Symbolic Regression

Follow this topic Ask about this topic

Latest papers 137

19 papers in the last four weeks, up 58% on the four weeks before. 0.1% of all new papers.

May 15, 2026

Diversified Residual Symbolic Regression

Symbolic regression (SR) aims to discover explicit mathematical expressions that explain observed data and is widely used in domains where interpretability is essential. Because interpretability requires expressions to reflect meaningful regularities, SR is sensitive to observations that deviate from the dominant relationship. Such irregular observations, or outliers, are common in real-world data and can hinder SR from identifying underlying regularities. Robust regression mitigates this by downweighting observations with large residuals. However, deciding which observations should be treated as outliers is often ambiguous and depends on user interpretation and domain knowledge, a perspective largely overlooked in existing SR studies. This motivates approaches that present multiple candidate expressions, allowing users to examine different residual patterns and choose expressions consistent with their expertise. We propose diversified residual symbolic regression (DRSR), which achieves high predictive accuracy while promoting diversity with respect to residual patterns based on the Quality-Diversity paradigm. DRSR collects multiple expressions that fit the data well but differ in how residuals are distributed, enabling post-search selection aligned with domain knowledge. On a synthetic mixture dataset, DRSR produces more diverse expressions than conventional SR while capturing multiple underlying relationships. On a real-world astronomical dataset, DRSR discovers multiple expressions consistent with known physical relationships.

Koki Ikeda, Masahiro Nomura, Ryoki Hamano

CyberAgent Shibuya, Tokyo, Japan · Institute of Science Tokyo Meguro, Tokyo, Japan

Symbolic Regression

May 12, 2026

FePySR: A Neural Feature Extraction Framework for Efficient and Scalable Symbolic Regression

A fundamental challenge in symbolic regression (SR) is efficiently recovering complex mathematical expressions from observational data. Although this problem is NP-hard, many expressions of practical interest decompose naturally into combinations of nonlinear feature modules, concentrating structural complexity into a small number of reusable components. Here, we introduce FePySR, a two-stage framework that reduces the SR search space by extracting valid features prior to equation search. FePySR first employs a heterogeneous neural network to constrain observational data to a set of candidate expressions, then performs structural optimization within this refined expression space using PySR. Across five standard benchmarks, FePySR outperforms state-of-the-art methods by achieving higher equation recovery rates. On a set of 75 highly complex synthesized equations, FePySR recovers 36 equations, while producing substantially smaller mean squared errors on the remaining unrecovered cases, with reduced computation time compared to PySR. FePySR's first stage also maintains consistent performance under varying numbers of selected top features and increasing levels of noise in the observational data. Applied to ordinary differential equations governing biological systems, FePySR successfully identifies governing equations in 24 out of 100 tests where PySR recovers none. Taken together, FePySR is a generalizable framework that can enhance the SR solvers, enabling the efficient and reliable recovery of symbolic expressions across scientific domains.

Zhiming Yu, Wangtao Lu, Xin Lai

School of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang, China · Faculty of Medicine and Health Technology, Tampere University, Tampere, Finland

Nonlinear System Identification Automated Feature Engineering Symbolic Regression

May 11, 2026

A Comparative Study of Model Selection Criteria for Symbolic Regression

Effective model selection is critical in symbolic regression (SR) to identify mathematical expressions that balance accuracy and complexity, and have low expected error on unseen data. Many modern implementations of genetic programming (GP) for SR generate a set of Pareto optimal candidate solutions, but reliable automatic selection of solutions that generalize well remains an open issue. Current literature offers various information-theoretic and Bayesian approaches, yet comprehensive comparisons of their performance across different data regimes are limited. This study presents a systematic empirical comparison of widely used selection criteria: the Akaike information criterion (AIC), the corrected AIC (AICc), the Bayesian information criterion (BIC), minimum description length (MDL), as well as Efron's bootstrap estimate for the in-sample prediction error on seven synthetic datasets with Gaussian noise. We rank candidate expressions generated by perturbing ground-truth functions to assess generalization error and selection probability of the ground-truth expression. Our findings reveal that MDL consistently identifies models with the lowest test error and the shortest length across most datasets. While no single criterion dominates all results, MDL and BIC produced the highest probability of selecting the ground-truth expressions.

Ali Soltani, Gabriel Kronberger, Fabricio Olivetti de Franca +2

Aarhus University Department of Mechanical Engineering Aarhus, Denmark · Heuristic and Evolutionary Algorithms Laboratory (HEAL) University of Applied Sciences Upper Austria Hagenberg, Austria · Federal University of ABC Center for Mathematics, Computing, and Cognition Brazil +1

Model Selection Symbolic Regression

May 11, 2026

The finite expression method for turbulent dynamics with high-order moment recovery

Turbulent dynamical systems are characterized by nonlinear interactions and stochastic effects that generate coupled statistical quantities, such as non-zero higher-order moments, which are difficult to capture from data with accuracy. We propose a two-stage data-driven modeling framework that combines symbolic regression with generative models to jointly identify the governing dynamics and predict their key statistical quantities. In Stage I of the framework, the Finite Expression Method (FEX) is adopted to discover closed-form expressions of the deterministic dynamics, recovering nonlinear interaction terms and external forcing without predefined libraries. In Stage II, generative models are introduced to learn the residual stochastic components as a refined correction to the model error from the Stage I approximation, enabling accurate characterization of higher-order statistics. Theoretical analysis establishes the consistency of the symbolic estimator and quantifies the estimation error in terms of data size and numerical discretization. The model performance is verified through detailed numerical experiments on the stochastic triad models across multiple regimes, demonstrating that the framework successfully recovers interaction terms and forcing expressions, and accurately predicts statistical moments up to order five. These results highlight the potential of integrating interpretable symbolic discovery with data-driven stochastic modeling for complex turbulent systems.

