process in the models we have illustrated-with forces and (directional) reactions rather than a statistical mechanics process (random variation and selection) and because they are different mechanisms, it means that natural induction has different necessary and suff i cient conditions to natural selection, and may apply in cases where natural selection does not. However, in terms of adaptive competence, that is not the important difference between natural induction and natural selection. The problem-solving competence of natural induction is not the same as that of natural selection because they are different algorithms (not because they are different mechanisms for implementing the same algorithm). Adaptation by natural induction f i nds better solutions than a local gradient-following optimisation process, a.k.a. a hill-climber. If biological evolution is algorithmically equivalent to a hill-climber (f i rst-order optimisation only), then its adaptive competence is inferior to adaptation by natural induction. Conversely, if biological evolution is a more sophisticated optimiser than a hill climber, then the substrate-independent algorithm of random variation and selection does not describe it.
It might be appropriate to conceive biological evolution as a simple gradient process when natural selection acts on a simple vector of genes or vector of phenotypic traits individually determined by corresponding genes in a one-to-one fashion [66]. In this case, the action of evolution by natural selection is analogous to the bed of clay-a univariate model; that is, each selective coeff i cient is responsible for the change in frequencies of alleles at one locus. Other work has illustrated, however, that an evolutionary process operating on heritable variation in the connections of a dynamical network can exhibit the same kind of adaptive competence as natural induction. This includes gene-regulation networks [35,48,66]. In this case, the gene network constitutes a dynamical developmental process that generates phenotypes indirectly (and ‘disturbances’ are provided by a lifecycle that resets developmental states to a neutral, undifferentiated, epigenetic state). This gives it the possibility of representing relational interactions between traits (i.e., pleiotropic inter-actions in the genotype-phenotype map of such a developmental process may constitute an associative model of selected phenotypes [34]). Previous work shows that under these circumstances, the outcome is superior to local gradient optimisation [66]; that is, evolution by natural selection acting on the parameters of a developmental process (second-order) can solve problems that evolution by natural selection acting on a directly encoded phenotype (f i rst-order) cannot.
One interpretation of this is that natural selection can exhibit the same adaptive com-petence as natural induction, after all. However, adaptation by natural induction is not the algorithm that Darwin described (evolution by natural selection does not describe a process that provides model induction, a network of viscoelastic connections, an ability to learn to adapt better with experience, nor the signif i cance of ‘pulse’ or disturbances, amongst other things). So, even though the basic gradient-following process might, in biological cases, be provided by a variation and selection process, in cases where natural selection operates on the parameters of a developmental process constituted by a network of interactions [35,48,66], it might be more correct to attribute the adaptive competence to natural induction and not to natural selection. After all, whereas the theory of evolu-tion by natural selection focuses on the mechanism of variation and selection, this paper demonstrates (using purely Newtonian processes) that this superior adaptive competence arises from the dynamical feedback between model induction and optimisation, and not from random variation and selection. Whereas natural selection depends on the differential survival and reproduction of things, natural induction fundamentally depends on the differential easing of frustrated relationships between things. Although adaptation by natural induction is therefore fully compatible with a Darwinian model of evolutionary change (in suitable network contexts), these are different adaptive algorithms with different necessary conditions, different algorithmic principles, and different adaptive competencies.
We also note that the relationship between evolution and learning has been recognised and developed by many. At a suitable level of abstraction, evolution and some learning methods appear to be the same algorithm (i.e., ‘trial and error’ plus reinforcement equates to random variation and selection). Both can be understood as processes that optimise a function by, to a f i rst approximation, following local gradients. It has often been noted that reinforcement learning and evolution by natural selection are closely analogous [34,82], and indeed, the replicator equation (an abstraction of biological evolution under natural selection) and Bayesian updating (a learning optimisation process) have been shown to be formally equivalent [83,84]. See also [17,48,49,85-87] for the relationship between learning and evolution. These works expand and deepen our understanding of the adaptation provided by natural selection; however, note that adaptation by natural induction involves a two-way feedback between an optimisation process and an inductive learning process- the latter on its own is simply an optimisation process in model parameters, and not suff i cient to demonstrate an improvement in problem-solving competency.
