Inference of compact nonlinear dynamic models by epigenetic local search

William La Cava, Kourosh Danai, Lee Spector
2016 Engineering applications of artificial intelligence  
6 We introduce a method to enhance the inference of meaningful dynamic models from observational data by genetic programming (GP). This method incorporates an inheritable epigenetic layer that specifies active and inactive genes for a more effective local search of the model structure space. We define several GP implementations using different features of epigenetics, such as passive structure, phenotypic plasticity, and inheritable gene regulation. To test these implementations, we use
more » ... of data sets generated from nonlinear ordinary differential equations (ODEs) in several fields of engineering and from randomly constructed nonlinear ODE models. The results indicate that epigenetic hill climbing consistently produces more compact dynamic equations with better fitness values, and that it identifies the exact solution of the system more often, validating the categorical improvement of GP by epigenetic local search. The results further indicate that when faced with complex dynamics, epigenetic hill climbing reduces the computational effort required to infer the correct underlying dynamics. We then apply the method to the identification of three real-world systems: a cascaded tanks system, a chemical distillation tower, and an industrial wind turbine. We analyze its solutions in comparison to theoretical and black-box approaches in terms of accuracy and intelligibility. Finally, we analyze population homology to evaluate the efficiency of the method. The results indicate that the epigenetic implementations provide protection from premature convergence by maintaining diversity in silenced portions of programs. 7 symbolic regression 8 10 ena associated with biological, ecological, social, and economic systems, as well as the dynamics of 11 artifacts such as wind turbines, robots, and aircraft. Dynamic behaviors are usually characterized 12 by differential equations which in aggregate represent the dynamic model of the system. These 13 dynamic models are the essence of the simulations that estimate/predict system behavior for pol-14 icy decisions, design, optimization, control, and/or automation. This paper presents a method for 15 construction of concise and mechanistically meaningful dynamic models from observations. 16 Dynamic models are preferably formulated according to first principles, to embody the knowl-17 edge of the process. However, first-principles models cannot often fully characterize the nonlinear 18 dynamics of the process, as represented by process observations. In regress, first-principles models may be abandoned in favor of empirical models such as neural networks [40, 17], linear or nonlinear 20 autoregressive moving average (ARMAX) models [36, 3], or others [41, 47], that have the structural 21 flexibility to accommodate the measured process observations. Although these empirical models 22 provide an effective basis for estimation/prediction, they have two major drawbacks. One is their 23 'black-box' format which obscures the knowledge of the process acquired through adaptation. The 24 second is their case-specificity which makes them potentially deficient in representing the process 25 under conditions (inputs) not encompassed by the measured observations. To remedy the black-box 26 nature of these empirical models, dynamic models consisting of differential equations can be de-27 fined in algebraic form by symbolic regression [16, 6, 4], wherein both the structure (topology) and 28 parameters (constants) are inferred from measured observations. Since these symbolic models are 29 intelligible, they have the capacity to elucidate the process physics. Symbolic regression is typically 30 conducted using genetic programming (GP) [29], which is a bio-inspired machine learning technique 31 that constructs candidate models from mathematical building blocks and proceeds with selection, 32 recombination and mutation over several generations before converging on a model that best fits 33 the process observations. 34 In comparison to system identification methods that presume fixed model structures, symbolic 35 regression can be computationally expensive because of its expanded search space. Furthermore, 36 when guided solely by an error metric, it can yield unwieldy equations that are elusive to physical 37 interpretation. To remedy these shortcomings, this paper introduces a new method of symbolic 38 regression that fine-tunes candidate model structures by local search [32]. This fine tuning is 39 enabled by the addition of an epigenetic layer for selection of program components (consisting of 40 variables and instructions) to be included in the model. The incorporation of this epigenetic layer 41 is motivated by two hypotheses: first, that the benefits of epigenetic regulation observed in biology 42 may confer analogous improvements on GP systems; and second, that generalized local search 43 methods enabled by epigenetics may improve the ability of GP to find correct model structures. 44 As to the first hypothesis, despite the highly regulated nature of biological genes, the role of 45 epigenetics in regulating gene expressions is traditionally ignored in GP (with some exceptions, 46 e.g. [12]). However, epigenetic processes may provide several evolutionary benefits. For exam-47 ple, because epigenetic processes allow the underlying genotype to encode various expressions and 48 lead to neutral variation through crossover and mutation of non-coding segments, they may allow 49 populations to avoid evolutionary bottlenecks or let them respond to changing evolutionary pres-50 sures [22]. Also, because they provide for phenotypic plasticity that enables gene expression to 51 change in response to environmental pressure [10], they may allow gene expression adaptations to 52 be inherited in offspring without explicit changes to the genotype. This property legitimizes, via 53 epigenetic processes, once discredited ideas of Lamarck pertaining to the inheritability of lifetime 54 adaptations [22, 20]. 