Xingjian Xu, Di Qi, Chunmei Wang

Department of Mathematics, University of Florida, Gainesvile, FL 32611 · Department of Mathematics, Purdue University, West Lafayette, IN 47907

Nonlinear System Identification Computational Fluid Dynamics Symbolic Regression

May 11, 2026

GESR: A Genetic Programming-Based Symbolic Regression Method with Gene Editing

Mathematical formulas serve as a language through which humans communicate with nature. Discovering mathematical laws from scientific data to describe natural phenomena has been a long-standing pursuit of humanity for centuries. In the field of artificial intelligence, this challenge is known as the symbolic regression problem. Among existing symbolic regression approaches, Genetic Programming (GP) based on evolutionary algorithms remains one of the most classical and widely adopted methods. GP simulates the evolutionary process across generations through genetic mutation and crossover. However, mutations and crossovers in GP are entirely random. While this randomness effectively mimics natural evolution, it inevitably produces both beneficial and detrimental variations. If there existed a metaphorical God capable of foreseeing which genetic mutations or crossovers would yield superior outcomes and performing targeted gene editing accordingly, the efficiency of evolution could be substantially improved. Motivated by this idea, we propose in this paper a symbolic regression approach based on gene editing, termed GESR. In GESR, we trained two "hands of God" (two BERT models). Among them, the first leverages the BERT's masked language modeling capability to guide the mutation of genes (expression symbols). The other BERT model guides the crossover of individual genes by predicting the crossover point. Experimental results demonstrate that GESR significantly improves computational efficiency compared with traditional GP algorithms and achieves strong overall performance across multiple symbolic regression tasks.

Yanjie Li, Liping Zhang, Min Wu +6

AnnLab, Institute of Semiconductors, Chinese Academy of Sciences, Beijing, China. · School of Electronic, Electrical and Communication Engineering,2026 University of Chinese Academy of Sciences, Beijing, China.

Evolutionary Optimization Genetic Programming Symbolic Regression

May 10, 2026

Discovery of Nonlinear Dynamics with Automated Basis Function Generation

Discovering governing equations from observational data remains a fundamental challenge in scientific modeling, particularly when the underlying mathematical structure is unknown. Traditional sparse identification methods like SINDy excel at discovering parsimonious models but require researchers to specify candidate basis functions a priori, a limitation that often leads to model failure when critical terms are omitted or when systems exhibit unconventional dynamics. Purely symbolic regression approaches offer unlimited flexibility but struggle with noise sensitivity and frequently produce overly complex, unstable equations. We present AutoSINDy, a hybrid Discovery-then-Solve framework that combines the exploratory power of symbolic regression with the robust sparsity-promoting capabilities of SINDy. Our method operates in three stages: (1) PySR-based symbolic regression discovers candidate functional forms from bootstrapped data chunks; (2) a curation pipeline decomposes, expands, and filters these expressions using collinearity analysis to construct a minimal yet comprehensive library; and (3) SINDy identifies sparse governing equations from this custom-tailored library. Extensive experiments across canonical nonlinear systems demonstrate that AutoSINDy consistently recovers ground-truth equations even under high observational noise, achieving a ground-truth recovery rate of 92.8% across all trials. Compared with standard SINDy using enriched libraries and standalone symbolic regression, AutoSINDy achieves higher predictive accuracy, superior generalization to unseen trajectories, and substantially lower symbolic complexity.

Mohammad Amin Basiri, Charles Nicholson

Data Science and Analytics Institute, University of Oklahoma, Norman, OK, USA · School of Industrial and Systems Engineering, University of Oklahoma, Norman, OK, USA

Sparse Identification of Nonlinear Dynamics Nonlinear System Identification Autonomous Scientific Discovery ODE Discovery Symbolic Regression

May 8, 2026

Discovering Ordinary Differential Equations with LLM-Based Qualitative and Quantitative Evaluation

Discovering governing differential equations from observational data is a fundamental challenge in scientific machine learning. Existing symbolic regression approaches rely primarily on quantitative metrics; however, real-world differential equation modeling also requires incorporating domain knowledge to ensure physical plausibility. To address this gap, we propose DoLQ, a method for discovering ordinary differential equations with LLM-based qualitative and quantitative evaluation. DoLQ employs a multi-agent architecture: a Sampler Agent proposes dynamic system candidates, a Parameter Optimizer refines equations for accuracy, and a Scientist Agent leverages an LLM to conduct both qualitative and quantitative evaluations and synthesize their results to iteratively guide the search. Experiments on multi-dimensional ordinary differential equation benchmarks demonstrate that DoLQ achieves superior performance compared to existing methods, not only attaining higher success rates but also more accurately recovering the correct symbolic terms of ground truth equations. Our code is available at https://github.com/Bon99yun/DoLQ.

Sum Kyun Song, Bong Gyun Shin, Jae Yong Lee

Department of Artificial Intelligence, Chung-Ang University, Seoul, Republic of Korea · Department of AI Big Data, Daejin University, Pocheon-si, Republic of Korea.

