Curvature-Aware Optimization for High-Accuracy Physics-Informed Neural Networks

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  • Authors: Anas Jnini, Elham Kiyani, Khemraj Shukla, Jorge F. Urban, Nazanin Ahmadi Daryakenari, Johannes Muller, Marius Zeinhofer, George Em Karniadakis
  • Preprint year: 2026
  • Status: Published
  • Publication type: Journal article
  • Publication year: 2026
  • Published online: 2026-09-09
  • Journal issue date: 2026-12-01
  • Journal: Computer Methods in Applied Mechanics and Engineering 462, 119289 (2026)
  • DOI: 10.1016/j.cma.2026.119289
  • First public date: 2026-04-06
  • arXiv: 2604.05230

Abstract

Efficient and robust optimization is essential for neural networks, enabling scientific machine learning models to converge rapidly to very high accuracy — faithfully capturing complex physical behavior governed by differential equations. In this work, we present advanced optimization strategies to accelerate the convergence of physics-informed neural networks (PINNs) for challenging partial (PDEs) and ordinary differential equations (ODEs). Specifically, we provide efficient implementations of the Natural Gradient (NG) optimizer, Self-Scaling BFGS and Broyden optimizers, and demonstrate their performance on problems including the Helmholtz equation, Stokes flow, inviscid Burgers equation, Euler equations for high-speed flows, and stiff ODEs arising in pharmacokinetics and pharmacodynamics. Beyond optimizer development, we also propose new PINN-based methods for solving the inviscid Burgers and Euler equations, and compare the resulting solutions against high-order numerical methods to provide a rigorous and fair assessment. Finally, we address the challenge of scaling these quasi-Newton optimizers for batched training, enabling efficient and scalable solutions for large data-driven problems.

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Research areas

BibTeX

@article{arxiv260405230,
  title = {Curvature-Aware Optimization for High-Accuracy Physics-Informed Neural Networks},
  author = {Anas Jnini and Elham Kiyani and Khemraj Shukla and Jorge F. Urban and Nazanin Ahmadi Daryakenari and Johannes Muller and Marius Zeinhofer and George Em Karniadakis},
  year = {2026},
  journal = {Computer Methods in Applied Mechanics and Engineering 462, 119289 (2026)},
  doi = {10.1016/j.cma.2026.119289},
  eprint = {2604.05230},
  archivePrefix = {arXiv},
  primaryClass = {cs.LG}
}

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