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FIND: A Framework for Discovering Formulas in Data
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Proposes FIND, a latent-layer plus polynomial-expression network that discovers formulas from data under dimensional constraints inspired by Buckingham's Pi theorem, using a coarse-to-fine grid search over latent weights.
Problem
Discovering interpretable, physically meaningful formulas from data is inefficient with symbolic regression (huge search space, non-physical outputs) and dimensional analysis (no effective analysis of latent variables, large search space).
Approach
Decompose the mapping into a Pi-theorem-style latent layer of power-law monomials constrained by dimension matrices and a polynomial expression layer; estimate latent structure from correlations of x_j·dy/dx_j ratios and search the weight coefficients with a coarse-to-fine grid search.
Key results
- On a 5-variable toy function, C2F search explores 186 candidates versus 413 for linear search on the first experiment, reaching R2 0.99.
- On the keyhole dataset, FIND discovered a dimensionless latent z with R2 = 0.9810 that corresponds to the known keyhole number Ke.
- On Navier-Stokes data, FIND recovered the Reynolds number 1/Re as the dynamic coefficient, and for planets it recovered Kepler's third law form up to constant (to = 5.43e-10 rs^1.5).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use C2F search instead of linear search | FIND · Toy function search | search candidates: 413 vs 186 | Helped |
| Increase coefficient accuracy setting from 0.1 to 0.5 | FIND · Laser–metal keyhole | R2: 0.9865 -> 0.9810 | Hurt |
Surprising findings
- Gradient optimization over the latent weights gets stuck in local optima and yields irregular decimals, motivating the grid-based C2F search.
Limitations
- Fixed structure yields only polynomial-form solutions.
- Grid search becomes costly for high-dimensional data.
- Accuracy degrades when input data are highly dispersed or sparse.
Experimental setup
Datasets: keyhole dataset (Xie et al. 2022) Navier-Stokes simulation dataset (Xie et al. 2022) NASA Planetary Fact Sheet
Benchmarks: toy functions with 1% Gaussian noise keyhole dataset Navier-Stokes dataset solar system planet data
Metrics: R2 number of candidate points searched running time
Baselines: symbolic regression linear search
Builds on: Buckingham Pi theorem SINDy
Closest papers by meaning
- Unsupervised Discovery of Formulas for Mathematical Constants (NeurIPS 2024 · similarity 0.88)
- LLM-SR: Scientific Equation Discovery via Programming with Large Language Models (ICLR 2025 · similarity 0.88)
- MOOSE-Chem2: Exploring LLM Limits in Fine-Grained Scientific Hypothesis Discovery via Hierarchical Search (NeurIPS 2025 · similarity 0.86)
- LLM-SRBench: A New Benchmark for Scientific Equation Discovery with Large Language Models (ICML 2025 · similarity 0.86)
- Position: Data-driven Discovery with Large Generative Models (ICML 2024 · similarity 0.85)
- FunctionEvolve: Structure-Guided Symbolic Regression with LLMs (NeurIPS 2026 · similarity 0.85)
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