SEVerA uses Formally Guarded Generative Models and a three-stage Search-Verification-Learning process to synthesize self-evolving agents that satisfy hard formal constraints while improving task performance.
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Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl
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abstract
PySR is an open-source library for practical symbolic regression, a type of machine learning which aims to discover human-interpretable symbolic models. PySR was developed to democratize and popularize symbolic regression for the sciences, and is built on a high-performance distributed back-end, a flexible search algorithm, and interfaces with several deep learning packages. PySR's internal search algorithm is a multi-population evolutionary algorithm, which consists of a unique evolve-simplify-optimize loop, designed for optimization of unknown scalar constants in newly-discovered empirical expressions. PySR's backend is the extremely optimized Julia library SymbolicRegression.jl, which can be used directly from Julia. It is capable of fusing user-defined operators into SIMD kernels at runtime, performing automatic differentiation, and distributing populations of expressions to thousands of cores across a cluster. In describing this software, we also introduce a new benchmark, "EmpiricalBench," to quantify the applicability of symbolic regression algorithms in science. This benchmark measures recovery of historical empirical equations from original and synthetic datasets.
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representative citing papers
The SDE benchmark shows LLMs lag on scientific discovery tasks relative to general science tests, with diminishing scaling returns and shared weaknesses across models.
KANs with learnable univariate spline activations on edges achieve better accuracy than MLPs with fewer parameters, faster scaling, and direct visualization for scientific discovery.
A self-explainable operator learning method reformulates operators as decomposable integral equations to reveal spatial input contributions to predictions in blood flow and aerodynamics problems.
AgentODE uses LLMs to discover ODE structures and infer parameter distributions from aggregate data, recovering consistent structures on benchmarks and RDEB clinical data with 231 observations from 46 patients.
Long-range exchange frustration in atomistic spin-lattice models can double skyrmion collapse barriers while keeping micromagnetic parameters fixed, revealing a limitation of continuum approximations.
Hybrid simulation and non-Euclidean elasticity theory demonstrate that clathrin coats develop adaptive rigidity and memory during growth, producing flat, stalled, or closed outcomes through two energy-landscape gates and matching experiments without fitted parameters.
Proves that Rademacher complexity of depth-d compositional trees over finite operator vocabulary is controlled by (K b L)^{d} / sqrt(n) under Lipschitz conditions on operators.
DT² trains digital twins to preserve pairwise policy rankings from fitted Q-evaluation on offline data rather than minimizing one-step transition errors, improving policy ranking and reducing decision regret.
Equilibrium pressure anisotropy modifies the tearing-mode growth-rate prefactor through parameters A and R0 while retaining the S^{-1/2} Lundquist scaling in gyrotropic MHD.
Introduces Centauric 1-jettiness in DIS, derives N3LL resummation matched to NLO, and establishes universal non-perturbative power corrections scaling as 1/R via reduction to rescaled hemisphere soft function.
Bayesian reconstruction and exhaustive symbolic regression on CMB, BAO, and supernova data yield the one-parameter dark energy parametrization w(a) = w0 / sqrt(a) that fits observations comparably to CPL and better than LambdaCDM.
Four parameters suffice to describe dust attenuation curve diversity in TNG simulations, yielding a new symbolic-regression model that recovers curves and fluxes better than existing parameterizations while linking parameters to SFR surface density, metallicity, and geometry.
FunctionEvolve recovers 107 exact symbolic forms out of 129 synthetic tasks (82.9% SA@50) by using expression-tree structure for evolutionary search, parent selection, mutation, and coefficient scoring with LLMs.
A convexity-preserving grammar enables symbolic regression to discover thermodynamically admissible dissipation potentials for generalized standard materials from noisy data.
LEE performs iterative amortized inference in a functionally grounded latent space to produce 2-10x simpler symbolic expressions than strong baselines on SRBench.
The Neural Compiler converts symbolic programs into exact differentiable PyTorch modules for hybrid scientific machine learning, enabling precise encoding of known physics with few trainable parameters.
Symbolic regression produces an approximate classifier for LHC exclusion limits that enables their direct inclusion during pMSSM global fits.
A graph-based automated model discovery framework identifies new concise soil hydraulic functions from data that outperform the Mualem-van Genuchten model across 249 soil samples.
DRSR uses Quality-Diversity to produce diverse symbolic regression expressions differing in residual distributions, enabling post-search selection on synthetic and astronomical data.
A two-stage symbolic regression plus generative model framework recovers governing interaction terms and forcing in stochastic triad models while accurately predicting statistical moments up to order five.