AI Agents for Scientific Discovery Autonomous Scientific Discovery Large Language Model-Guided Scientific Discovery ODE Discovery Symbolic Regression

May 5, 2026

Complex Equation Learner: Rational Symbolic Regression with Gradient Descent in Complex Domain

Symbolic regression aims to discover interpretable equations from data, yet modern gradient-based methods fail for operators that introduce singularities or domain constraints, including division, logarithms, and square roots. As a result, Equation Learner-type models typically avoid these operators or impose restrictions, e.g. constraining denominators to prevent poles, which narrows the hypothesis class. We propose a complex weight extension of the Equation Learner that mitigates real-valued optimization pathologies by allowing optimization trajectories to bypass real-axis degeneracies. The proposed approach converges stably even when the target expression has real-domain poles, and it enables unconstrained use of operations such as logarithm and square root. We Validate the method on symbolic regression benchmarks and show it can recover singular behavior from experimental frequency response data.

Sergei Garmaev, Maurice Gauché, Olga Fink

Intelligent Maintenance and Operations Systems Laboratory, EPFL, Lausanne, Switzerland · Mathematics Section, EPFL, Lausanne, Switzerland

Complex-Valued Neural Networks Symbolic Regression

May 4, 2026

Programmatic Context Augmentation for LLM-based Symbolic Regression

Symbolic regression (SR), the task of discovering mathematical expressions that best describe a given dataset, remains a fundamental challenge in scientific discovery. Traditional approaches, primarily based on genetic algorithms and related evolutionary methods, have proven useful but suffer from scalability and expressivity limitations. Recently, large language model (LLM)-based evolutionary search methods have been introduced into SR and show promise. However, existing LLM-based approaches typically rely on scalar evaluation metrics, such as mean squared error, as the sole source of feedback during the search process, thereby overlooking the rich information embedded in the dataset. To address this limitation, we propose a novel LLM-based evolutionary search framework that incorporates programmatic context augmentation. By enabling code-based interactions with the dataset, our method can actively perform data analysis and extract informative signals, beyond aggregated evaluation scores. We evaluate our framework on advanced benchmarks, such as LLM-SRBench, and demonstrate superior efficiency and accuracy compared to strong baselines.

Hao Liu, Xiao-Wen Yang, Atharva Sehgal +4

Caltech · Nanjing University · University of Texas at Austin +1

Large Language Model-Guided Optimization Large Language Model-Guided Scientific Discovery Symbolic Regression

May 1, 2026

Deep Variational Inference Symbolic Regression

Symbolic regression discovers explicit, interpretable equations without assuming a functional form in advance. A Bayesian approach strengthens this through probability distributions over candidate expressions, thus quantifying uncertainty in the presence of noisy and limited data. Deep Symbolic Regression (DSR) uses a neural network to generate symbolic expressions, but it is designed to identify a single best-fitting expression rather than infer a posterior distribution over models. We introduce Deep Variational Inference Symbolic Regression (DVISR), a variational Bayesian extension of DSR. DVISR replaces the original reward with the integrand of the evidence lower bound. It also extends the network architecture to output distributions over constants within expressions, enabling posterior inference over both expression trees and their associated constants. We show that DVISR can recover the true posterior in simple settings, both with and without constant tokens, and we examine how its performance changes as the size of the expression space increases. These results position DVISR as a step toward scalable Bayesian symbolic regression with uncertainty over full symbolic models.

James Butterworth, Gevik Grigorian, Alejandro DiazDelaO

Clinical Operational Research Unit University College London London, United Kingdom · Department of Computer Science University College London London, United Kingdom

Uncertainty Quantification Variational Inference Symbolic Regression

May 1, 2026

Additive Atomic Forests for Symbolic Function and Antiderivative Discovery

We present a framework for the simultaneous symbolic recovery of a function and its antiderivative from data. The framework rests on three ideas. First, a derivative algebra: the observation that the product rule

\frac{d}{dx}[f \cdot g] = f'g + fg'

and the chain rule, applied to a seed set of elementary functions, generate a self-expanding system of function-derivative pairs -- a living library that grows each time a new function is discovered. Second, two complementary primitives -- EML

\,(e^u - \ln v)

, which is theoretically complete for all elementary functions, and SOL

\,(\sin u - \cos v)

, introduced here, which makes trigonometric atoms available at depth~~1 instead of depth~~

\sim

8 -- that seed the library with core atoms cheaply. Third, additive atomic forests: finite sums of primitive trees, optionally composed via multiplicative nodes, whose derivatives are fitted to data by continuous optimisation or by exhaustive search over the library. Because differentiation of each atom is determined by construction, the forest simultaneously encodes a symbolic expression

F

and its derivative

F'

; no symbolic integration step is required. The library is not a fixed object: it self-constructs from a small seed set by recursive application of the product rule, chain rule, and the two primitives, and it can grow as newly discovered functions are folded back in. The larger the library, the richer the expressible class of candidate functions. We give conditional completeness, additive-depth, and analytic simultaneous-recovery results for the framework. Empirically, in our reported runs on 17 classification benchmarks, sparse atom combinations match or exceed XGBoost on 13 datasets while producing interpretable formulas.

