Fig. 2: The detailed process of PSRN forward propagation for obtaining the optimal expression.

Step 1: the input data X and the base expression set s, together with optional token constants sampled from a predefined distribution p(c), are fed into PSRN for forward propagation. Based on the designated operator categories (for example, {I, +, ×, sin, exp}, where I is the identity operator), a large number of distinct subtree values h, for example, \(\hat{{\mathscr{F}}}({X})\) are rapidly calculated layer by layer. Step 2: the MSE is computed between the final layer’s output and the broadcast target tensor y, resulting in the loss tensor. Step 3: the position of the minimum value in the loss tensor is selected. Step 4: starting from the optimal position, a recursive symbolic deducing is performed using the offset tensor Θ generated when building the network, ultimately yielding the optimal expression. If necessary, the coefficients of this expression will be adjusted in post-processing. The algorithmic procedure for PSRN evaluation and symbolic reconstruction is presented in Supplementary Algorithm 1.