To address the fit-complexity trade-off, we obtain the posterior probability of each expression from basic probabilistic arguments and explicit approximations. Here, we propose a Bayesian approach to the definition of machine scientists. For more general situations, sparse regression can be combined with genetic programs that automatically generate the basis functions, thus relaxing the need to know them a priori ( 12, 13). This approach is particularly suited to learn differential equations ( 8– 11), whose form often follows the assumption of linearity on relatively simple basis functions. In this approach, closed-form mathematical models are assumed to be linear combinations of some (linear or nonlinear) “basis functions” of the independent variables, and sparse regression is used to select and weigh the relevant basis functions. Another successful approach is based on sparse regression ( 7– 11). In this approach, closed-form mathematical expressions are represented as graphs, and, given a goodness-of-fit metric, populations of expressions are created and evolved in such a way that high-fitness expressions are selected for further exploration. One of these approaches is based on genetic programming ( 5, 6). Attempts to design machine scientists date back to, at least, the 1970s and have led to very successful approaches in recent years.
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