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.
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.
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
Latent Grammar Flow discovers ODEs by placing grammar-based equation representations in a discrete latent space, using a behavioral loss to cluster similar equations, and sampling via a discrete flow model guided by data fit and constraints.
First direct constraints on total cosmic backreaction over a significant redshift range are consistent with vanishing backreaction within 1 sigma but are too weak to exclude meaningful backreaction.
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.
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.
Symbolic rational-function networks recover an admissible PDE from noiseless complete measurements and select the regularization-minimizing parameterization within the architecture.
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.
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.
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.
Symbolic regression on mobility data recovers gravity and distance-decay models while identifying a new exponential-power-law form linked to maximum entropy.
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.
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.
Tensor perturbations from first-order phase transitions and domain wall annihilation induce curvature fluctuations at second order that form primordial black holes, allowing asteroid-mass PBHs to comprise all dark matter for specific parameter ranges with associated gravitational wave peaks in LISA,
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.
-
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.
-
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.
-
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.
-
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.
-
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.
-
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.
-
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.
-
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.
-
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
-
Neuro-Symbolic ODE Discovery with Latent Grammar Flow
Latent Grammar Flow discovers ODEs by placing grammar-based equation representations in a discrete latent space, using a behavioral loss to cluster similar equations, and sampling via a discrete flow model guided by data fit and constraints.
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First observational constraints on cosmic backreaction over an extended redshift range
First direct constraints on total cosmic backreaction over a significant redshift range are consistent with vanishing backreaction within 1 sigma but are too weak to exclude meaningful backreaction.
-
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.
-
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.
-
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.
-
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.
-
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.
-
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.
-
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.
-
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.
-
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.
-
Primordial Black Hole from Tensor-induced Density Fluctuation: First-order Phase Transitions and Domain Walls
Tensor perturbations from first-order phase transitions and domain wall annihilation induce curvature fluctuations at second order that form primordial black holes, allowing asteroid-mass PBHs to comprise all dark matter for specific parameter ranges with associated gravitational wave peaks in LISA,
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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.
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Discovering Ordinary Differential Equations with LLM-Based Qualitative and Quantitative Evaluation
DoLQ employs a sampler agent, parameter optimizer, and LLM-based scientist agent to iteratively propose, refine, and evaluate ODE candidates, yielding higher success rates and better symbolic term recovery than prior symbolic regression methods on multi-dimensional benchmarks.
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Programmatic Context Augmentation for LLM-based Symbolic Regression
Programmatic context augmentation lets LLM-based symbolic regression perform code-driven data analysis during search, yielding superior efficiency and accuracy over baselines on LLM-SRBench.
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Interpretable Analytic Formulae for GWTC-4 Binary Black Hole Population Properties via Symbolic Regression
Symbolic regression on GWTC-4 posteriors yields closed-form analytic formulae for merger-rate evolution, effective-spin dependencies on mass ratio and redshift, and conditional mass-ratio distributions at specific primary mass peaks.
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Physics-Informed Neural Networks for Biological $2\mathrm{D}{+}t$ Reaction-Diffusion Systems
BINNs are extended to 2D+t systems and combined with symbolic regression to recover reaction-diffusion models of lung cancer cell dynamics from time-lapse microscopy data.
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Machine Learning Hamiltonian Dynamical Systems with Sparse and Noisy Data
ASRNNs recover Hamiltonian dynamics and symbolic equations from trajectories with only two irregularly spaced noisy points by preserving symplectic structure without derivative estimation.
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Discovering quantum phenomena with Interpretable Machine Learning
Variational autoencoders combined with symbolic regression extract physically meaningful representations and order parameters from raw quantum measurement data, revealing new phenomena such as corner-ordering in Rydberg arrays.
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Into the Gompverse: A robust Gompertzian reionization model for CMB analyses
A Gompertzian reionization model with three nuisance parameters demotes optical depth to a derived quantity, reducing its uncertainty by a factor of three and revealing potential neutrino mass tension in CMB analyses.
