Controlling the residual’s curvature
Physics-informed neural networks learn solutions to differential equations by penalizing violations of those equations. In this work, we also penalize the curvature of that residual across the input domain. CuPINN and its variants encourage a flatter residual surface. We also combine this approach with gradient-enhanced PINNs.
What CuPINN adds to training
- Equation residual
- Penalize violations of the differential equation.
- Residual curvature
- Also discourage sharp changes in the residual across the input domain.
- Computation
- Use trace estimation and finite differences without constructing the full Hessian.
The curvature is measured with respect to the inputs, not the model parameters.
Results
Across seven PDE benchmarks, including linear and nonlinear problems, the methods improve solution accuracy and residual error over the tested PINN and GPINN baselines. We also see improvements with relatively few training points.
Current work at EPFL
E3 research internship with Prof. Olga Fink.
I’m working on graph neural networks that use the structure of conventional numerical solvers to predict fluid motion. The goal is to take larger time steps than the solvers can reliably take on their own, while keeping predictions accurate over long periods. Taking fewer steps would make these simulations less expensive.
Related paper
CuPINN: Optimizing PINNs through Curvature Minimization and Residual Landscape Flattening
BibTeX
@article{raoofzareCU,
title = {{CuPINN: Optimizing PINNs through Curvature Minimization and Residual Landscape Flattening}},
author = {Zare Moayedi, Raoof and Abbaszadeh, Mostafa and Dehghan, Mehdi},
year = {2025},
journal = {Computer Methods in Applied Mechanics and Engineering},
doi = {10.1016/j.cma.2025.118180}
}