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A Neural-Guided Dynamic Symbolic Network for Exploring Mathematical Expressions from Data
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Proposes DySymNet, an RL-controlled (RNN, risk-seeking policy gradient) search over dynamically generated symbolic networks for symbolic regression, evaluated on Nguyen-style standard benchmarks, SRBench, and a free-fall air-resistance experiment.
Problem
Symbolic regression struggles with high-dimensional problems and constant fitting because the expression search space is huge; existing deep generative and transformer SR methods scale poorly to unseen problems.
Approach
Instead of searching expression trees directly, an RNN controller samples the architecture (layers, operators per layer, operator types) of a sparse symbolic network, which is trained by backpropagation with L0.5 regularization and pruning; constants are refined by BFGS and the controller is updated with risk-seeking policy gradient.
Key results
- On SRBench Black-box, DySymNet reaches mean R^2 0.8908 vs 0.6697 for uDSR and 0.6086 for NGGP.
- On the free-fall experiment, DySymNet gives lowest test MSE on 10 of 11 balls with average MSE 0.1736 vs 1.5902 for the best physical model (Model-C).
- On the standard benchmarks DySymNet recovers Jin and Koza expressions with R^2 1.0000.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use DySymNet instead of the strongest listed baseline | Standard benchmarks · Standard benchmarks (Nguyen, Nguyen*, Constant, Keijzer, Livermore, R, Jin, Koza) | R² | Mixed |
| Use DySymNet instead of the strongest listed baseline | SRBench · Feynman, Strogatz, and Black-box | R² | Helped |
| Use DySymNet rather than the physical-model baselines | Free-falling ball prediction · Free-falling balls test set (average) | MSE: 0.1736 vs 2.2698, 71.94, and 1.5902 | Helped |
| Remove constant refinement (RC) (ablation) | DySymNet · Nguyen and Feynman | Accuracy solution rate | Hurt |
| Remove policy gradient (PG) (ablation) | DySymNet · Nguyen and Feynman | Accuracy solution rate | Hurt |
| Remove adaptive gradient clipping (AGC) (ablation) | DySymNet · Nguyen and Feynman | Accuracy solution rate | Hurt |
| Add Gaussian noise to the dependent variable | Symbolic regression methods · Standard benchmarks | Accuracy solution rate | Mixed |
Surprising findings
- On the Nguyen-style benchmarks, uDSR and NGGP match or exceed DySymNet on R (0.9999 vs 0.9895).
Limitations
- Some standard benchmarks (e.g., R rationals) are not solved as well as uDSR/NGGP.
- Pruning and interpretability depend on hand-set thresholds (weight pruning threshold 0.01) and structure library limits (max 5 layers, width 6).
- Free-fall experiment uses a small dataset (20-30 points per training set) and 11 balls.
Experimental setup
Datasets: Nguyen benchmark Nguyen* benchmark Constant Keijzer Livermore R rationals Jin Koza SRBench Feynman SRBench Strogatz SRBench Black-box free-fall ball measurements (11 balls)
Benchmarks: Nguyen Nguyen* Constant Keijzer Livermore R Jin Koza SRBench Feynman SRBench Strogatz SRBench Black-box free-fall balls
Metrics: R^2 MSE accuracy solution rate
Baselines: uDSR NGGP NeSymReS EQL AI Feynman SRBench methods
Builds on: deep symbolic regression equation learner (EQL) NGGP uDSR risk-seeking policy gradient
Closest papers by meaning
- Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients (ICLR 2021 · similarity 0.92)
- MetaSymNet: A Tree-like Symbol Network with Adaptive Architecture and Activation Functions (AAAI 2025 · similarity 0.88)
- SMILE: Bridging Continuous Optimization and Discrete Symbolic Recovery (NeurIPS 2026 · similarity 0.87)
- Neuro-Symbolic Data Generation for Math Reasoning (NeurIPS 2024 · similarity 0.87)
- SNIP: Bridging Mathematical Symbolic and Numeric Realms with Unified Pre-training (ICLR 2024 · similarity 0.87)
- Symbolic regression via MDLformer-guided search: from minimizing prediction error to minimizing description length (ICLR 2025 · similarity 0.86)
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