# Learning functional components of PDEs from data using neural networks

arXiv: https://arxiv.org/abs/2602.13174 · HTML: https://mlsanity.com/paper/2602.13174/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 41 references that are in the map.
- First posted on arXiv 13 Feb 2026.

## Builds on (most cited first, 6 of 6)

- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [On structural and practical identifiability](https://mlsanity.com/paper/2102.05100/) (2021)
- [Scientific Machine Learning of Flow Resistance Using Universal Shallow Water Equations with Differentiable Programming](https://mlsanity.com/paper/2502.12396/) (2025)
- [HyPer-EP: Meta-Learning Hybrid Personalized Models for Cardiac Electrophysiology](https://mlsanity.com/paper/2403.15433/) (2024)
- [Parameter identifiability, parameter estimation and model prediction for differential equation models](https://mlsanity.com/paper/2405.08177/) (2024)
- [A cautionary tale of model misspecification and identifiability](https://mlsanity.com/paper/2507.04894/) (2025)

## Abstract (arXiv)

Partial differential equation (PDE) models frequently contain unknown functional terms that cannot be measured directly, limiting their predictive utility. While data-driven methods for estimating scalar PDE parameters are well established, the recovery of unknown functions remains comparatively underexplored. Here, we show that standard parameter estimation workflows can be extended to infer functional components of PDEs directly from data. Our approach embeds neural networks within the PDE framework, allowing unknown functions to be learned during training with high accuracy. Using nonlocal aggregation-diffusion equations as a case study, we infer interaction kernels and external potentials from steady-state observations. We systematically examine how reconstruction accuracy depends on factors such as the number and diversity of available solutions, sampling density, and measurement noise. The resulting framework retains the advantages of conventional PDE calibration approaches while extending them to functional inference: once trained, the PDE model can be used in the standard way to analyse system behaviour and generate predictions.