  Vertical Symbolic Regression (VSR) recently has been proposed to expedite the
discovery of symbolic equations with many independent variables from
experimental data. VSR reduces the search spaces following the vertical
discovery path by building from reduced-form equations involving a subset of
independent variables to full-fledged ones. Proved successful by many symbolic
regressors, deep neural networks are expected to further scale up VSR.
Nevertheless, directly combining VSR with deep neural networks will result in
difficulty in passing gradients and other engineering issues. We propose
Vertical Symbolic Regression using Deep Policy Gradient (VSR-DPG) and
demonstrate that VSR-DPG can recover ground-truth equations involving multiple
input variables, significantly beyond both deep reinforcement learning-based
approaches and previous VSR variants. Our VSR-DPG models symbolic regression as
a sequential decision-making process, in which equations are built from
repeated applications of grammar rules. The integrated deep model is trained to
maximize a policy gradient objective. Experimental results demonstrate that our
VSR-DPG significantly outperforms popular baselines in identifying both
algebraic equations and ordinary differential equations on a series of
benchmarks.
