GENSR: Symbolic Regression Based in Equation Generative Space

Symbolic regression (SR) seeks closed-form mathematical expressions that fit observed data. Neural SR methods amortize the search by training an encoder to map observations directly to expressions in a single pass, but this amortized inference leaves a residual amortization gap between its one-shot prediction and the true posterior. We propose Latent Equation Embedding (LEE), a framework that closes this gap through iterative amortized inference in a functionally grounded latent space. LEE learns a shared latent space Z equipped with three components: an encoder f_theta that jointly embeds symbolic tokens and numerical observations into a single latent vector z; an expression decoder g_expr that reconstructs formulas from z; and an evaluation decoder g_eval that predicts function values from z, explicitly grounding the latent space in functional behavior. At inference, LEE performs iterative refinement by re-encoding decoded expressions jointly with observations, progressively improving the latent estimate. LEE uses the encoder itself as a learned inference optimizer: each re-encoding step implicitly computes the mismatch between the candidate and the data. Because g_eval is differentiable in z, we additionally interleave continuous gradient descent with discrete re-encoding, yielding a hybrid iterative and gradient refinement procedure. On SRBench across three noise levels, against 19 baselines spanning genetic programming, symbolic-neural hybrids, and pre-trained Transformers, LEE produces expressions 2--10x simpler than the strongest accuracy-oriented baselines, including Operon, GP-GOMEA, TPSR, RAG-SR, and GenSR, with complexity 8--11 versus 20--90. These results advance the low-complexity region of the accuracy-complexity Pareto frontier and show graceful degradation as noise increases.

Symbolic regression (SR) seeks to discover parsimonious mathematical laws from observational data, yet conventional approaches often struggle with the vast combinatorial search space of physically meaningful expressions. We present InsightSR, a framework that embeds Large Language Models (LLMs) as a guiding layer around the PySR genetic programming engine. Rather than relying on LLMs to generate expressions directly, InsightSR uses LLMs to progressively transform the search space itself through two complementary pathways: a Semantic Seed Pathway that proposes dimensionally consistent functional skeletons, and a Structural Feature Pathway that recommends nonlinear feature transformations. These transformations accumulate over iterations, broadening the input space and shifting the symbolic search from constructing deep expression trees over raw variables to assembling shallow trees over a rich, semantically informed feature set. A post-generation feedback loop evaluates candidates, categorizes features by their empirical utility, and refines the guidance for the next iteration, transforming the discovery process from open-ended generation into iterative, self-correcting refinement. Across three benchmarks, InsightSR achieves a 95% exact recovery rate on the Feynman benchmark and 80.18% accuracy on the LLM-SRBench LSR-Transform task, substantially outperforming state-of-the-art genetic programming and neural-symbolic methods while maintaining strong out-of-distribution generalization on real-world datasets.

Symbolic regression aims to discover human-interpretable equations that explain observational data. However, existing approaches rely heavily on discrete structure search (e.g., genetic programming), which often leads to high computational cost, unstable performance, and limited scalability to large equation spaces. To address these challenges, we propose SRCO, a unified embedding-driven framework for symbolic regression that transforms symbolic structures into a continuous, optimizable representation space. The framework consists of three key components: (1) structure embedding: we first generate a large pool of exploratory equations using traditional symbolic regression algorithms and train a Transformer model to compress symbolic structures into a continuous embedding space; (2) continuous structure search: the embedding space enables efficient exploration using gradient-based or sampling-based optimization, significantly reducing the cost of navigating the combinatorial structure space; and (3) coefficient optimization: for each discovered structure, we treat symbolic coefficients as learnable parameters and apply gradient optimization to obtain accurate numerical values. Experiments on synthetic and real-world datasets show that our approach consistently outperforms state-of-the-art methods in equation accuracy, robustness, and search efficiency. This work introduces a new paradigm for symbolic regression by bridging symbolic equation discovery with continuous embedding learning and optimization.