Symbolic regression is a type of discrete optimization problem that involves searching expressions that fit given data points. In many cases, other mathematical constraints about the unknown expression not only provide more information beyond just values at some inputs, but also effectively constrain the search space. We identify the asymptotic constraints of leading polynomial powers as the function approaches 0 and ∞ as useful constraints and create a system to use them for symbolic regression. The first part of the system is a conditional production rule generating neural network which preferentially generates production rules to construct expressions with the desired leading powers, producing novel expressions outside the training domain. The second part, which we call Neural-Guided Monte Carlo Tree Search, uses the network during a search to find an expression that conforms to a set of data points and desired leading powers. Lastly, we provide an extensive experimental validation on thousands of target expressions showing the efficacy of our system compared to exiting methods for finding unknown functions outside of the training set.
This paper considers a task of symbolic regression (SR) when additional information called 'asymptotic constraints' on the target expression is given. For example, when the groundtruth expression is 3 x^2 + 5 x, it behaves like x^2 in the limit x -> infinity, thus the power '2' is given as additional information 'asymptotic constraint'. In the paper's setting, for SR with univariate groundtruth functions f(x), 'asymptotic constraints' for x-> infinity and x -> 0 are given. For this situation, the paper proposes a method called NG-MCTS with an RNN-based generator and MCTS guided by it to consider asymptotic constraints. In SR, a learner is asked to acquire an explicit symbolic expression \hat{f}(x) from a given set of datapoints {x_i, f(x_i)}, but unlike the parameter optimization aspect of a standard supervised ML setting, the problem is essentially combinatorial optimization over exponentially large space of symbolic expressions of a given context-free grammar (CFG). For a given symbolic space with a prespecified CFG, extensive experimental evaluations are performed and demonstrated significant performance gain over existing alternatives based on EA. Also, quantitative evaluations about extrapolative performance and detailed evaluation of the RNN generator are also reported.
Though it includes a lot of quite interesting methods and results, the paper has two major issues: 
(1) the proposed method NG-MCTS explicitly uses the fact that the target expressions are generated from a CFG, but this assumption sounds unnatural if the target problem is SR (unlike the case of GVAE). Apparently, all results except a toy case study in 5.2 depend on artificial datasets from a CFG, and any practical advantages or impacts are still unclear because the experimental settings are very artificial. 
(2) the most important claim of this paper would be the proposal to use 'asymptotic constraints', but the availability of this information sounds too strong and again artificial in practical situations. A one-line motivation saying that 'this property is known a priori for some physical systems before the detailed law is derived' is not convincing enough.