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Model-independent constraints on generalized FLRW consistency relations with bootstrap-based symbolic regression
Bootstrap-based symbolic regression on supernova and BAO data finds mild 2-4 sigma deviations from FLRW consistency relations, which if real would rule out most FLRW-based solutions to cosmological tensions.
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Automatic Construction of Clinical Scoring Systems with LLM Agents
AgentScore uses LLM agents for semantically guided search over clinical scoring rules combined with data-driven verification, outperforming prior score generation methods on eight tasks and established guidelines on two externally validated tasks.
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Generating Literature-Driven Scientific Theories at Scale
Literature-grounded LLM synthesis of theories from 13.7k papers yields 2.9k theories that better match evidence and predict future results from 4.6k subsequent papers than parametric baselines.
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HoloNet: Toward a Unified Einstein-Maxwell-Dilaton Framework of QCD
A neural network learns holographic bulk functions from lattice QCD data at zero chemical potential and embeds them into an EMD model to describe finite-density QCD and locate the critical end point.
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statmorph-lsst: Quantifying and correcting morphological biases in galaxy surveys
Morphological metrics in galaxy images suffer systematic biases from resolution, depth, and noise that can be quantified and corrected empirically, with new metrics proposed to reduce those effects.
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ASP-Assisted Symbolic Regression: Uncovering Hidden Physics in Fluid Mechanics
A hybrid symbolic regression and answer set programming framework derives compact, physically plausible equations for velocity and pressure in 3D laminar channel flow from simulation data.
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On the definition and importance of interpretability in scientific machine learning
Interpretability in SciML requires mechanistic understanding rather than sparsity, and prior knowledge is often essential for interpretable scientific discovery.
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LLM-FE: Automated Feature Engineering for Tabular Data with LLMs as Evolutionary Optimizers
LLM-FE is a framework that treats feature engineering as LLM-driven program search with data feedback, reporting consistent gains over baselines on classification and regression tabular tasks.
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SINDyG: Sparse Identification of Nonlinear Dynamical Systems from Graph-Structured Data, with Applications to Stuart-Landau Oscillator Networks
SINDyG extends SINDy by adding a graph-informed penalty to sparse regression, yielding more accurate and simpler models of network dynamics on Stuart-Landau oscillator networks than standard SINDy.
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Guiding Multi-Objective Genetic Programming with Description Length Improves Symbolic Regression Solutions
Post-selection with DL or FBF after multi-objective GP search improves test-set performance over AIC/BIC baselines on noisy synthetic and real regression tasks, while using DL directly as fitness often causes premature convergence to overly simple models.
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Prediction Is Not Physics: Learning and Evaluating Conserved Quantities in Neural Simulators
Conservation Discovery Networks recover analytical energy with R² ≥ 0.996 in Hamiltonian systems using temporal consistency and λ_align=0.2, but collapse without alignment and show mixed noise robustness.
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Discovering interpretable low-dimensional dynamics using maximum entropy
Edwin integrates dynamic maximum entropy dimensionality reduction with symbolic regression to recover physically interpretable low-dimensional dynamics from high-dimensional observations that generalize to unseen conditions.
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Discovery of Interpretable Surrogates via Agentic AI: Application to Gravitational Waves
GWAgent agentic workflow produces analytic surrogates for eccentric BBH waveforms with 6.9e-4 median mismatch and 8.4x speedup, outperforming baselines, and infers eccentricity for GW200129.
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Balance-Guided Sparse Identification of Multiscale Nonlinear PDEs with Small-coefficient Terms
BG-SINDy reformulates l0-constrained regression as term-level l2,0 regularization and uses progressive pruning guided by balance contributions to recover small-coefficient terms in multiscale PDEs.
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Singularity Formation: Synergy in Theoretical, Numerical and Machine Learning Approaches
The work introduces a modulation-based analytical method for singularity proofs in singular PDEs and refines ML techniques like PINNs and KANs to identify blowup solutions, with application to the open 3D Keller-Segel problem.
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What is the diatomic molecule with the largest dipole moment?
A machine learning model based on atomic properties predicts diatomic dipole moments, screens the periodic table for the largest values, and condenses into an analytical expression.