Equation discovery, also known as symbolic regression, is a type of automated modeling that discovers scientific laws, expressed in the form of equations, from observed data and expert knowledge. Deterministic grammars, such as context-free grammars, have been used to limit the search spaces in equation discovery by providing hard constraints that specify which equations to consider and which not. In this paper, we propose the use of probabilistic context-free grammars in equation discovery. Such grammars encode soft constraints, specifying a prior probability distribution on the space of possible equations. We show that probabilistic grammars can be used to elegantly and flexibly formulate the parsimony principle that favors simpler equations, through probabilities attached to the rules in the grammars. We demonstrate that the use of probabilistic, rather than deterministic grammars, in the context of a Monte-Carlo algorithm for grammar-based equation discovery, leads to more efficient equation discovery. Finally, by specifying prior probability distributions over equation spaces, the foundations are laid for Bayesian approaches to equation discovery.
COBISS.SI-ID: 61709059
The central task in modeling complex dynamical systems is parameter estimation. This task is an optimization task that involves numerous evaluations of a computationally expensive objective function. Surrogate-based optimization introduces a computationally efficient predictive model that approximates the value of the objective function. The standard approach involves learning a surrogate from training examples that correspond to past evaluations of the objective function. Current surrogate-based optimization methods use static, predefined substitution strategies to decide when to use the surrogate and when the true objective. We introduce a meta-model framework where the substitution strategy is dynamically adapted to the solution space of the given optimization problem. The meta model encapsulates the objective function, the surrogate model and the model of the substitution strategy, as well as components for learning them. The framework can be seamlessly coupled with an arbitrary optimization algorithm without any modification: It replaces the objective function and autonomously decides how to evaluate a given candidate solution. We test the utility of the framework on three tasks of estimating parameters of real-world models of dynamical systems. The results show that the meta model significantly improves the efficiency of optimization, reducing the total number of evaluations of the objective function up to an average of 77%.
COBISS.SI-ID: 33102631
Equation discovery methods enable modelers to combine domain-specific knowledge and system identification to construct models most suitable for a selected modeling task. The method described and evaluated in this paper can be used as a nonlinear system identification method for gray-box modeling. It consists of two interlaced parts of modeling that are computer-aided. The first performs computer-aided identification of a model structure composed of elements selected from user-specified domain-specific modeling knowledge, while the second part performs parameter estimation. In this paper, recent developments of the equation discovery method called process-based modeling, suited for nonlinear system identification, are elaborated and illustrated in two continuous-time case studies. The first case study illustrates the use of the process-based modeling on synthetic data while the second case-study evaluates process-based modeling on measured data for a standard system-identification benchmark. The experimental results clearly demonstrate the ability of process-based modeling to reconstruct both model structure and parameters from measured data.
COBISS.SI-ID: 33208103