# On the post-hoc Evaluation of PDE Discovery: A Multifaceted Challenge of Scientific Advancement

arXiv: https://arxiv.org/abs/2607.23753 · HTML: https://mlsanity.com/paper/2607.23753/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 60 of its 115 references that are in the map.
- First posted on arXiv 26 Jul 2026.

## Builds on (most cited first, 30 of 60)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control](https://mlsanity.com/paper/2111.10992/) (2021)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
- [Lie Point Symmetry Data Augmentation for Neural PDE Solvers](https://mlsanity.com/paper/2202.07643/) (2022)
- [LLM-SRBench: A New Benchmark for Scientific Equation Discovery with Large Language Models](https://mlsanity.com/paper/2504.10415/) (2025)
- [Error estimates for physics informed neural networks approximating the Navier-Stokes equations](https://mlsanity.com/paper/2203.09346/) (2022)
- [Learning Physical Models that Can Respect Conservation Laws](https://mlsanity.com/paper/2302.11002/) (2023)
- [LieGG: Studying Learned Lie Group Generators](https://mlsanity.com/paper/2210.04345/) (2022)
- [Learning Sparse Nonlinear Dynamics via Mixed-Integer Optimization](https://mlsanity.com/paper/2206.00176/) (2022)
- [Exhaustive Symbolic Regression](https://mlsanity.com/paper/2211.11461/) (2022)
- [Latent Space Symmetry Discovery](https://mlsanity.com/paper/2310.00105/) (2023)
- [Discovering a reaction-diffusion model for Alzheimer's disease by combining PINNs with symbolic regression](https://mlsanity.com/paper/2307.08107/) (2023)
- [PDE-READ: Human-readable Partial Differential Equation Discovery using Deep Learning](https://mlsanity.com/paper/2111.00998/) (2021)
- [Lie Point Symmetry and Physics Informed Networks](https://mlsanity.com/paper/2311.04293/) (2023)
- [A Unified Framework to Enforce, Discover, and Promote Symmetry in Machine Learning](https://mlsanity.com/paper/2311.00212/) (2023)
- [Learning interaction kernels in mean-field equations of 1st-order systems of interacting particles](https://mlsanity.com/paper/2010.15694/) (2020)
- [PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Data](https://mlsanity.com/paper/2212.04971/) (2022)
- [Symbolic regression via MDLformer-guided search: from minimizing prediction error to minimizing description length](https://mlsanity.com/paper/2411.03753/) (2024)
- [Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterion](https://mlsanity.com/paper/2208.03322/) (2022)
- [Think like a Scientist: Physics-guided LLM Agent for Equation Discovery](https://mlsanity.com/paper/2602.12259/) (2026)
- [Tanimoto Random Features for Scalable Molecular Machine Learning](https://mlsanity.com/paper/2306.14809/) (2023)

## Abstract (arXiv)

Partial differential equation (PDE) discovery aims to identify from data the governing law of a physical system. Constituting a cornerstone of scientific advancement, it has become during the past decade a major line of research in the rapidly evolving field of Physics-informed Machine Learning (PiML). Among the remaining open problems to address in this domain, the post-hoc evaluation of discovered PDEs raises the particular difficulty of being multifaceted. Indeed, it requires jointly considering predictive accuracy, physical consistency, interpretability, and out-of-distribution generalization capacity. Given that some of these properties are conflicting, it is worth noting that the wide range of existing evaluation metrics only partially address the overall problem, potentially leading to overly interpreted conclusions about the validity of a presumed new physical theory. From an abundant literature spanning machine learning, numerical analysis, information theory or symbolic regression, we propose, to our knowledge, the first taxonomy of PDE evaluation metrics, and discuss their advantages and limitations in depth. Based on the observation that evaluation is often achieved on a case-by-case basis and that a universally accepted methodology remains elusive, we further provide recommendations with the aim of promoting standardized and reliable practices, before sketching promising future lines of research in this field. We argue that this paper is intended both for ML experts who design new PDE discovery algorithms and for users of these methods aiming, in real applications, to discover and validate well-founded scientific laws.