Symbolic Program Discovery

• Symbolic program discovery is the process of deriving explicit, closed-form expressions that fit observed data and adhere to scientific constraints.
• It integrates diverse methodologies such as genetic programming, deep reinforcement learning, sparse regression, and Bayesian search to balance model fidelity and simplicity.
• Applications include uncovering governing equations, physical laws, and optimization algorithms, facilitating automated scientific hypothesis generation.

Symbolic program discovery is the computational task of inferring explicit, human-interpretable mathematical expressions (or programs) consistent with observed data or scientific constraints. Unlike traditional regression, which yields fixed parametric mappings, or black-box models such as neural networks, symbolic discovery returns closed-form, often minimalistic analytic expressions, exposing structure and invariances underlying physical, engineered, or biological systems. This functionally unifies diverse methodologies from genetic programming and deep reinforcement learning to sparse dictionary regression, Bayesian search, and neural operator surrogates. Major applications include discovering governing equations, closed-form physical laws, optimal algorithms, and candidate models for scientific hypothesis generation and automated theory building.

1. Symbolic Program Discovery: Problem Formalization and Scope

The central goal in symbolic program discovery is to recover a symbolic model $f$—typically expressed as an expression tree built from a grammar $\mathcal{G}$ of variables, constants, and operators—such that it accurately fits observed data, or satisfies constraints imposed by system dynamics or invariants. Formally, given:

• Input–output pairs $(x_n, y_n)$ sampled from an unknown function, or an unlabeled dataset $\{x_n\}$ constrained by an implicit relation $f(x) = 0$
• A symbolic language $\mathcal{G}: E ::= v | c | E+E | E-E | \dots$ with primitive set $O$ (e.g., $\{+, -, \times, \div, \sin, \exp\}$)
• (Optionally) priors over model structures $p(m)$ and parameters $p(\theta_m)$

the task is to search $\mathcal{G}$ for the minimal $f$ such that fidelity to the data and/or scientific plausibility is optimized, subject to constraints such as physical units, invariance, or model sparsity (Makke et al., 2022, Clarkson et al., 2022, Yufei et al., 6 May 2025).

This problem encompasses several variants:

• Supervised symbolic regression: $f: \mathbb{R}^d \rightarrow \mathbb{R}$, mapping inputs to outputs with fit + complexity balance
• Implicit equation discovery: recover $f(x) = 0$ or $f(x, y) = 0$ given point clouds lying on a manifold, as in PIE (Yufei et al., 6 May 2025)
• Programmatic discovery of algorithms: search for program snippets (e.g., update rules) optimizing learning objectives (Chen et al., 2023)
• Symbolic PDE/system identification: learn expressions for dynamical system evolution (e.g., SINDy, NOMTO) (Garmaev et al., 14 Jan 2025)

2. Algorithmic Paradigms: GP, Deep Learning, Bayesian, and Hybrid Approaches

Methodologies in symbolic program discovery are classified into several families:

• Genetic programming (GP)-based: Population-based evolution over expression trees; crossover, mutation, parsimony pressure, and complexity control are primary tools. Best for highly flexible search but prone to bloat and expensive evaluation (Makke et al., 2022).
• Deep learning-based: Sequence models (RNNs, Transformers) generate symbolic tokens; objectives are REINFORCE-based, maximizing expected rewards combining data fit and symbolic complexity; architectures include DSO (Hayes et al., 16 May 2025), SymbolicGPT, and Transformer-based symbolic regression (Lalande et al., 2023).
• Hybrid/dictionary sparse regression: Enumerate a massive feature library (e.g., FFX, SINDy, SyMANTIC (Muthyala et al., 5 Feb 2025), DISCOVER (Gajera et al., 27 Jan 2026)), select concise subsets via $\ell_0$ or $\ell_1$ constraint, sometimes integrating domain knowledge, dimensionality, and physical invariants.
• Bayesian model selection/experimental design: Bayesian frameworks treat both structure $m$ and parameters $\theta$ probabilistically, drive data acquisition via optimal design (mutual information for maximal expected model discrimination) (Clarkson et al., 2022).
• Neural operator or hybrid surrogate approaches: Substitute neural surrogates for analytic operations, extend symbolic discovery to classes of functions lacking closed forms (e.g., NOMTO's use of neural operators for singularities, PDEs, special functions) (Garmaev et al., 14 Jan 2025).
• Meta-learned translation: Formulate symbolic discovery as a translation task from data manifolds to symbolic skeletons (notably for implicit equations), leveraging pretraining to eliminate degenerate forms (Yufei et al., 6 May 2025).

