# MorphingNet: Symbolic Regression Where the Network Is the Formula

arXiv: https://arxiv.org/abs/2610.19942 · HTML: https://mlsanity.com/paper/2610.19942/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 47 references that are in the map.
- First posted on arXiv 14 Oct 2026.

## Builds on (most cited first, 28 of 28)

- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Contemporary Symbolic Regression Methods and their Relative Performance](https://mlsanity.com/paper/2107.14351/) (2021)
- [End-to-end symbolic regression with transformers](https://mlsanity.com/paper/2204.10532/) (2022)
- [Neural Symbolic Regression that Scales](https://mlsanity.com/paper/2106.06427/) (2021)
- [Learning Equations for Extrapolation and Control](https://mlsanity.com/paper/1806.07259/) (2018)
- [AI Feynman 2.0: Pareto-optimal symbolic regression exploiting graph modularity](https://mlsanity.com/paper/2006.10782/) (2020)
- [LLM-SR: Scientific Equation Discovery via Programming with Large Language Models](https://mlsanity.com/paper/2404.18400/) (2024)
- [KAN 2.0: Kolmogorov-Arnold Networks Meet Science](https://mlsanity.com/paper/2408.10205/) (2024)
- [Extrapolation and learning equations](https://mlsanity.com/paper/1610.02995/) (2016)
- [Pad'e Activation Units: End-to-end Learning of Flexible Activation Functions in Deep Networks](https://mlsanity.com/paper/1907.06732/) (2019)
- [Deep Symbolic Regression for Recurrent Sequences](https://mlsanity.com/paper/2201.04600/) (2022)
- [Learning outside the Black-Box: The pursuit of interpretable models](https://mlsanity.com/paper/2011.08596/) (2020)
- [A Neural-Guided Dynamic Symbolic Network for Exploring Mathematical Expressions from Data](https://mlsanity.com/paper/2309.13705/) (2023)
- [ParFam -- (Neural Guided) Symbolic Regression Based on Continuous Global Optimization](https://mlsanity.com/paper/2310.05537/) (2023)
- [GINN-LP: A Growing Interpretable Neural Network for Discovering Multivariate Laurent Polynomial Equations](https://mlsanity.com/paper/2312.10913/) (2023)
- [SymbolNet: Neural Symbolic Regression with Adaptive Dynamic Pruning for Compression](https://mlsanity.com/paper/2401.09949/) (2024)
- [PruneSymNet: A Symbolic Neural Network and Pruning Algorithm for Symbolic Regression](https://mlsanity.com/paper/2401.15103/) (2024)
- [Transformation-Interaction-Rational Representation for Symbolic Regression](https://mlsanity.com/paper/2205.06807/) (2022)
- [DGEMM on Integer Matrix Multiplication Unit](https://mlsanity.com/paper/2306.11975/) (2023)
- [Enabling Population-Level Parallelism in Tree-Based Genetic Programming for GPU Acceleration](https://mlsanity.com/paper/2501.17168/) (2025)
- [Kozax: Flexible and Scalable Genetic Programming in JAX](https://mlsanity.com/paper/2502.03047/) (2025)
- [A continuum among logarithmic, linear, and exponential functions, and its potential to improve generalization in neural networks](https://mlsanity.com/paper/1602.01321/) (2016)
- [Symbolic regression based genetic approximations of the Colebrook equation for flow friction](https://mlsanity.com/paper/1808.10394/) (2018)
- [Performance Enhancement of the Ozaki Scheme on Integer Matrix Multiplication Unit](https://mlsanity.com/paper/2409.13313/) (2024)
- [Sparse Interpretable Deep Learning with LIES Networks for Symbolic Regression](https://mlsanity.com/paper/2506.08267/) (2025)
- [Towards symbolic regression for interpretable clinical decision scores](https://mlsanity.com/paper/2512.07961/) (2025)
- [Efficient Constant Optimization for Symbolic Regression with GPU-Accelerated Tree-Based Genetic Programming](https://mlsanity.com/paper/2609.03352/) (2026)

## Abstract (arXiv)

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.