# Symbolic recovery of PDEs from measurement data

arXiv: https://arxiv.org/abs/2602.15603 · HTML: https://mlsanity.com/paper/2602.15603/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 38 of its 102 references that are in the map.
- First posted on arXiv 17 Feb 2026.

## Builds on (most cited first, 30 of 38)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [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)
- [Extrapolation and learning equations](https://mlsanity.com/paper/1610.02995/) (2016)
- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [Rational neural networks](https://mlsanity.com/paper/2004.01902/) (2020)
- [Symbolic Physics Learner: Discovering governing equations via Monte Carlo tree search](https://mlsanity.com/paper/2205.13134/) (2022)
- [DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems](https://mlsanity.com/paper/2101.07206/) (2020)
- [Neural networks and rational functions](https://mlsanity.com/paper/1706.03301/) (2017)
- [Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning](https://mlsanity.com/paper/2402.10926/) (2024)
- [Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems](https://mlsanity.com/paper/2212.03130/) (2022)
- [Deep Generative Symbolic Regression](https://mlsanity.com/paper/2401.00282/) (2023)
- [PDE-constrained Models with Neural Network Terms: Optimization and Global Convergence](https://mlsanity.com/paper/2105.08633/) (2021)
- [ParFam -- (Neural Guided) Symbolic Regression Based on Continuous Global Optimization](https://mlsanity.com/paper/2310.05537/) (2023)
- [Symbolic Recovery of Differential Equations: The Identifiability Problem](https://mlsanity.com/paper/2210.08342/) (2022)
- [Neural Power Units](https://mlsanity.com/paper/2006.01681/) (2020)
- [Learning-informed parameter identification in nonlinear time-dependent PDEs](https://mlsanity.com/paper/2202.10915/) (2022)
- [Identifiability Challenges in Sparse Linear Ordinary Differential Equations](https://mlsanity.com/paper/2506.09816/) (2025)
- [When is a System Discoverable from Data? Discovery Requires Chaos](https://mlsanity.com/paper/2511.08860/) (2025)
- [A descent algorithm for the optimal control of ReLU neural network informed PDEs based on approximate directional derivatives](https://mlsanity.com/paper/2210.07900/) (2022)
- [Optimization with learning-informed differential equation constraints and its applications](https://mlsanity.com/paper/2008.10893/) (2020)
- [On uniqueness in structured model learning](https://mlsanity.com/paper/2410.22009/) (2024)
- [First-order conditions for the optimal control of learning-informed nonsmooth PDEs](https://mlsanity.com/paper/2206.00297/) (2022)

## Abstract (arXiv)

Models based on partial differential equations (PDEs) are powerful for describing a wide range of complex phenomena in the natural sciences. Accurately identifying the PDE model, which represents the underlying physical law, is essential for a proper understanding of the problem. This reconstruction typically relies on indirect and noisy measurements of the system's state and, without specifically tailored methods, rarely yields symbolic expressions, thereby limiting interpretability. In this work, we address this limitation by considering neural network architectures based on rational functions for the symbolic representation of physical laws. These networks combine the approximation power of rational functions with the flexibility to represent arithmetic operations, and generalize ParFam and EQL-type architectures used in symbolic regression for physical law learning. We further establish regularity results for these symbolic networks. Our main contribution is a reconstruction result showing that, if there exists an admissible physical law that is expressible within the symbolic network architecture, then in the limit of noiseless and complete measurements, symbolic networks recover a physical law within the PDE model that is representable by the architecture. Moreover, the recovered law corresponds to a regularization-minimizing parameterization, promoting interpretability and sparsity in case of L1-regularization. Under an additional identifiability condition, the unique true physical law is recovered. These reconstruction and regularity results are derived at the continuous level prior to discretization due to a formulation in function space. Empirical results using the ParFam architecture are consistent with the theoretical findings and suggest the feasibility of reconstructing interpretable physical laws in practice.