A tabular overview of representative frameworks:

Family	Example Methods	Key Attributes
GP-based	Classic GP, Gplearn, GP-GOMEA	Flexible, population-based, interpretable
Deep learning (RL)	DSO, SymbolicGPT, PIE, Mix-Encoder SR	Sequence modeling, RL-based, scalable
Sparse/dictionary regression	FFX, SINDy, SyMANTIC, DISCOVER	Convex/NP-hard subproblems, rapid evaluation
Bayesian	Bayesian SR, Bayesian OED for SPD	Probabilistic model/prior integration
Neural surrogate/hybrid	NOMTO, neural-operator SR models	Handles singular/special functions, PDEs

3. Optimization, Inference, and Search Techniques

Optimization and search strategies span evolutionary, gradient-based, probabilistic, and combinatorial approaches.

• Evolutionary Search (GP etc.): Tree-structured search with tournament selection, crossover, and mutation. Suffering from local optima and bloat, advances include semantic-guided crossover and bloat control (Makke et al., 2022).
• Gradient-based RL/Sequence Models: Autoregressive token generators trained by policy gradient (REINFORCE, risk-seeking variants, priority-queue training) or actor-critic, often with inside-loop constant optimization. Constraints and priors are folded into the action logits, and best-in-batch strategies ("risk-seeking PG", PQT) help escape reward sparsity (Hayes et al., 16 May 2025).
• Sparse Regression and Feature Construction: SISSO, SyMANTIC, DISCOVER, and related frameworks enumerate or generate extremely large feature sets via adaptive expansion, then select compact models via SIS filtering, recursive OMP, or MIQP within complexity constraints (Gajera et al., 27 Jan 2026, Muthyala et al., 5 Feb 2025).
• Bayesian Experimental Design: Optimal information-gain-driven point selection to maximize model discrimination; parameter posteriors sampled via HMC, experimental settings chosen to maximize mutual information between response and model index (Clarkson et al., 2022).
• Neural Operator Discovery: Pretrain neural surrogates for analytic/differential/special operators, then learn sparse DAGs via $\ell_{1/2}$ penalization and energy-based pruning; final symbolic model is compiled by replacing operator nodes with analytic forms (Garmaev et al., 14 Jan 2025).

4. Constraints, Prior Knowledge, and Handling Degeneracy

Symbolic program discovery methods employ an array of constraints and priors to restrict the vast search space and enforce meaningful results:

• Physical and dimensional constraints: Enforced via unit tracking in ASTs (DISCOVER), shape constraints, or domain-specific invariants, ensuring discovered models adhere to physical laws (Gajera et al., 27 Jan 2026).
• Grammar and arity constraints: Models like DSO and Transformer SR enforce grammar legality and limit functional arity by logit masking in decoding.
• Semantic inductive bias: Pretraining paradigms (PIE, Transformer-based SR) intentionally exclude degenerate solutions (e.g., $x-x$, $0\cdot f(x)$) to prevent the models from learning trivial equations—degeneracies are low-probability under their learned priors (Yufei et al., 6 May 2025).
• Information-theoretic selection: SyMANTIC uses mutual information for feature screening, and information-theoretic complexity for pareto-optimal front selection (Muthyala et al., 5 Feb 2025).
• Program simplification and selection: Automated elimination of dead code, functional equivalence hashing, and algebraic simplification (e.g., Lion discovery (Chen et al., 2023)) are critical for collapsing redundant or over-parameterized candidate programs.

5. Evaluation and Benchmarking: Metrics, Datasets, Performance

Evaluation of symbolic program discovery comprises both data-fit and structural complexity, typically against established benchmarks.

