Physics and Equality Constrained Artificial Neural Networks with the Conditionally Adaptive Penalty Update Algorithm (PECANN-CAPU)

Summary:

This version introduces several key advances to the original Physics and Equality Constrained Artificial Neural Networks (PECANN) framework, substantially improving its capacity and efficiency to solve challenging partial differential equations (PDEs):

1. Generalized ALM — extends the Augmented Lagrangian Method (ALM) to support multiple, independent penalty parameters for enforcing heterogeneous constraints.
2. Constraint aggregation — addresses inefficiencies associated with point-wise enforcement of PDE constraints.
3. Fourier feature mapping — a single Fourier feature mapping captures highly oscillatory solutions with multi-scale features, where alternative physics-informed methods often require multiple mappings or costlier architectures.
4. Time-windowing — enables seamless long-time evolution of transport equations without relying on discrete time models.
5. Conditionally adaptive penalty update (CAPU) — accelerates the growth of Lagrange multipliers for constraints with larger violations, while enabling coordinated updates of multiple penalty parameters.

We demonstrate the effectiveness of PECANN-CAPU across diverse problems, including:

• Transonic rarefaction problem,
• Reversible scalar advection by a vortex,
• Helmholtz and Poisson's equations with high-wavenumber solutions,
• Inverse heat source identification.

The framework achieves competitive accuracy across all cases when compared with established methods and recent approaches based on Kolmogorov-Arnold networks. An important implication of our investigation is that pure regression is insufficient to evaluate network architecture in physics-informed learning. Collectively, these advances improve the robustness, computational efficiency, and applicability of PECANN to demanding problems in scientific computing.

Paper Link: [https://doi.org/10.1016/j.cma.2026.118953]

Installation

From GitHub (recommended)

pip install git+https://github.com/HiPerSimLab/PECANN-CAPU.git

From source (editable / development mode)

git clone https://github.com/HiPerSimLab/PECANN-CAPU.git
cd PECANN-CAPU
pip install -e .

Required packages

• PyTorch
• numpy
• scipy
• matplotlib

Citation

Please cite us if you find our work useful for your research:

@article{caalm2026hu,
title = {Conditionally adaptive augmented Lagrangian method for physics-informed learning of forward and inverse problems},
journal = {Computer Methods in Applied Mechanics and Engineering},
volume = {456},
pages = {118953},
year = {2026},
issn = {0045-7825},
doi = {https://doi.org/10.1016/j.cma.2026.118953},
url = {https://www.sciencedirect.com/science/article/pii/S0045782526002264},
author = {Qifeng Hu and Shamsulhaq Basir and Inanc Senocak},
}

Funding Acknowledgment

This material is based upon work supported by the National Science Foundation under Grant No. 1953204 and in part by the University of Pittsburgh Center for Research Computing​ and Data, RRID:SCR_022735, through the resources provided. Specifically, this work used the H2P cluster, which is supported by NSF award number OAC-2117681.

Questions and feedback?

For questions or feedback feel free to reach us at Qifeng Hu, Inanc Senocak, Shams Basir