Synthesizing Physics Formulae with Transformers

We introduce MorphingNet, a network for symbolic regression (SR) whose units change their function continuously. Instead of searching over expressions built from a fixed operator library, MorphingNet trains the form of each function as a parameter, so SR becomes one continuous optimization problem. Its units are either pole-free rational functions or complex exponential--logarithm functions, which move between exponentials, sinusoids, damped oscillations, power laws and logarithms as their parameters change. The network is a readable formula at every step of training, and no operator library is needed to train it; a structure-recovery step afterwards rewrites the trained formula into short elementary expressions. Training is dense linear algebra, so it is GPU-friendly and scales to large data. MorphingNet achieves state-of-the-art accuracy on SRBench, recovers most generating formulas on synthetic benchmarks, and on large data is up to two orders of magnitude faster than the strongest search method. By turning the search for a formula into the training of a network, MorphingNet offers a new paradigm for SR and for scientific discovery from data.

Symbolic regression aims to recover closed-form equations from observations, providing interpretable models for scientific discovery. Existing approaches struggle to combine flexible structural search with efficient inference. Search-based methods can refine expression structure but often rely on costly combinatorial optimization with random initialization. Pretrained neural models generate formulas almost instantly, but their predictions often contain symbolic errors. We introduce MOSAIC-SR, which uses a pretrained Transformer to propose multiple initial sketches. These sketches initialize searches in several promising regions, avoiding random starts in the vast expression space. Each search jointly recovers structure and constants through scale-aware constant optimization and local symbolic repair. We evaluate MOSAIC-SR on the SRSD-Feynman dataset with and without dummy variables and on six additional benchmarks. MOSAIC-SR obtains the highest symbolic solution rate on every dataset while ranking among the top two methods in predictive accuracy. This advantage persists in the presence of irrelevant dummy inputs. The results show that learned priors can focus search on promising equation structures, and that numerical optimization and symbolic repair are important for recovery.

Scientific equation discovery must combine broad domain priors with strict numerical testing. Symbolic regression supplies numerical grounding but faces a combinatorial search space, whereas many language-model systems ask the model to propose or select formulas directly. We test a different division of labour. We compare role specifications in which the language model acts as equation author, candidate decider or search controller, alongside end-to-end language-model and purely numerical baselines. In the controller setting we propose here, implemented as LLM-PySR, language models specify variables, operators, transformations and search depth; symbolic regression enumerates and fits expressions; and deterministic metrics govern retention. Across 74 AI-Feynman equations and seven complex formula-recovery tasks, search control achieved the strongest observed balance of accuracy, complexity, stability and cost. On an independent battery dataset, LLM-PySR identified a compact piecewise-linear relation between early voltage-curve displacement and cycle life. The results suggest that language models should shape hypothesis exploration rather than decide which equations survive.