Symbolic Machine Learning for Vapor-Liquid Equilibrium Prediction in Cx-N2 Binary Mixtures

Symbolic regression provides analytical expressions, but it is usually applied one output at a time. This is limiting in process systems, where state variables are often coupled through shared physical parameters. Independent symbolic regression can give accurate individual equations that are difficult to interpret as one model. We present a neuro-evolutionary symbolic regression method for coupled multi-output systems. The method searches for a shared symbolic backbone: a set of latent symbolic units that is discovered once and reused by several outputs through sparse additive or multiplicative read-outs. The discrete model structure is evolved by mutation and crossover, whereas the continuous parameters are tuned by gradient descent and inherited by the offspring. The method is assessed on a set of benchmarks with known ground truth and on a hydrothermal liquefaction yield case. The results show that coupling is not a general route to lower prediction error. Its main contribution is the enforcement and diagnosis of cross-output consistency when a physically shared factor is embedded in a latent expression and is weakly identifiable from the data. This occurs for Langmuir-Hinshelwood and site-coverage denominators, for which independent PySR does not close the consistency gap or recover the same shared form. Conversely, when each output is already identifiable, as in the Van de Vusse benchmark, independent symbolic regression matches or improves the coupled model. The proposed framework, rather than a general purpose predictor, is a structured shared-mechanism extractor. Its value is highest when the target structure is sparse, shared, weakly identifiable or constrained by closure.

Scientific equation discovery must combine broad domain priors with strict numerical testing. Symbolic regression supplies numerical grounding but faces a combinatorial search space, whereas many language-model systems ask the model to propose or select formulas directly. We test a different division of labour. We compare role specifications in which the language model acts as equation author, candidate decider or search controller, alongside end-to-end language-model and purely numerical baselines. In the controller setting we propose here, implemented as LLM-PySR, language models specify variables, operators, transformations and search depth; symbolic regression enumerates and fits expressions; and deterministic metrics govern retention. Across 74 AI-Feynman equations and seven complex formula-recovery tasks, search control achieved the strongest observed balance of accuracy, complexity, stability and cost. On an independent battery dataset, LLM-PySR identified a compact piecewise-linear relation between early voltage-curve displacement and cycle life. The results suggest that language models should shape hypothesis exploration rather than decide which equations survive.

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.