Reda Belaiche

Université Paris-Est Créteil, Vitry-sur-Seine, France

Symbolic Regression

Apr 30, 2026

Machine Collective Intelligence for Explainable Scientific Discovery

Deriving governing equations from empirical observations is a longstanding challenge in science. Although artificial intelligence (AI) has demonstrated substantial capabilities in function approximation, the discovery of explainable and extrapolatable equations remains a fundamental limitation of modern AI, posing a central bottleneck for AI-driven scientific discovery. Here, we present machine collective intelligence, a unified paradigm that integrates two fundamental yet distinct traditions in computational intelligence--symbolism and metaheuristics--to enable autonomous and evolutionary discovery of governing equations. It orchestrates multiple reasoning agents to evolve their symbolic hypotheses through coordinated generation, evaluation, critique, and consolidation, enabling scientific discovery beyond single-agent inference. Across scientific systems governed by deterministic, stochastic, or previously uncharacterized dynamics, machine collective intelligence autonomously recovered the underlying governing equations without relying on hand-crafted domain knowledge. Furthermore, the resulting equations reduced extrapolation error by up to six orders of magnitude relative to deep neural networks, while condensing 0.5-1 million model parameters into just 5-40 interpretable parameters. This study marks an important shift in AI toward the autonomous discovery of principled scientific equations.

Gyoung S. Na, Chanyoung Park

Korea Research Institute of Chemical Technology (KRICT) · Korea Advanced Institute of Science and Technology (KAIST)

Explainable Artificial Intelligence Autonomous Scientific Discovery Symbolic Regression

Apr 25, 2026

Architecture-Induced Recoverability Bias in Differentiable Symbolic Regression

Symbolic regression aims to recover closed-form expressions from numerical data, but in differentiable symbolic regression the recovered expression depends not only on the grammar but also on the fixed architecture through which variables are routed during training. This is relevant to signal-processing settings in which closed-form models and interpretable nonlinear structure are useful. This architecture-specific effect has rarely been isolated directly, because existing comparisons often vary architecture together with operator family, grammar, or search procedure. Three depth-3 architectures are compared across twenty-four operator--shape--leaf combinations, holding operator family, grammar, and training protocol fixed as far as possible while varying the variable-routing architecture. Recovery changes from

0/64

to

64/64

trials on the same target under an architecture-plus-native-training-protocol comparison. The best architecture on one target is the worst on another, and trees with two equal-depth subtrees fail in every configuration tested (

0/3{,}776

). As a proof-of-concept mitigation, a small architecture set is trained and the hardened expression with the lowest held-out RMSE is selected. On the jointly-run subset, this improves recovery from

34.4\%

for the only architecture present in all three configurations to

50.1\%

. On a Shockley diode target, the validation selector recovers cases missed by that baseline architecture, which by itself recovers

0/32

seeds. Since the jointly-run subset contains only three configurations, the selector result is evidence that validation-based architecture selection is promising, not a complete benchmark. These results support treating architecture as a measurable design variable that should be reported, stress-tested, and selected using held-out validation rather than fixed a priori.

Chakshu Gupta, Theodore J. LaGrow

⋆College of Computing, Georgia Institute of Technology, Atlanta, GA, USA · † College of Lifetime Learning, Georgia Institute of Technology, Atlanta, GA, USA

Algorithm Selection Symbolic Regression

Apr 20, 2026

AC-SINDy: Compositional Sparse Identification of Nonlinear Dynamics

We present AC-SINDy, a compositional extension of the Sparse Identification of Nonlinear Dynamics (SINDy) framework that replaces explicit feature libraries with a structured representation based on arithmetic circuits. Rather than enumerating candidate basis functions, the proposed approach constructs nonlinear features through compositions of linear functions and multiplicative interactions, yielding a compact and scalable parameterization and enabling sparsity to be enforced directly over the computational graph. We also introduce a formulation that separates state estimation from dynamics identification by combining latent state inference with shared dynamics and multi-step supervision, improving robustness to noise while preserving interpretability. Experiments on nonlinear and chaotic systems demonstrate that the method recovers accurate and interpretable governing equations while scaling more favorably than standard SINDy.

Peter Racioppo

Independent Researcher Los Angeles, CA 90731

Latent Dynamics Modeling Sparse Identification of Nonlinear Dynamics Nonlinear System Identification Symbolic Regression

Apr 19, 2026

On the Generalization Bounds of Symbolic Regression with Genetic Programming

Symbolic regression (SR) with genetic programming (GP) aims to discover interpretable mathematical expressions directly from data. Despite its strong empirical success, the theoretical understanding of why GP-based SR generalizes beyond the training data remains limited. In this work, we provide a learning-theoretic analysis of SR models represented as expression trees. We derive a generalization bound for GP-style SR under constraints on tree size, depth, and learnable constants. Our result decomposes the generalization gap into two interpretable components: a structure-selection term, reflecting the combinatorial complexity of choosing an expression-tree structure, and a constant-fitting term, capturing the complexity of optimizing numerical constants within a fixed structure. This decomposition provides a theoretical perspective on several widely used practices in GP, including parsimony pressure, depth limits, numerically stable operators, and interval arithmetic. In particular, our analysis shows how structural restrictions reduce hypothesis-class growth while stability mechanisms control the sensitivity of predictions to parameter perturbations. By linking these practical design choices to explicit complexity terms in the generalization bound, our work offers a principled explanation for commonly observed empirical behaviors in GP-based SR and contributes towards a more rigorous understanding of its generalization properties.

