Neural Network–Guided Symbolic Regression

• NN-guided symbolic regression is a method that integrates neural networks with symbolic search to efficiently discover analytic mathematical expressions from data.
• It employs techniques like neural generators, expression encoders, and differentiable networks to navigate the vast space of symbolic expressions with structural and domain-informed guidance.
• The approach enhances model interpretability and performance across scientific domains by incorporating feature selection, complexity regularization, and scalable optimization.

Neural network–guided symbolic regression (NN-guided SR) is a family of computational techniques for discovering analytic mathematical expressions from data, in which neural networks play a central role in informing, biasing, or accelerating the search for symbolic equations. These approaches address the prohibitive size and complexity of the symbolic expression space by leveraging neural network architectures for search guidance, structural proposal, feature selection, and regularized optimization, yielding models that are both accurate and interpretable. The recent proliferation of NN-guided SR methods encompasses pre-training, hybrid neuro-evolutionary schemes, domain-informed guidance, and scalable compression, all aimed at improving model quality, interpretability, and applicability to high-dimensional and data-scarce scientific problems.

1. Core Methodologies in NN-Guided Symbolic Regression

NN-guided SR approaches fall into several architectural and algorithmic paradigms, often combining neural models with symbolic expression search:

• Sequence- or Tree-Based Neural Generators: Transformer (Biggio et al., 2021) and LSTM (Mundhenk et al., 2021) models are trained to generate symbolic expressions, using data (input–output pairs) as conditioning context. The neural generator proposes expression skeletons that are further analyzed or refined via symbolic regression or constant optimization steps.
• Neural Networks as Expression Encoders: In neuro-encoded expression programming (Anjum et al., 2019), a recurrent neural network (RNN) encodes candidate expressions as continuous parameters. Small changes in weights yield smooth variations in expression structure, enabling efficient continuous optimization by algorithms such as CMA-ES or PSO.
• End-to-End Differentiable Symbolic Networks: Approaches such as the Equation Learner (EQL) (Kim et al., 2019) and SymbolNet (Tsoi et al., 18 Jan 2024) replace standard NN activation functions with symbolic primitives (e.g., sin, exp, multiplication), trained via gradient descent with sparsity-inducing regularization. These models can be unrolled post hoc into compact symbolic expressions.
• NN-Guided Architecture Search: Dynamic symbolic networks (Li et al., 2023) and neuro-evolutionary methods (Kubalík et al., 23 Apr 2025) use reinforcement learning or evolutionary algorithms to explore diverse NN topologies where each "neuron" encodes a symbolic operator. The search is guided by neural controllers or genetic algorithms, while the network parameters (constants and weights) are fit using standard gradient methods.
• Feature Selection and Search Space Pruning via NN Importance: Neural networks pre-train on available data to perform feature importance analysis (via permutation or learned saliencies), which then restricts symbolic regression to compact, physically meaningful descriptors (Xian et al., 16 Jul 2025).
• Hybrid Strategies with Global and Local Optimization: In continuous global optimization approaches (Scholl et al., 2023), NN-based pre-trained models select promising families of analytic forms or parameterizations, further optimized by continuous algorithms (e.g., basin-hopping, BFGS) to produce closed-form expressions.

2. Mathematical Formulations and Training Objectives

NN-guided SR frameworks impose joint objectives tailored to both symbolic correctness and data fit:

• Symbolic Loss: Cross-entropy or tree-edit distance between predicted and ground-truth expressions is minimized, typically over token sequences generated by the neural network (Bertschinger et al., 24 Feb 2025).
• Behavioral/Numeric Loss: Mean squared error (MSE) or normalized root-mean-square error (NRMSE) between the output of the generated expression and the ground-truth outputs is included in the objective (Bertschinger et al., 24 Feb 2025).
• Complexity Regularization: The majority of differentiable symbolic networks include sparsity penalties on weights or activations, such as L₀, L₁, or smoothed L₀.₅, to induce simple, interpretable models (Kim et al., 2019, Tsoi et al., 18 Jan 2024).
• Constraint Incorporation: Prior knowledge and desired properties (symmetry, monotonicity, dimensionality, boundary conditions) are encoded as additional terms in the loss or as explicit constraints in optimization-based formulations (Kubalík et al., 2020, Kubalík et al., 2023).
• Multiobjective Evolution: Pareto-front or multiobjective evolutionary algorithms are applied to simultaneously optimize for symbolic match and predictive fit (Bertschinger et al., 24 Feb 2025). An example objective balancing symbolic and behavioral loss is:

$$\min_\theta \quad \text{SymbolicLoss}(\hat{e}_\theta, e) + \lambda \,\text{MSE}(\hat{e}_\theta(X), Y) + \mu\,\text{Complexity}(\hat{e}_\theta)$$

3. Incorporation of Prior Knowledge and Domain Constraints

A central innovation in NN-guided SR is the explicit integration of domain-specific knowledge to focus the search:

• Asymptotic Constraints: Neural models are conditioned to generate expressions that match specified asymptotic behaviors (leading-order exponents as $x \to 0$ or $\infty$) (Li et al., 2019), implemented via conditioning vectors or grammar constraints in the generator and the use of NN-guided Monte Carlo Tree Search for efficient exploration.
• Domain Symbol Priors: Tree-structured RNN agents (with KL-divergence regularization) enforce symbol usage statistics learned from domain corpora (e.g., physics or biology), biasing the sampling toward plausible operator combinations and blocks (Huang et al., 12 Mar 2025).
• Feature Selection as Search Space Reduction: Neural models identify highly predictive features or combinations; subsequent symbolic regression is restricted to these, yielding concise, physically interpretable formulae and avoiding overfitting, as in materials informatics (Xian et al., 16 Jul 2025).
• Conditioned Generation: Transformers with dedicated encoders are conditioned on user-provided hypotheses or properties (e.g., symmetry, complexity constraints, known sub-expressions), guiding the generation to equations with desired structure or properties (Bendinelli et al., 2023).