55 Regarding the second hypothesis, although local search methods have been developed and in-56 tegrated into evolutionary algorithms [18, 63, 23, 46, 15], especially in genetic algorithms (GAs) 57 through prescribed changes to the genotype, the role of structure optimization in symbolic regres-58 sion is typically left to the GP process. Aside from some recent developments [1], local search is 59 traditionally conducted at the genome level. More generic local search methods, like tree snip-60 ping [4], focus on improving secondary metrics like size or legibility, whereas the traditional search 61 methods, like stochastic hill-climbing [4], linear [21] or non-linear regression [58] are confined to con-62 stant optimization. Although these local search methods improve symbolic regression performance, 63 they cannot aid the search for program topology. 64 Epigenetics, on the other hand, provide a natural basis for performing local search at the 65 2 structural (i.e., program topology) level. Motivated by this benefit of epigenetics, we introduce 66 in this paper an epigenetics-enabled GP system to conduct topological optimization of programs 67 at the level of gene expression. The contributions of this method are twofold: first, it introduces a 68 generic method of topological search of the space of individual genotypes via modifications to gene 69 expression. Second, it improves programs without affecting the genotype and without discarding the 70 acquired knowledge gained through evolution, thereby lowering the risk of premature convergence 71 observed in previous studies [63]. These contributions are achieved by conducting local search on 72 the epigenome rather than the genome and making these adaptations inheritable via evolutionary 73 processes. 74 The proposed Epigenetic Linear Genetic Programming (ELGP) method is tested on a large 75 array of data generated from nonlinear ordinary differential equations (ODEs), as well as from 76 three real-world processes, to evaluate the quality of its solutions. The paper is organized as 77 follows. We formulate in §2 the identification problem and describe in §3 the ELGP method and 78 its application to inference of dynamic models. We also review the relevant work in the context of 79 GP and nonlinear dynamics modeling in §4. We then present the experimental analysis of different 80 epigenetic implementations on a series of increasingly complex problems in §5. We begin by testing 81 the method on a large set of data obtained from simulated nonlinear ODEs in different engineering 82 fields, in order to illustrate its breadth of application. We then perform identification on hundreds of 83 randomly constructed nonlinear systems, varying in complexity and dimensionality, to evaluate the 84 scalability of the method in comparison to traditional GP approaches. Finally, we apply the ELGP 85 method to three real-world problems, including the identification of 1) a benchmark cascaded tanks 86 system [66], 2) a chemical distillation tower, and 3) an industrial wind turbine. The results are 87 sitivity of simulation to different model structures [34] and the computational cost of numerical 105 3 integration, the alternative approach of algebraically estimating candidate model outputs is pre-106 ferred for symbolic regression [4, 49]. In the algebraic approach, un-measured states, denotedx, 107 are estimated from measurements via numerical differentiation together with smoothing functions. 108 In the case of first-order differential equations with un-measured state derivatives, the target is 109 estimated numerically as y(t k , u) =x, such that the prediction error of a candidate model has the The fitness metric f in Eq. (2) for individuals is often defined using the mean absolute error 112 (MAE) or mean squared error (MSE), although some prefer using the correlation coefficient due to 113 its insensitivity to linear scaling [25, 28]. We use a fitness metric [34] designed to minimize error 114 and maximize correlation so that both the prediction error and the closeness of shapes defined by 115 the coefficient of determination (R 2 ), are accounted for in the results. This fitness metric takes 116 advantage of the covariance comparison afforded by R 2 and avoids the need for post-hoc linear 117 scaling of the solutions which could lead to increased model complexity. For target y and output 118ŷ , f is defined as: 119 127 Afterwards, the population undergoes selection, recombination and mutation, as in standard GP, 128 to produce an updated population. The process repeats until an adequate solution is produced. 129 The ELGP method has two salient features that improve its performance: (i) its use of linear, 130 stack-based programs to represent equations, and (ii) its incorporation of local search of the model 131 structures space to improve the candidate model's fitness and to reduce its complexity. These 132 features have been shown to bolster traditional GP's performance on several benchmark regression 133 problems in terms of the conciseness of the developed models, their fitness, and efficiency of the 134 search [33, 32]. The effectiveness of these features is evaluated here in construction of nonlinear 135 dynamic models. 136 3.1. GP Representation 137
doi:10.1016/j.engappai.2016.07.004 fatcat:pce73g5byzfdvcbxjpz42zvamu