Masahiro Nomura, Ryoki Hamano, Isao Ono

Institute of Science Tokyo, Yokohama, Japan · CyberAgent, Shibuya, Japan

Genetic Programming Statistical Learning Theory Generalization Bounds Symbolic Regression

Apr 17, 2026

Neuro-Symbolic ODE Discovery with Latent Grammar Flow

Understanding natural and engineered systems often relies on symbolic formulations, such as differential equations, which provide interpretability and transferability beyond black-box models. We introduce Latent Grammar Flow (LGF), a neuro-symbolic generative framework for discovering ordinary differential equations from data. LGF embeds equations as grammar-based representations into a discrete latent space and forces semantically similar equations to be positioned closer together with a behavioural loss. Then, a discrete flow model guides the sampling process to recursively generate candidate equations that best fit the observed data. Domain knowledge and constraints, such as stability, can be either embedded into the rules or used as conditional predictors.

Karin Yu, Eleni Chatzi, Georgios Kissas

Institute of Structural Engineering, ETH Zurich, Zurich, Switzerland · ETH AI Center, ETH Zurich, Zurich, Switzerland · Swiss Data Science Center, Zurich, Switzerland

Autonomous Scientific Discovery Neurosymbolic Learning ODE Discovery Symbolic Regression

Apr 14, 2026

Interpretable Relational Inference with LLM-Guided Symbolic Dynamics Modeling

Inferring latent interaction structures from observed dynamics is a fundamental inverse problem in many-body interacting systems. Most neural approaches rely on black-box surrogates over trainable graphs, achieving accuracy at the expense of mechanistic interpretability. Symbolic regression offers explicit dynamical equations and stronger inductive biases, but typically assumes known topology and a fixed function library. We propose COSINE (Co-Optimization of Symbolic Interactions and Network Edges), a differentiable framework that jointly discovers interaction graphs and sparse symbolic dynamics. To overcome the limitations of fixed symbolic libraries, COSINE further incorporates an outer-loop large language model that adaptively prunes and expands the hypothesis space using feedback from the inner optimization loop. Experiments on synthetic systems and large-scale real-world epidemic data demonstrate robust structural recovery and compact, mechanism-aligned dynamical expressions. Code: https://anonymous.4open.science/r/COSINE-6D43.

Xiaoxiao Liang, Juyuan Zhang, Liming Pan +1

School of Cyber Science and Technology, University of Science and Technology of China, Hefei, Anhui, China.

Latent Dynamics Modeling Symbolic Regression Dynamic Graph Learning Large Language Model-Guided Scientific Discovery

Apr 9, 2026

Multi-Modal Learning meets Genetic Programming: Analyzing Alignment in Latent Space Optimization

Symbolic regression (SR) aims to discover mathematical expressions from data, a task traditionally tackled using Genetic Programming (GP) through combinatorial search over symbolic structures. Latent Space Optimization (LSO) methods use neural encoders to map symbolic expressions into continuous spaces, transforming the combinatorial search into continuous optimization. SNIP (Meidani et al., 2024), a contrastive pre-training model inspired by CLIP, advances LSO by introducing a multi-modal approach: aligning symbolic and numeric encoders in a shared latent space to learn the phenotype-genotype mapping, enabling optimization in the numeric space to implicitly guide symbolic search. However, this relies on fine-grained cross-modal alignment, whereas literature on similar models like CLIP reveals that such an alignment is typically coarse-grained. In this paper, we investigate whether SNIP delivers on its promise of effective bi-modal optimization for SR. Our experiments show that: (1) cross-modal alignment does not improve during optimization, even as fitness increases, and (2) the alignment learned by SNIP is too coarse to efficiently conduct principled search in the symbolic space. These findings reveal that while multi-modal LSO holds significant potential for SR, effective alignment-guided optimization remains unrealized in practice, highlighting fine-grained alignment as a critical direction for future work.

Benjamin Léger, Kazem Meidani, Christian Gagné

IID / Mila, Université Laval, Québec, Canada · Department of Mechanical Engineering, Carnegie Mellon University Canada-CIFAR AI Chair

Latent Space Optimization Symbolic Regression Cross-Modal Alignment Genetic Programming Multimodal Contrastive Learning

Apr 7, 2026

A machine learning framework for uncovering stochastic nonlinear dynamics from noisy data

Modeling real-world systems requires accounting for noise - whether it arises from unpredictable fluctuations in financial markets, irregular rhythms in biological systems, or environmental variability in ecosystems. While the behavior of such systems can often be described by stochastic differential equations, a central challenge is understanding how noise influences the inference of system parameters and dynamics from data. Traditional symbolic regression methods can uncover governing equations but typically ignore uncertainty. Conversely, Gaussian processes provide principled uncertainty quantification but offer little insight into the underlying dynamics. In this work, we bridge this gap with a hybrid symbolic regression-probabilistic machine learning framework that recovers the symbolic form of the governing equations while simultaneously inferring uncertainty in the system parameters. The framework combines deep symbolic regression with Gaussian process-based maximum likelihood estimation to separately model the deterministic dynamics and the noise structure, without requiring prior assumptions about their functional forms. We verify the approach on numerical benchmarks, including harmonic, Duffing, and van der Pol oscillators, and validate it on an experimental system of coupled biological oscillators exhibiting synchronization, where the algorithm successfully identifies both the symbolic and stochastic components. The framework is data-efficient, requiring as few as 100-1000 data points, and robust to noise - demonstrating its broad potential in domains where uncertainty is intrinsic and both the structure and variability of dynamical systems must be understood.