4. Scalability, Efficiency, and Model Compression

NN-guided SR approaches address key practical barriers:

• High-Dimensional Inputs: Methods such as SymbolNet (Tsoi et al., 18 Jan 2024) leverage neural architectures with dynamic, end-to-end pruning capable of compressing and extracting symbolic models from data with hundreds to thousands of input features (e.g., MNIST and SVHN).
• Parallelization: Many hybrid NN-guided frameworks exploit parallel exploration (e.g., multiple symbolic architectures or expression trees searched concurrently), and can use GPGPU hardware to scale to larger datasets.
• Efficiency Gains: Neural pre-training, dynamic threshold-based pruning, and NN-guided parameter initialization result in several orders-of-magnitude reduction in symbolic regression runtime compared to traditional genetic programming or pure enumeration approaches (Scholl et al., 2023, Tsoi et al., 18 Jan 2024).
• Device Deployment: Compact symbolic expressions automatically extracted from dynamic pruning networks are suitable for deployment on custom hardware (e.g., FPGA), achieving nanosecond-scale inference and significant resource and power savings (Tsoi et al., 18 Jan 2024).

5. Practical Applications Across Scientific Domains

NN-guided symbolic regression is already being deployed in real-world and scientific contexts:

• Scientific Law Discovery: The global optimality and exactness of certain methods enable rediscovery of canonical physical laws (e.g., Kepler’s laws, the pendulum equation) and support the search for novel relationships in physics and engineering (Austel et al., 2017, Kubalík et al., 2023).
• Materials Informatics: Frameworks combining neural feature selection (permutation importance in NNs) with symbolic regression have been applied to discover interpretable, physically meaningful descriptors in perovskite catalyst design, yielding formulas that accurately predict oxygen evolution activity from just a few geometric and electronic parameters (Xian et al., 16 Jul 2025).
• Program Synthesis and Control Engineering: NN-guided symbolic regression is leveraged to generate or verify control laws and programs, with constraints on program syntax and semantics integrated into the neural generation process (Li et al., 2019).
• Model Compression for Custom Hardware: In high-energy physics (such as at the CERN LHC), symbolic models generated through dynamic pruning networks provide faster, lower-latency alternatives to deep neural networks for inference on specialized hardware (Tsoi et al., 18 Jan 2024).
• System Identification: Tasks such as identifying models for robotic motion, magnetic manipulation, and mechanical systems benefit from NN-guided regression that incorporates prior invariances and constraint satisfaction (Kubalík et al., 2023, Kubalík et al., 23 Apr 2025).

6. Performance, Advantages, and Limitations

Empirical evaluations consistently indicate that NN-guided symbolic regression:

• Outperforms Heuristic and GP-Only Methods: Across benchmarks (Nguyen, Feynman, ODEbase), NN-guided approaches—especially when combined with search-space pruning and domain priors—solve more expressions, converge faster, and yield simpler models than either GP-only or unconditioned neural methods (Biggio et al., 2021, Mundhenk et al., 2021, Bertschinger et al., 24 Feb 2025, Huang et al., 12 Mar 2025).
• Handles Data Scarcity and Extrapolation: By explicitly encoding prior knowledge or constraints (asymptotic behavior, monotonicity, physical validity), these methods generalize beyond the training regime, outperforming black-box or purely data-driven ML in extrapolation (Li et al., 2019, Kubalík et al., 2023, Kubalík et al., 2020).
• Balances Interpretability and Predictive Accuracy: Pruning, sparsity regularization, dual-objective optimization, and projection into space of interpretable features yield expressions that are both numerically accurate and physically meaningful (Tsoi et al., 18 Jan 2024, Xian et al., 16 Jul 2025).
• Limitations and Challenges: Despite their utility, NN-guided SR methods face challenges with very high-dimensional, noisy, or underconstrained problems where the priors are weak or in conflict. The choice of architecture, operator set, and regularization strongly influences performance, and some methods remain computationally intensive, especially in the hybrid or neuro-evolutionary variants (Kubalík et al., 23 Apr 2025).

7. Prospective Directions and Open Problems

• Richer and Adaptive Priors: Expanding the set of conditioning constraints (complexity, symmetry, domain blocks) and integrating adaptive or learned priors to further steer or diversify symbolic discovery (Bendinelli et al., 2023, Huang et al., 12 Mar 2025).
• Full Neuro-Symbolic Integration: Moving toward fully differentiable hybrid pipelines where NN architectures not only guide but also continually adapt symbolic representations, possibly in an online or interactive discovery loop.
• Scalable Global Optimization: Merging large-scale pre-training, dynamic architecture generation, and continuous parameter optimization to address problems with hundreds or thousands of potential variables (Tsoi et al., 18 Jan 2024).
• Automated Domain Knowledge Extraction: Leveraging scientific corpora at scale to construct statistical priors for symbol and block usage, automating the domain adaptation process for NN-guided SR agents (Huang et al., 12 Mar 2025).
• Interpretability in Practice: Enhancing transparency tools for model selection, sparsity tuning, and complexity control to ensure discovered formulas remain physically interpretable and practically useful.

Neural network–guided symbolic regression occupies a crucial intersection of machine learning and scientific modeling, enabling interpretable model discovery at scale and in data regimes inaccessible to traditional methods. Through the combination of deep learning, probabilistic modeling, and symbolic optimization, it provides an extensible and powerful framework for interpretable machine reasoning.