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SMILE: Bridging Continuous Optimization and Discrete Symbolic Recovery

NeurIPS 2026 · Probabilistic and scientific modeling › Bayesian optimization · approach: Offline black-box optimization with surrogates · 0 citations (OpenAlex)

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Proposes SMILE, a hybrid symbolic regression framework using a shallow feedforward network with five fixed symbolic activations (sine, multiplication, identity, logarithm, exponential), structural data analysis, gated training, and a pruning/refit/rounding recovery pipeline, evaluated on SRBench.

Problem

Symbolic regression suffers from slow discrete combinatorial search and does not exploit compositional structure in data, while continuous methods often lack exact symbolic recovery and interpretability.

Approach

A neural network with fixed interpretable activations (S, M, I, L, E) trained with gating and sparsity penalties, preceded by data-driven variable analysis (log-linear R^2 scan, inversion test, window-based decomposition) and followed by greedy gate pruning, least-squares refit, and gradient-based rounding to exact constants.

Key results

• On noisy Feynman data (highest noise level), SMILE achieves the highest symbolic solution rate, with SSR dropping by at most 8% versus over 30% for ParFam and PySR.
• On noise-free Feynman data SMILE ranks third in SSR behind PySR and ParFam, with lower accuracy solution rate ($R^2>0.999$) than top baselines.
• SMILE recovers expressions in minutes (average 43.9 s per problem on an RTX A5000) versus several hours for ParFam, PySR, and AI Feynman.

Surprising findings

• Increasing network depth improves R^2 but does not improve symbolic solution rate, since added degrees of freedom make pruning harder.
• SMILE's median R^2 on black-box datasets is lower than several baselines despite very low complexity.

Limitations

• Favors shallow architectures to keep pruning tractable, limiting accuracy on some problems.
• Data-driven structural analysis assumes compositional patterns common in physical laws, which may not hold for general real-world relationships.
• Lower accuracy solution rate and median R^2 than top baselines on Feynman and black-box datasets.

Experimental setup

Datasets: SRBench Feynman dataset Strogatz ODE dataset PMLB black-box datasets

Benchmarks: SRBench Feynman SRBench Strogatz SRBench black-box (PMLB)

Metrics: symbolic solution rate accuracy solution rate (R^2>0.999) median R^2 expression complexity training time

Baselines: PySR uDSR E2E ParFam AI Feynman SINDy DSR

Builds on: AI Feynman EQL PySR ParFam uDSR

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