Matteo Bosso, Giovanni Franzese, Kushal Swamy +3

Nonlinear System Identification Symbolic Regression Uncertainty Quantification

Apr 1, 2026

Bias Inheritance in Neural-Symbolic Discovery of Constitutive Closures Under Function-Class Mismatch

We investigate the data-driven discovery of constitutive closures in nonlinear reaction-diffusion systems with known governing PDE structures. Our objective is to robustly recover diffusion and reaction laws from spatiotemporal observations while avoiding the common pitfall where low residuals or short-horizon predictions are conflated with physical recovery. We propose a three-stage neural-symbolic framework: (1) learning numerical surrogates under physical constraints using a noise-robust weak-form-driven objective; (2) compressing these surrogates into restricted interpretable symbolic families (e.g., polynomial, rational, and saturation forms); and (3) validating the symbolic closures through explicit forward re-simulation on unseen initial conditions. Extensive numerical experiments reveal two distinct regimes. Under matched-library settings, weak polynomial baselines behave as correctly specified reference estimators, showing that neural surrogates do not uniformly outperform classical bases. Conversely, under function-class mismatch, neural surrogates provide necessary flexibility and can be compressed into compact symbolic laws with minimal rollout degradation. However, we identify a critical "bias inheritance" mechanism where symbolic compression does not automatically repair constitutive bias. Across various observation regimes, the true error of the symbolic closure closely tracks that of the neural surrogate, yielding a bias inheritance ratio near one. These findings demonstrate that the primary bottleneck in neural-symbolic modeling lies in the initial numerical inverse problem rather than the subsequent symbolic compression. We underscore that constitutive claims must be rigorously supported by forward validation rather than residual minimization alone.

Hanbing Liang, Ze Tao, Fujun Liu

Nanophotonics and Biophotonics Key Laboratory of Jilin Province, School of Physics, Changchun University of Science and Technology, Changchun 130022, P.R. China

PDE Discovery Neurosymbolic Learning Dynamical Systems Neural Surrogate Modeling Symbolic Regression

Mar 31, 2026

PASM: Population Adaptive Symbolic Mixture-of-Experts Model for Cross-location Hurricane Evacuation Decision Prediction

Accurate prediction of evacuation behavior is critical for disaster preparedness, yet models trained in one region often fail elsewhere. Using a multi-state hurricane evacuation survey, we show this failure goes beyond feature distribution shift: households with similar characteristics follow systematically different decision patterns across states. As a result, single global models overfit dominant responses, misrepresent vulnerable subpopulations, and generalize poorly across locations. We propose Population-Adaptive Symbolic Mixture-of-Experts (PASM), which pairs large language model guided symbolic regression with a mixture-of-experts architecture. PASM discovers human-readable closed-form decision rules, specializes them to data-driven subpopulations, and routes each input to the appropriate expert at inference time. On Hurricanes Harvey and Irma data, transferring from Florida and Texas to Georgia with 100 calibration samples, PASM achieves a Matthews correlation coefficient of 0.607, compared to XGBoost (0.404), TabPFN (0.333), GPT-5-mini (0.434), and meta-learning baselines MAML and Prototypical Networks (MCC

\leq

0.346). The routing mechanism assigns distinct formula archetypes to subpopulations, so the resulting behavioral profiles are directly interpretable. A fairness audit across four demographic axes finds no statistically significant disparities after Bonferroni correction. PASM closes more than half the cross-location generalization gap while keeping decision rules transparent enough for real-world emergency planning.

Xiao Qian, Shangjia Dong

Dept. of Civil, Environmental, and Construction Engineering, University of Delaware, Newark, DE, USA

Domain Generalization Mixture-of-Experts Models Interpretable ML Symbolic Regression

Mar 25, 2026

Symbolic--KAN: Kolmogorov-Arnold Networks with Discrete Symbolic Structure for Interpretable Learning

Symbolic discovery of governing equations is a long-standing goal in scientific machine learning, yet a fundamental trade-off persists between interpretability and scalable learning. Classical symbolic regression methods yield explicit analytic expressions but rely on combinatorial search, whereas neural networks scale efficiently with data and dimensionality but produce opaque representations. In this work, we introduce Symbolic Kolmogorov-Arnold Networks (Symbolic-KANs), a neural architecture that bridges this gap by embedding discrete symbolic structure directly within a trainable deep network. Symbolic-KANs represent multivariate functions as compositions of learned univariate primitives applied to learned scalar projections, guided by a library of analytic primitives, hierarchical gating, and symbolic regularization that progressively sharpens continuous mixtures into one-hot selections. After gated training and discretization, each active unit selects a single primitive and projection direction, yielding compact closed-form expressions without post-hoc symbolic fitting. Symbolic-KANs further act as scalable primitive discovery mechanisms, identifying the most relevant analytic components that can subsequently inform candidate libraries for sparse equation-learning methods. We demonstrate that Symbolic-KAN reliably recovers correct primitive terms and governing structures in data-driven regression and inverse dynamical systems. Moreover, the framework extends to forward and inverse physics-informed learning of partial differential equations, producing accurate solutions directly from governing constraints while constructing compact symbolic representations whose selected primitives reflect the true analytical structure of the underlying equations. These results position Symbolic-KAN as a step toward scalable, interpretable, and mechanistically grounded learning of governing laws.