A knowledge-first approach to LLM-driven automatic heuristic design in combinatorial optimization yields better discovery efficiency, transfer, and generalization than code-centric baselines by formalizing a distortion-compression trade-off.
Transformers reconstruct the constituent RCFTs in tensor-product theories from low-energy spectra, reaching 98% accuracy on WZW models and generalizing to larger central charges with few out-of-domain examples.
A derivative algebra with EML and SOL primitives plus additive atomic forests enables simultaneous symbolic recovery of functions and antiderivatives from data, matching or exceeding XGBoost on 13 of 17 benchmarks with interpretable formulas.
citing papers explorer
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SEVerA: Verified Synthesis of Self-Evolving Agents
SEVerA uses Formally Guarded Generative Models and a three-stage Search-Verification-Learning process to synthesize self-evolving agents that satisfy hard formal constraints while improving task performance.
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Evaluating Large Language Models in Scientific Discovery
The SDE benchmark shows LLMs lag on scientific discovery tasks relative to general science tests, with diminishing scaling returns and shared weaknesses across models.
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KAN: Kolmogorov-Arnold Networks
KANs with learnable univariate spline activations on edges achieve better accuracy than MLPs with fewer parameters, faster scaling, and direct visualization for scientific discovery.
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Self-explainable Operator Learning for Discovering Spatial Patterns in Functional Data
A self-explainable operator learning method reformulates operators as decomposable integral equations to reveal spatial input contributions to predictions in blood flow and aerodynamics problems.
-
LLM-Guided ODE Discovery and Parameter Inference from Small-Cohort Aggregate Data
AgentODE uses LLMs to discover ODE structures and infer parameter distributions from aggregate data, recovering consistent structures on benchmarks and RDEB clinical data with 231 observations from 46 patients.
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Beating micromagnetic limits on skyrmion stability by long-range frustration
Long-range exchange frustration in atomistic spin-lattice models can double skyrmion collapse barriers while keeping micromagnetic parameters fixed, revealing a limitation of continuum approximations.
-
Pathway variability, coat stiffening and mechanical adaptation during clathrin-mediated endocytosis
Hybrid simulation and non-Euclidean elasticity theory demonstrate that clathrin coats develop adaptive rigidity and memory during growth, producing flat, stalled, or closed outcomes through two energy-landscape gates and matching experiments without fitted parameters.
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Sample Complexity of Scientific Discovery: PAC Learnability of Compositional Function Trees
Proves that Rademacher complexity of depth-d compositional trees over finite operator vocabulary is controlled by (K b L)^{d} / sqrt(n) under Lipschitz conditions on operators.
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$\text{DT}^2$: Decision-Targeted Digital Twins
DT² trains digital twins to preserve pairwise policy rankings from fitted Q-evaluation on offline data rather than minimizing one-step transition errors, improving policy ranking and reducing decision regret.
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Tearing Instability in Gyrotropic MHD: Effects of Equilibrium Pressure Anisotropy
Equilibrium pressure anisotropy modifies the tearing-mode growth-rate prefactor through parameters A and R0 while retaining the S^{-1/2} Lundquist scaling in gyrotropic MHD.
-
Centauric 1-Jettiness in DIS and Universal Power Corrections
Introduces Centauric 1-jettiness in DIS, derives N3LL resummation matched to NLO, and establishes universal non-perturbative power corrections scaling as 1/R via reduction to rescaled hemisphere soft function.
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Data-Driven Discovery of a Simple Phantom-Crossing Dark Energy Parametrization
Bayesian reconstruction and exhaustive symbolic regression on CMB, BAO, and supernova data yield the one-parameter dark energy parametrization w(a) = w0 / sqrt(a) that fits observations comparably to CPL and better than LambdaCDM.
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Learning the Universe: The Structure of Dust Attenuation Curves in Galaxy Simulations
Four parameters suffice to describe dust attenuation curve diversity in TNG simulations, yielding a new symbolic-regression model that recovers curves and fluxes better than existing parameterizations while linking parameters to SFR surface density, metallicity, and geometry.
-
FunctionEvolve: Structure-Guided Symbolic Regression with LLMs
FunctionEvolve recovers 107 exact symbolic forms out of 129 synthetic tasks (82.9% SA@50) by using expression-tree structure for evolutionary search, parent selection, mutation, and coefficient scoring with LLMs.
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Discovering Thermodynamically Admissible Dissipation Potentials via Grammar-Based Symbolic Regression
A convexity-preserving grammar enables symbolic regression to discover thermodynamically admissible dissipation potentials for generalized standard materials from noisy data.
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Symbolic Regression via Latent Iterative Refinement
LEE performs iterative amortized inference in a functionally grounded latent space to produce 2-10x simpler symbolic expressions than strong baselines on SRBench.