The results in this paper thus demonstrate that a dynamical system described by a network of viscoelastic connections, and subject to occasional disturbances, exhibits adaptation in the more stringent sense of learning to optimise better with experience or improving its problem-solving competency over time-and this is not the same mechanism, algorithm, or competency as natural selection.
Where might networks of suitable connections occur naturally? By far the most fa-miliar examples of viscoelastic networks are in fact biological ones. Ecological networks, protein networks, cytoskeletal networks, metabolic networks, bio-electrical networks, social networks, and the biosphere as a whole are all networks at least partially characterised by linkages that are likely to give way under stress and are subject to at least occasional perturbations. Although these networks all involve biological individuals and materials, most of them are not (always) evolutionary units, so natural selection does not straightfor-wardly apply. This suggests that the interaction of natural selection and natural induction may be complex and possibly widespread. Outside of systems that we already recognise as biological, another obvious candidate where natural induction may be important is the origins of life and origins of evolution [88,89]. To the extent that a pre-biotic chemical network has an internal conformation structure that gives way under stress, we speculate that it has the potential to induce a model of its past experience that can anticipate and generalise without having properties suff i cient to be a bonaf i de evolutionary unit.
Whilst we wish to make the case that the conditions for this are quite natural and not onerous (i.e., do not require selection or design), we do not claim that these conditions are ubiquitous or even frequently or commonly met. Bear in mind that the conditions for evolution by natural selection, namely self-replicating systems with heritable variation in reproductive success, are hardly ubiquitous in the physical world (even though they may be realised in various substrates [90,91]) and their origin is not known. Instead, we claim that the contrary assumption, that natural selection is the only possible naturally occurring mechanism of spontaneous adaptation, is not correct. Who would make such a claim? Actually, this assumption is quite widely adopted, usually implicitly, with very wide-reaching and important implications [4].
Reasoning in many biological domains often depends on this assumption implicitly or explicitly. For example, the assertion that loose ecological communities (or the biosphere as a whole) cannot possibly be adapted because they are not evolutionary units (or members of a population) is an argument that presupposes natural selection to be the only possible source of adaptation. Biological systems are replete with dynamical systems described by networks in many different substrates (e.g., chemical reaction networks, metabolic networks, protein networks, gene-regulatory networks, bio-electric networks, ecological networks, and social networks). Linkages in many networks give way under stress (perfect elasticity is an idealisation) and biological systems are rarely isolated from shocks or disturbances. To the extent that these come together in natural systems, natural induction has the potential to provide a mechanism of spontaneous adaptation that is relevant to biological systems in many domains. This includes those where natural selection does not apply and those where natural selection also applies. How natural induction interacts with natural selection, and the broader implications for biological thought and the origins of adaptive complexity, will be analysed in future work.
Conclusions It has been argued strongly, and widely assumed across biological thinking, that natu-ral selection is the only possible mechanism capable of producing spontaneous adaptation in natural systems. Here we show that this assumption is false; adaptive organisation occurs spontaneously in physical systems with suitable natural properties, through an effect we call adaptation by natural induction. This occurs in dynamical systems described by a network of viscoelastic connections subject to disturbances. A viscoelastic connection is simply one that ‘gives way’ slightly under stress, which is a natural property of many physical materials, biological networks, and complex systems more generally. When dis-turbances cause the system to visit a distribution of locally optimal solutions, the changes to the connections in the network learn a generalised associative model of the local so-lutions visited, which causes the system to adapt in the rigorous sense of improving its problem-solving competency; that is, it provides optimisation ability that is superior to local optimisation and improves with experience. The simplicity of natural induction, and its necessary and suff i cient conditions, offer a solution to both the chicken and the egg problems-i.e., natural selection is not involved at run time nor in the construction or setup of the system. This has important implications for our understanding of biological evolution and adaptation in other complex systems-not least that adaptation can occur spontaneously in systems that are not units of selection.