Learning functional components of PDEs from data using neural networks
cs.LG, math.AP
Submitted: 2026-02-13
Updated: 2026-09-08
Comments: 25 pages with 6 figures. Additional 25 pages and 21 figures supplementary information
License: http://creativecommons.org/licenses/by/4.0/
The gist: Partial differential equation (PDE) models frequently contain unknown functional terms that cannot be measured directly, limiting their predictive utility.
Terminology
Abstract
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.
Sources
- A cautionary tale of model misspecification and identifiability
- Well-posedness of aggregation-diffusion systems with irregular kernels
- Numerical stationary states for nonlocal Fokker-Planck equations via fixed points of consistency maps
- HyPer-EP: Meta-Learning Hybrid Personalized Models for Cardiac Electrophysiology
- Adam: A Method for Stochastic Optimization
- Scientific Machine Learning of Flow Resistance Using Universal Shallow Water Equations with Differentiable Programming
- Parameter identifiability, parameter estimation and model prediction for differential equation models
Related papers
- Polynomial-Augmented Neural Networks (PANNs) with Weak Orthogonality Constraints for Enhanced Function and PDE Approximation
- AIRL-S: Unifying Reinforcement Learning and Search-Based Test-Time Scaling via Adversarial Inverse Reinforcement Learning
- Transformers as Bayesian In-Context Experimenters: Smoothness-Adaptive Efficient ATE Estimation
- Convergence issues in Relational Concept Analysis based on AOC-posets
- Beliefs Beyond Posteriors: Local-Consistency Optimisation for Bayesian Neural Networks
- Understanding Diffusion Models via Ratio-Based Function Approximation with SignReLU Networks