Salah A Faroughi, Farinaz Mostajeran, Amirhossein Arzani +1

Energy & Intelligence Lab, Department of Chemical Engineering, University of Utah, Salt Lake City, Utah, 84112, USA · Department of Mechanical Engineering, University of Utah, Salt Lake City, Utah, 84112, USA · Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, Utah, 84112, USA +2

Interpretable ML Scientific ML Kolmogorov-Arnold Networks Symbolic Regression PDE Discovery Physics-Informed ML

Mar 24, 2026

Weak-PDE-Net: Discovering Open-Form PDEs via Differentiable Symbolic Networks and Weak Formulation

Discovering governing Partial Differential Equations (PDEs) from sparse and noisy data is a challenging issue in data-driven scientific computing. Conventional sparse regression methods often suffer from two major limitations: (i) the instability of numerical differentiation under sparse and noisy data, and (ii) the restricted flexibility of a pre-defined candidate library. We propose Weak-PDE-Net, an end-to-end differentiable framework that can robustly identify open-form PDEs. Weak-PDE-Net consists of two interconnected modules: a forward response learner and a weak-form PDE generator. The learner embeds learnable Gaussian kernels within a lightweight MLP, serving as a surrogate model that adaptively captures system dynamics from sparse observations. Meanwhile, the generator integrates a symbolic network with an integral module to construct weak-form PDEs, avoiding explicit numerical differentiation and improving robustness to noise. To relax the constraints of the pre-defined library, we leverage Differentiable Neural Architecture Search strategy during training to explore the functional space, which enables the efficient discovery of open-form PDEs. The capability of Weak-PDE-Net in multivariable systems discovery is further enhanced by incorporating Galilean Invariance constraints and symmetry equivariance hypotheses to ensure physical consistency. Experiments on several challenging PDE benchmarks demonstrate that Weak-PDE-Net accurately recovers governing equations, even under highly sparse and noisy observations.

Xinxin Li, Xingyu Cui, Jin Qi +3

School of Mathematical Sciences, East China Normal University, Shang Hai, 200241, China · Institute of Applied Physics and Computational Mathematics, 100094, Beijing, China · Shanghai Zhangjiang Institute of Mathematics, 201203, Shanghai, China +2

PDE Discovery Symbolic Regression

Mar 23, 2026

Neural Structure Embedding for Symbolic Regression via Continuous Structure Search and Coefficient Optimization

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.

Fateme Memar, Tao Zhe, Dongjie Wang

Department of Electrical Engineering and Computer Science, University of Kansas USA

Symbolic Regression

Mar 23, 2026

Instruction Set and Language for Symbolic Regression

A fundamental but largely unaddressed obstacle in Symbolic regression (SR) is structural redundancy: every expression DAG with admits many distinct node-numbering schemes that all encode the same expression, each occupying a separate point in the search space and consuming fitness evaluations without adding diversity. We present IsalSR (Instruction Set and Language for Symbolic Regression), a representation framework that encodes expression DAGs as strings over a compact two-tier alphabet and computes a pruned canonical string -- a complete labeled-DAG isomorphism invariant -- that collapses all the equivalent representations into a single canonical form.

Ezequiel Lopez-Rubio, Mario Pascual-Gonzalez

Department of Computer Languages and Computer Science University of Málaga Bulevar Louis Pasteur, 35 29071 Málaga, Spain · ITIS Software. Universidad de Málaga. C/ Arquitecto Francisco Peñalosa 18, 29010, Málaga, Spain

Symbolic Regression

Mar 21, 2026

LLM-ODE: Data-driven Discovery of Dynamical Systems with Large Language Models

Discovering the governing equations of dynamical systems is a central problem across many scientific disciplines. As experimental data become increasingly available, automated equation discovery methods offer a promising data-driven approach to accelerate scientific discovery. Among these methods, genetic programming (GP) has been widely adopted due to its flexibility and interpretability. However, GP-based approaches often suffer from inefficient exploration of the symbolic search space, leading to slow convergence and suboptimal solutions. To address these limitations, we propose LLM-ODE, a large language model-aided model discovery framework that guides symbolic evolution using patterns extracted from elite candidate equations. By leveraging the generative prior of large language models, LLM-ODE produces more informed search trajectories while preserving the exploratory strengths of evolutionary algorithms. Empirical results on 91 dynamical systems show that LLM-ODE variants consistently outperform classical GP methods in terms of search efficiency and Pareto-front quality. Overall, our results demonstrate that LLM-ODE improves both efficiency and accuracy over traditional GP-based discovery and offers greater scalability to higher-dimensional systems compared to linear and Transformer-only model discovery methods.

