[Paper] Learning functional components of PDEs from data using neural networks

Source: arXiv - 2602.13174v1

Overview

The paper demonstrates how to learn unknown functional components of partial differential equations (PDEs) directly from data by embedding neural networks inside the governing equations. By treating these networks as flexible function approximators, the authors show that they can recover interaction kernels and external potentials in non‑local aggregation‑diffusion models with high accuracy, opening a new pathway for data‑driven scientific modeling.

Key Contributions

• Neural‑augmented PDE framework: Introduces a systematic way to replace unknown functions in a PDE with trainable neural networks.
• Recovery of functional terms: Shows that, given only steady‑state observations, the method can reconstruct interaction kernels and external potentials to arbitrary precision (subject to data quality).
• Comprehensive sensitivity analysis: Quantifies how the number of observed solutions, sampling density, noise level, and solution diversity influence recovery success.
• Compatibility with existing pipelines: The trained PDE can be used like any conventional model—for simulation, prediction, or control—without bespoke inference tools.
• Open‑source implementation: Provides code and notebooks that let practitioners reproduce the experiments on their own datasets.

Methodology

1. Problem setup – Start with a PDE that contains one or more unknown scalar functions (e.g., a kernel (K(x)) governing non‑local interaction).
2. Neural embedding – Replace each unknown function with a small feed‑forward neural network ( \hat{K}_\theta(x) ) parameterized by weights (\theta). The PDE now becomes a parameterized equation.
3. Data collection – Gather steady‑state snapshots of the system (e.g., density fields) under various boundary conditions or initial states.
4. Loss construction – For each snapshot, compute the residual of the PDE using the neural‑augmented model. The loss is the mean‑squared residual across all spatial points and all snapshots.
5. Training – Optimize the network weights with standard gradient‑based optimizers (Adam, L‑BFGS). Because the loss is differentiable with respect to (\theta), back‑propagation automatically updates the functional approximations.
6. Validation – After training, the learned networks are evaluated on held‑out snapshots or used to simulate the PDE forward in time, confirming that the recovered functions produce realistic dynamics.

The workflow mirrors classic parameter‑estimation pipelines (e.g., fitting diffusion coefficients) but extends them to functions rather than scalar constants.

Results & Findings

• Accurate kernel recovery: On synthetic aggregation‑diffusion problems, the method reconstructed interaction kernels with relative errors below 1 % when provided with as few as 5 distinct steady‑state profiles.
• Robustness to noise: Even with up to 5 % Gaussian measurement noise, the learned functions remained within 3 % of the ground truth, thanks to the regularizing effect of the PDE residual loss.
• Sampling density matters: Finer spatial grids (≥ 100 points per domain length) dramatically reduced error, while coarse grids introduced aliasing artifacts that the network struggled to correct.
• Multiple solutions improve identifiability: When only a single steady state was available, the recovered function was under‑determined; adding diverse solutions (different boundary conditions or external forces) resolved ambiguities.
• Post‑training utility: The trained PDE, now equipped with the learned functions, could predict transient dynamics for unseen initial conditions with errors comparable to a fully known model.

Practical Implications

• Accelerated model discovery: Engineers can embed neural nets into legacy PDE codes to automatically infer missing physics from experimental measurements, cutting down on costly trial‑and‑error modeling.
• Real‑time calibration: In fields like fluid dynamics, materials science, or epidemiology, the approach enables on‑the‑fly updating of interaction laws as new sensor data streams in.
• Plug‑and‑play simulation: Once trained, the neural‑augmented PDE behaves like any standard solver—developers can reuse existing numerical libraries (finite elements, spectral methods) without redesigning inference algorithms.
• Enhanced inverse‑design workflows: Designers of self‑assembling materials or swarm robotics can retrieve the underlying interaction rules from observed collective patterns, then reuse those rules to synthesize new behaviors.
• Cross‑disciplinary reuse: The method is agnostic to the specific PDE form; any domain that models phenomena with unknown functional terms (e.g., climate sub‑grid parameterizations, finance diffusion models) can adopt it.

Limitations & Future Work

• Dependence on steady‑state data: The current study focuses on equilibrium observations; extending to time‑dependent data could broaden applicability but introduces additional training complexity.
• Identifiability constraints: When the unknown function appears only through certain integral transforms, multiple functional forms can produce identical residuals; incorporating physics‑based priors or regularizers may be necessary.
• Scalability to high dimensions: Experiments were performed in 1‑D and 2‑D settings; scaling neural embeddings to 3‑D or higher may require more sophisticated architectures (e.g., convolutional or Fourier neural operators).
• Computational cost: Training involves repeatedly solving the PDE residual across the whole domain, which can be expensive for large‑scale industrial simulations. Future work could explore surrogate models or multi‑fidelity training schemes.

Overall, the paper offers a compelling bridge between classical PDE modeling and modern deep learning, equipping developers with a practical tool to uncover hidden functional relationships directly from data.

Authors

• Torkel E. Loman
• Yurij Salmaniw
• Antonio Leon Villares
• Jose A. Carrillo
• Ruth E. Baker

Paper Information

• arXiv ID: 2602.13174v1
• Categories: cs.LG, math.AP
• Published: February 13, 2026
• PDF: Download PDF