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The Neural Compiler: Program-to-Network Translation for Hybrid Scientific Machine Learning
The Neural Compiler converts symbolic programs into exact differentiable PyTorch modules for hybrid scientific machine learning, enabling precise encoding of known physics with few trainable parameters.
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Symbolic Classification-Enabled LHC Limits Online BSM Global Fits
Symbolic regression produces an approximate classifier for LHC exclusion limits that enables their direct inclusion during pMSSM global fits.
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Graph-based automated discovery of concise soil hydraulic functions from data: beyond the Mualem - van Genuchten model
A graph-based automated model discovery framework identifies new concise soil hydraulic functions from data that outperform the Mualem-van Genuchten model across 249 soil samples.
-
Diversified Residual Symbolic Regression
DRSR uses Quality-Diversity to produce diverse symbolic regression expressions differing in residual distributions, enabling post-search selection on synthetic and astronomical data.
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The finite expression method for turbulent dynamics with high-order moment recovery
A two-stage symbolic regression plus generative model framework recovers governing interaction terms and forcing in stochastic triad models while accurately predicting statistical moments up to order five.
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Back to the Beginning of Heuristic Design: Bridging Code and Knowledge with LLMs
A knowledge-first approach to LLM-driven automatic heuristic design in combinatorial optimization yields better discovery efficiency, transfer, and generalization than code-centric baselines by formalizing a distortion-compression trade-off.
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Reconstructing conformal field theoretical compositions with Transformers
Transformers reconstruct the constituent RCFTs in tensor-product theories from low-energy spectra, reaching 98% accuracy on WZW models and generalizing to larger central charges with few out-of-domain examples.
-
Additive Atomic Forests for Symbolic Function and Antiderivative Discovery
A derivative algebra with EML and SOL primitives plus additive atomic forests enables simultaneous symbolic recovery of functions and antiderivatives from data, matching or exceeding XGBoost on 13 of 17 benchmarks with interpretable formulas.
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Machine Collective Intelligence for Explainable Scientific Discovery
Machine collective intelligence uses coordinated AI agents to evolve symbolic hypotheses and recover governing equations from observations in deterministic, stochastic, and uncharacterized systems, achieving up to six orders of magnitude better extrapolation than neural networks with 5-40 parameters
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LLM-ODE: Data-driven Discovery of Dynamical Systems with Large Language Models
LLM-ODE integrates large language models into genetic programming to guide symbolic search for governing equations of dynamical systems, outperforming classical GP on 91 test cases in efficiency and solution quality.
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In-Context Symbolic Regression for Robustness-Improved Kolmogorov-Arnold Networks
In-context symbolic regression methods improve robustness of symbolic formula recovery from KANs, cutting median OFAT test MSE by up to 99.8 percent across hyperparameter sweeps.
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Symbolic recovery of PDEs from measurement data
Symbolic rational-function networks recover an admissible PDE from noiseless complete measurements and select the regularization-minimizing parameterization within the architecture.
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Kolmogorov-Arnold Chemical Reaction Neural Networks for learning pressure-dependent kinetic rate laws
KA-CRNNs learn pressure-dependent and collider-specific kinetic rate laws from data using Kolmogorov-Arnold activations inside a CRNN framework, outperforming interpolative methods by 2.88x in MSE on two proof-of-concept reactions.
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Using Symbolic Regression to Emulate the Radial Fourier Transform of the S\'ersic profile for Fast, Accurate and Differentiable Galaxy Profile Fitting
Symbolic regression yields an emulator for the radial Fourier transform of the Sérsic profile that enables 2.5 times faster galaxy profile fitting with minimal accuracy loss.
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AlphaEvolve: A coding agent for scientific and algorithmic discovery
AlphaEvolve is an LLM-orchestrated evolutionary coding agent that discovered a 4x4 complex matrix multiplication algorithm using 48 scalar multiplications, the first improvement over Strassen's algorithm in 56 years, plus optimizations for Google data centers and hardware.
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Distilling human mobility models with symbolic regression
Symbolic regression on mobility data recovers gravity and distance-decay models while identifying a new exponential-power-law form linked to maximum entropy.
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Identifying Topological Invariants of Non-Hermitian Systems via Domain-Adaptive Multimodal Model for Mathematics
There are only finitely many k-vertex-critical graphs in the classes (P4+ℓP1, B4(m), B3(m)+)-free and (P4+ℓP1, 2P2)-free for all k, ℓ, m, with improved χ-bounds for (P4+ℓP1, Kk)-free graphs.