Amirmohammad Ziaei Bideh, Jonathan Gryak

The Graduate Center, CUNY New York, New York, USA · Queens College and The Graduate Center, CUNY New York, New York, USA

ODE Discovery Dynamical Systems Large Language Model-Guided Scientific Discovery Symbolic Regression Genetic Programming

Mar 21, 2026

Graph-based data-driven discovery of interpretable laws governing corona-induced noise and radio interference for high-voltage transmission lines

The global shift towards renewable energy necessitates the development of ultrahigh-voltage (UHV) AC transmission to bridge the gap between remote energy sources and urban demand. While UHV grids offer superior capacity and efficiency, their implementation is often hindered by corona-induced audible noise (AN) and radio interference (RI). Since these emissions must meet strict environmental compliance standards, accurate prediction is vital for the large-scale deployment of UHV infrastructure. Existing engineering practices often rely on empirical laws, in which fixed log-linear structures limit accuracy and extrapolation. Herein, we present a monotonicity-constrained graph symbolic discovery framework, Mono-GraphMD, which uncovers compact, interpretable laws for corona-induced AN and RI. The framework provides mechanistic insight into how nonlinear interactions among the surface gradient, bundle number and diameter govern high-field emissions and enables accurate predictions for both corona-cage data and multicountry real UHV lines with up to 16-bundle conductors. Unlike black-box models, the discovered closed-form laws are highly portable and interpretable, allowing for rapid predictions when applied to various scenarios, thereby facilitating the engineering design process.

Hao Xu, Yuntian Chen, Chongqing Kang +1

Zhejiang Key Laboratory of Industrial Intelligence and Digital Twin, Eastern Institute of Technology, Ningbo, Zhejiang 315200, China · Department of Electrical Engineering, Tsinghua University, Beijing 100084, P. R. China · Ningbo Institute of Digital Twin, Eastern Institute of Technology, Ningbo, Zhejiang 315200, P. R. China +1

Power Systems Symbolic Regression

Mar 19, 2026

SINDy-KANs: Sparse identification of non-linear dynamics through Kolmogorov-Arnold networks

Kolmogorov-Arnold networks (KANs) have arisen as a potential way to enhance the interpretability of machine learning. However, solutions learned by KANs are not necessarily interpretable, in the sense of being sparse or parsimonious. Sparse identification of nonlinear dynamics (SINDy) is a complementary approach that allows for learning sparse equations for dynamical systems from data; however, learned equations are limited by the library. In this work, we present SINDy-KANs, which simultaneously train a KAN and a SINDy-like representation to increase interpretability of KAN representations with SINDy applied at the level of each activation function, while maintaining the function compositions possible through deep KANs. We apply our method to a number of symbolic regression tasks, including dynamical systems, to show accurate equation discovery across a range of systems.

Amanda A. Howard, Nicholas Zolman, Bruno Jacob +2

Pacific Northwest National Laboratory, Richland, WA 99354 · University of Washington, Seattle, WA 98195

Nonlinear System Identification Sparse Identification of Nonlinear Dynamics Symbolic Regression Neural Network Interpretability Kolmogorov-Arnold Networks

Mar 16, 2026

Minimum-Action Learning: Energy-Constrained Symbolic Model Selection for Physical Law Identification from Noisy Data

Identifying physical laws from noisy observational data is a central challenge in scientific machine learning. We present Minimum-Action Learning (MAL), a framework that selects symbolic force laws from a pre-specified basis library by minimizing a Triple-Action functional combining trajectory reconstruction, architectural sparsity, and energy-conservation enforcement. A wide-stencil acceleration-matching technique reduces noise variance by 10,000x, transforming an intractable problem (SNR ~0.02) into a learnable one (SNR ~1.6); this preprocessing is the critical enabler shared by all methods tested, including SINDy variants. On two benchmarks -- Kepler gravity and Hooke's law -- MAL recovers the correct force law with Kepler exponent p = 3.01 +/- 0.01 at ~0.07 kWh (40% reduction vs. prediction-error-only baselines). The raw correct-basis rate is 40% for Kepler and 90% for Hooke; an energy-conservation-based criterion discriminates the true force law in all cases, yielding 100% pipeline-level identification. Basis library sensitivity experiments show that near-confounders degrade selection (20% with added r^{-2.5} and r^{-1.5}), while distant additions are harmless, and the conservation diagnostic remains informative even when the correct basis is absent. Direct comparison with noise-robust SINDy variants, Hamiltonian Neural Networks, and Lagrangian Neural Networks confirms MAL's distinct niche: interpretable, energy-constrained model selection that combines symbolic basis identification with dynamical rollout validation.

Martin G. Frasch

Institute on Human Development and Disability, University of Washington, Seattle, WA 98195, USA · Health Stream Analytics, LLC, Seattle, WA, USA

Scientific Discovery Model Selection Physics-Informed ML Nonlinear System Identification Symbolic Regression

Mar 16, 2026

EvoIQA - Explaining Image Distortions with Evolved White-Box Logic

Traditional Image Quality Assessment (IQA) metrics typically fall into one of two extremes: rigid, hand-crafted mathematical models or "black-box" deep learning architectures that completely lack interpretability. To bridge this gap, we propose EvoIQA, a fully explainable symbolic regression framework based on Genetic Programming that Evolves explicit, human-readable mathematical formulas for image quality assessment (IQA). Utilizing a rich terminal set from the VSI, VIF, FSIM, and HaarPSI metrics, our framework inherently maps structural, chromatic, and information-theoretic degradations into observable mathematical equations. Our results demonstrate that the evolved GP models consistently achieve strong alignment between the predictions and human visual preferences. Furthermore, they not only outperform traditional hand-crafted metrics but also achieve performance parity with complex, state-of-the-art deep learning models like DB-CNN, proving that we no longer have to sacrifice interpretability for state-of-the-art performance.

Ruchika Gupta, Illya Bakurov, Nathan Haut +1

Michigan State University

Image Quality Assessment Symbolic Regression