Vertical Symbolic Regression via Deep Policy Gradient

TL;DR

This work proposes Vertical Symbolic Regression using Deep Policy Gradient (VSR-DPG) and demonstrates that VSR-DPG can recover ground-truth equations involving multiple input variables, significantly beyond both deep reinforcement learning-based approaches and previous VSR variants.

Abstract

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.

Figures (6)

• Figure 1: Our Vsr-Dpg follows a vertical path (colored blue) better than the horizontal path (colored red), in the scientific discovery of Joule's first law. (Left) The vertical discovery starts by finding the relationship between two factors ($Q, T$) in a reduced hypothesis space with other factors held constant. It then finds models in extended hypothesis space with three factors ($Q, I, T$), and finally in the full hypothesis space. Searching following the vertical paths is way cheaper since the sizes of the reduced hypothesis spaces in the first few steps are exponentially smaller than the full hypothesis space. (Right) Our Vsr-Dpg extends the equation in each step. The placeholder symbol $A$ indicates a sub-expression.
• Figure 2: The proposed Vsr-Dpg for the discovery of expression $\phi=x_1\times x_3-x_2\times x_4$. (a) Initially, a reduced-form equation $\phi= x_1\times C_1 - C_2$ is found, in which $x_2, x_3, x_4$ are held constant and only $x_1$ is allowed to vary. $C_1$ and $C_2$ (colored blue) are summary constants, which are sub-expressions containing the controlled variables. The open constants in the expression are fitted by the corresponding controlled variable data. (b) In the second stage, this equation is expanded to $x_1\times C_3 - x_2\times C_4$. (c, d) This process continues until the ground-truth equation $\phi=x_1 x_3 - x_2 x_4$ is found. (e, f) Under those controlled variables, the deep recurrent neural network (RNN) predicts a categorical distribution over the available grammar rules, which only have the free variables (colored blue). The best-predicted expression in (e) is reformulated as the start symbol for in (f) $x_1\times A - A$.
• Figure 3: Convert a sequence of grammar rules into a valid expression. Each rule expands the first non-terminal symbol in the squared box. The parts that get expanded are color-highlighted.
• Figure 4: Visualization of Vsr-Dpg controlling variables $x_1$ (Left) and $x_2$ (Right) for the Lorenz attractor. The data of our Vsr-Dpg are drawn from the intersection of the mesh plane and the curve on the Lorenz attractor. In comparison, the ODEFormer draws data by picking a consecutive sequence $\{(t_i,\mathbf{x}(t_i))\}_{t=0}^N$ without knowing its time derivative on the curve.
• Figure 5: Constraint-based integration of deep reinforcement learning with vertical symbolic regression. The constraints enforce the output of RNN output the given token at each step. It has limitations in passing the gradient to the parameters of RNN and also requires heavy engineering of different constraints. (a) Initially, the RNN to learn a reduced form equation with variables $x_2,x_3,x_4$ controlled. The RNN learns to sample the best preorder traversal of the reduced form expression tree from the available tokens. No constraints are applied in the first round. (b, e) Given the best-predicted expression $\phi_1$ represented as $(-,\times, x_1, C_1, C_2)$ at the first round, the RNN is used to predict an expression with control variables $x_3,x_4$. For the first four steps, the constraints are applied to mask out other tokens in the output, to enforce that the output must be $-,\times, x_1,C_1$. Since $C_1$ is a summary constant, the RNN samples a sub-expression with no constraints starting at the 5th step, which is $C_3$. In 6-th step, with the termination of the prior sub-expression, constraints are applied to enforce the RNN outputs $C_2$. Starting at the 7th step, we sample a subexpression $x_2\times C_4$. (c,d) The rest steps in the pipeline of vertical symbolic regression using expression tree representation.