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GINN-LP: A Growing Interpretable Neural Network for Discovering Multivariate Laurent Polynomial Equations
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Proposes GINN-LP, a growing, end-to-end differentiable neural network built from log/exp "power-term approximator" blocks that recovers multivariate Laurent polynomial equations, and an ensemble with AIFeynman to handle non-LP equations on SRBench Feynman datasets.
Problem
Discovering interpretable closed-form equations (multivariate Laurent polynomials) that map inputs to outputs from data, which black-box ML does not provide and symbolic regression searches expensively.
Approach
A power-term approximator block computes a product of inputs raised to learned real exponents via log-linear-exp structure; parallel blocks are grown one at a time with sparsity regularization and an unregularized refinement phase, and a non-integer-exponent check triggers a secondary SR method for non-LP equations.
Key results
- On 48 LP Feynman datasets without noise, GINN-LP outperforms all 14 SRBench methods in median solution rate among the top five shown.
- Ensembling GINN-LP with AIFeynman on all 113 Feynman datasets gives a 61.9% solution rate, a 7.1% absolute improvement over standalone AIFeynman (54.9%).
- GINN-LP with MRGP ensemble achieves 99.1% R^2>0.99 accuracy, outperforming all other methods by 0.9%.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Ensemble GINN-LP with AIFeynman | AIFeynman · Feynman SRBench, 113 datasets | solution rate: 54.9% -> 61.9%; 7.1% | Helped |
| Ensemble GINN-LP with AIFeynman | AIFeynman · Feynman SRBench, 113 datasets | R² > 0.99 accuracy: 85.8% -> 87.6%; 1.8% | Helped |
| Ensemble GINN-LP with MRGP | MRGP · Feynman SRBench, 113 datasets | solution rate: 0.0% -> 38.1%; 38.1% | Helped |
| Ensemble GINN-LP with MRGP | MRGP · Feynman SRBench, 113 datasets | R² > 0.99 accuracy: 98.2% -> 99.1%; 0.9% | Helped |
| Ensemble GINN-LP with Operon | Operon · Feynman SRBench, 113 datasets | R² > 0.99 accuracy: 95.1% -> 94.7%; -0.4% | Hurt |
| Add Gaussian white noise to the target (ablation) | GINN-LP · Feynman LP subset | solution rate | Mixed |
| Add irrelevant input features (ablation) | GINN-LP · Feynman LP subset | solution rate | No clear effect |
| Vary the number of training data points (ablation) | GINN-LP · Feynman LP subset | solution rate | Mixed |
| Train each network growth stage for 500 epochs (ablation) | GINN-LP · Feynman LP subset | solution rate | Helped |
| Increase the number of independently initialized training instances (ablation) | GINN-LP · Feynman LP subset | solution rate | Helped |
Surprising findings
- GINN-LP's LP/non-LP classification produced zero false positives on the 113 Feynman datasets, per the confusion matrix in the appendix.
- AIFeynman's performance did not noticeably increase when its search space was reduced to LP operators.
Limitations
- Assumes all inputs are positive because of the logarithm in the PTA block; negative inputs are not handled.
- Growing the network sequentially to handle non-LP equations would risk overfitting, so it is not yet solved.
- Solution rate drops under high target noise (0.1), where recovered equations have slightly wrong constants.
Experimental setup
Datasets: Feynman symbolic regression benchmark (SRBench subset, 48 LP datasets; 113 total)
Benchmarks: SRBench Feynman datasets (48 LP subset) SRBench Feynman datasets (113 total)
Metrics: symbolic solution rate R^2 > 0.99 accuracy
Baselines: AIFeynman AFP AFP-FE BSR DSR EPLEX FEAT FFX GP-GOMEA ITEA MRGP Operon
Builds on: AIFeynman SRBench equation learner (EQL)
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