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DeepPySR -- A Symbolic Regression Framework with Dynamic Pruning, Pareto Selection, and Hierarchical Composition for Real-World Scientific Discovery
DeepPySR adds dynamic pruning, exponential Pareto selection, and hierarchical layers to PySR and reports better R²/F1 plus domain-aligned formulas on Feynman and seven real-world datasets.
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Skin friction prediction for attached flows based on two-dimensional inviscid solutions
A progressive symbolic regression framework discovers an interpretable analytical formula chain for skin friction prediction across subsonic to hypersonic attached flows using inviscid surface flow features.
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Learning Turbulence Closures with Physics-Informed Neural Networks for the Rayleigh-Taylor Transition to Turbulence
A PINN-derived analytical correction to the dissipation-production coefficient C_ε0 in a k-ε-b RANS model captures the Rayleigh-Taylor transition to turbulence by coupling dissipation to the Froude number and directed-to-total kinetic energy ratio.
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Calabi-Yau Metrics with Full Moduli Dependence
Approximate analytic Ricci-flat metrics on a one-parameter bi-cubic Calabi-Yau family with explicit moduli dependence obtained via symbolic regression on numerical data, achieving percent-level agreement.
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LLM-ACES: Closed-Loop Discovery of Dynamical Systems with LLM-Guided Adaptive Search
LLM-ACES is a closed-loop method that combines LLM-proposed operator priors with disagreement-driven adaptive data acquisition to discover governing ODEs, reporting lowest median NMSE and 46-52% symbolic accuracy on 122 systems.
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Synergizing Physically Constrained MCMC and Chemical-Informed Gaussian Processes for Reaction Network Discovery
PC-MCMC-CIGP integrates spike-and-slab MCMC with hard physical screening and CIGP residuals to discover reaction networks, distinguishing pathways on H2+Br2 and raising styrene epoxidation yield 12.5% over GP-BO.
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An interpretable closed form for entanglement entropy from bitstrings, guided by a graph neural network
Derives a six-term linear closed form for entanglement entropy from boundary correlators in Rydberg arrays via GNN-guided search, achieving 0.024 nats MAE and portable performance after size-dependent refitting.
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Towards Diverse Scientific Hypothesis Search with Large Language Models
A parallel-tempering evolutionary framework for LLM hypothesis search improves both quality and diversity of candidates in molecular, equation, and algorithm discovery under fixed validation budgets.
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Decision-Making under Combinatorial Risk
People navigate combinatorial risk by focusing on core features like post-investment success probabilities rather than computing the full induced distribution, unless the PMF is explicitly displayed.
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A universal model for the accretion rates and formation times of dark matter halos
A six-parameter function of peak height ν, power spectrum slope n_eff, and growth rate α_eff accurately describes median halo mass accretion rates from simulations in ΛCDM and Einstein-de Sitter cosmologies at z=0-14.
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CMBolic: Symbolic emulators for the Cosmic Microwave Background. I. Lensing
CMBolic supplies analytic emulators for CMB lensing spectra achieving 0.27-0.32% mean fractional error, validated against CLASS on ACT DR6 and Planck lensing data.
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LLM-AutoSciLab: Closed-Loop Scientific Discovery via Active Experimentation with LLMs
LLM-AutoSciLab proposes an LLM-driven closed-loop system for hypothesis generation and adaptive experiment selection that reports higher accuracy and 2-5x better sample efficiency than baselines on new chemistry and gene-network discovery benchmarks.
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Symbolic Density Estimation for Discrete Distributions
SDE recovers closed-form PMFs for discrete distributions via evolutionary search guided by domain priors, recovering all benchmark families with accurate parameters and improving mixture fits on real data.
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STRIDE: A Self-Reflective Agent Framework for Reliable Automatic Equation Discovery
STRIDE is a self-reflective agent framework that improves accuracy, OOD robustness, and structural recovery in LLM-based symbolic regression by integrating generation, evaluation, repair, and diversity-preserving memory.
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FePySR: A Neural Feature Extraction Framework for Efficient and Scalable Symbolic Regression
FePySR uses a neural network to pre-extract valid features before PySR search, recovering more equations than baselines on benchmarks and identifying governing ODEs in 24 of 100 biological cases where PySR finds none.
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GESR: A Genetic Programming-Based Symbolic Regression Method with Gene Editing
GESR uses two BERT models to intelligently direct mutations and crossovers inside genetic programming, yielding higher efficiency and competitive accuracy on symbolic regression benchmarks.
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Discovery of Nonlinear Dynamics with Automated Basis Function Generation
AutoSINDy automatically builds a tailored basis library from PySR symbolic regression and applies SINDy to recover ground-truth nonlinear dynamics with 92.8% success under noise.