• Metrics:
  • Data fit: MSE, NMSE, $R^2$, test RMSE
  • Program complexity: tree size, node count, information-theoretic metrics
  • Recovery rate: exact-match to ground truth analytical expression
  • Structural accuracy: e.g., normalized tree-edit distance (Lalande et al., 2023)
  • Pareto-optimality: accuracy-complexity frontiers
  • Runtime: wall-clock time, evaluations to solution
• Benchmarks:
  • Feynman SR database (physics equations), Nguyen, Keijzer, SRBench
  • EmpiricalBench (science-derived), synthetic functions, chaotic systems (Lorenz, PDEs)
• Comparative results:
  • SyMANTIC achieves ground-truth recovery $\approx 95\%$ on benchmark suites, with a median runtime of $\sim 10\,\mathrm{s}$ compared to next-best approaches at $\sim 50\%$ recovery and minutes-plus runtime (Muthyala et al., 5 Feb 2025).
  • DSO reports state-of-the-art symbolic and accuracy-solution rates on SRBench, outperforming genetic and prior neural methods (Hayes et al., 16 May 2025).
  • PIE achieves NMSE fitness $\approx 0.78$ (AI-Feynman), vastly surpassing GP/DSO ($\lesssim 0.31$) in implicit, unsupervised settings (Yufei et al., 6 May 2025).
  • DISCOVER and NOMTO demonstrate advanced scaling, physical interpretability, and inclusion of nonstandard operators, with GPU acceleration providing $10\times$–$20\times$ speed-up (Gajera et al., 27 Jan 2026, Garmaev et al., 14 Jan 2025).

6. Recent Innovations: Unsupervised Discovery, Higher-Order Operators, Scientific Applications

Contemporary work extends symbolic program discovery in fundamental directions:

• Unsupervised and translation-based discovery: PIE frames implicit equation recovery as a translation task from unstructured point clouds to symbolic skeletons, robust against degeneracy and noise (Yufei et al., 6 May 2025).
• Higher-order operators and PDEs: Neural operator-based approaches (NOMTO) generalize symbolic discovery to expressions including differential, singular, and special functions, e.g., rediscovery of nonlinear PDEs with exact coefficients (Garmaev et al., 14 Jan 2025).
• End-to-end deep symbolic models: Transformer models trained on synthetic symbolic corpora achieve state-of-the-art structural accuracy and near-instantaneous inference on scientific datasets (Lalande et al., 2023).
• Algorithmic innovation via program search: Symbolic discovery successfully yields optimization algorithms (e.g. Lion), surpassing hand-designed baselines in large-scale learning tasks (Chen et al., 2023).
• Physics-informed, scalable frameworks: DISCOVER systematizes unit-aware grammar pruning, GPU feature evaluation, and domain-aware linear modeling, facilitating reproducible scientific SR workflows (Gajera et al., 27 Jan 2026).

7. Limitations, Open Challenges, and Prospects

Persistent challenges include:

• Search complexity and scalability: Combinatorial explosion of search space in high dimensions or deep grammars, NP-hardness of sparse regression, and reliance on approximate or heuristic solvers (Clarkson et al., 2022, Muthyala et al., 5 Feb 2025).
• Generalization and robustness: Out-of-domain transfer depends critically on the pretraining regime and grammar design; robustness to heavy noise, missing data, or underdetermined systems remains imperfect (Lalande et al., 2023, Yufei et al., 6 May 2025).
• Degenerate or equivalent expressions: Normalization, canonicalization, and equivalence detection for symbolic programs are not fully automated; normalization via computer algebra (SymPy) is imperfect.
• Interpretability–complexity–accuracy tradeoff: Efficiently navigating the Pareto front of explainability and prediction remains active.
• Multi-output and systems identification: Many frameworks treat outputs independently; joint discovery for vector- or tensor-valued targets is incompletely solved (Muthyala et al., 5 Feb 2025, Garmaev et al., 14 Jan 2025).

Advances expected in hybridization (deep model + sparse regression), richer grammar learning, scientific-invariant integration, dynamic grammar adaptation, and multi-task or multi-output SR, as well as human-in-the-loop symbolic discovery interfaces.

---

Key references:

• (Clarkson et al., 2022) Bayesian Experimental Design for Symbolic Discovery
• (Makke et al., 2022) Interpretable Scientific Discovery with Symbolic Regression: A Review
• (Yufei et al., 6 May 2025) Advancing Symbolic Discovery on Unsupervised Data: A Pre-training Framework for Non-degenerate Implicit Equation Discovery
• (Hayes et al., 16 May 2025) Deep Symbolic Optimization: Reinforcement Learning for Symbolic Mathematics
• (Lalande et al., 2023) A Transformer Model for Symbolic Regression towards Scientific Discovery
• (Muthyala et al., 5 Feb 2025) SyMANTIC: An Efficient Symbolic Regression Method for Interpretable and Parsimonious Model Discovery in Science and Beyond
• (Gajera et al., 27 Jan 2026) DISCOVER: A Physics-Informed, GPU-Accelerated Symbolic Regression Framework
• (Garmaev et al., 14 Jan 2025) NOMTO: Neural Operator-based symbolic Model approximaTion and discOvery
• (Chen et al., 2023) Symbolic Discovery of Optimization Algorithms