Conditionally adaptive augmented Lagrangian method for physics-informed learning of forward and inverse problems

Qifeng Hu; Shamsulhaq Basir; Inanc Senocak

Home/Archive/arXiv

arXiv · 2508.15695

Conditionally adaptive augmented Lagrangian method for physics-informed learning of forward and inverse problems

Qifeng Hu·Shamsulhaq Basir·Inanc Senocak

View original source ↗Read source document ↗Export metadata ↓

Abstract

We present several key advances to the Physics and Equality Constrained Artificial Neural Networks (PECANN) framework, substantially improving its capacity to solve challenging partial differential equations (PDEs). Our enhancements broaden the framework's applicability and improve efficiency. First, we generalize the Augmented Lagrangian Method (ALM) to support multiple, independent penalty parameters for enforcing heterogeneous constraints. Second, we introduce a constraint aggregation technique to address inefficiencies associated with point-wise enforcement. Third, we incorporate a single Fourier feature mapping to capture highly oscillatory solutions with multi-scale features, where alternative methods often require multiple mappings or costlier architectures. Fourth, a novel time-windowing strategy enables seamless long-time evolution without relying on discrete time models. Fifth, and critically, we propose a conditionally adaptive penalty update (CAPU) strategy for ALM that accelerates the growth of Lagrange multipliers for constraints with larger violations, while enabling coordinated updates of multiple penalty parameters. CAPU accelerates the growth of Lagrange multipliers for selectively challenging constraints, enhancing constraint enforcement during training. We demonstrate the effectiveness of PECANN-CAPU across diverse problems, including the transonic rarefaction problem, reversible scalar advection by a vortex, high-wavenumber Helmholtz and Poisson's equations, and 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. Collectively, these advances improve the robustness, computational efficiency, and applicability of PECANN to demanding problems in scientific computing.

Explore related subjects

cs.LG math.OC physics.comp-ph

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Qifeng Hu, Shamsulhaq Basir, Inanc Senocak. 2025-12-07. Conditionally adaptive augmented Lagrangian method for physics-informed learning of forward and inverse problems. https://arxiv.org/abs/2508.15695

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Random Polytope Descriptors

We introduce a class of random polytopes which simultaneously generalizes several known constructions. While being fairly general, these polytopes are also computationally exceptionally benign. We indicate how these properties can be exploited for classification and clustering tasks in data analysis. Crucially, our construction lets users smoothly trade off between a tighter description of the data and faster computation.

cs.LG↗

arXiv2026

CurvFed: Curvature-Aligned Federated Learning for Fairness without Demographics

Modern human sensing applications often rely on data distributed across users and devices, where privacy concerns prevent centralized training. Federated Learning (FL) addresses this challenge by enabling collaborative model training without exposing raw data or attributes. However, achieving fairness in such settings remains difficult, as most human sensing datasets lack demographic labels, and FL's privacy guarantees limit the use of sensitive attributes. This paper introduces CurvFed: Curvature Aligned Federated Learning for Fairness without Demographics, a theoretically grounded framework that promotes fairness in FL without requiring any demographic or sensitive attribute information, a concept termed Fairness without Demographics (FWD), by optimizing the underlying loss landscape curvature. Building on the theory that equivalent loss landscape curvature corresponds to consistent model efficacy across sensitive attribute groups, CurvFed regularizes the top eigenvalue of the Fisher Information Matrix (FIM) as an efficient proxy for loss landscape curvature, both within and across clients. This alignment promotes uniform model behavior across diverse bias inducing factors, offering an attribute agnostic route to algorithmic fairness. CurvFed is especially suitable for real world human sensing FL scenarios involving single or multi user edge devices with unknown or multiple bias factors. We validated CurvFed through theoretical and empirical justifications, as well as comprehensive evaluations using three real world datasets and a deployment on a heterogeneous testbed of resource constrained devices. Additionally, we conduct sensitivity analyses on local training data volume, client sampling, communication overhead, resource costs, and runtime performance to demonstrate its feasibility for practical FL edge device deployment.

cs.LG↗

arXiv2026

Path Regularization: A Near-Complete and Optimal Nonasymptotic Generalization Theory for Multilayer Neural Networks and Double Descent Phenomenon

Path regularization has shown to be a very effective regularization to train neural networks, leading to a better generalization property than common regularizations i.e. weight decay, etc. We propose a first near-complete (as will be made explicit in the main text) nonasymptotic generalization theory for multilayer neural networks with path regularizations for general learning problems. In particular, it does not require the boundedness of the loss function, as is commonly assumed in the literature. Our theory goes beyond the bias-variance tradeoff and aligns with phenomena typically encountered in deep learning. It is therefore sharply different from other existing nonasymptotic generalization error bounds. More explicitly, we propose an explicit generalization error upper bound for multilayer neural networks with $σ(0)=0$ and sufficiently broad Lipschitz loss functions, without requiring the width, depth, or other hyperparameters of the neural network to approach infinity, a specific neural network architecture (e.g., sparsity), or boundedness of the loss function, while also taking approximation error into consideration. In particular, we solve an open problem proposed by Weinan E et. al. in 2020 regarding the approximation rates in generalized Barron spaces. Furthermore, we show the near-minimax optimality of our theory for regression problems with ReLU activations. Notably, our upper bound exhibits the famous double descent phenomenon for such networks, which is the most distinguished characteristic compared with other existing results. Our subsequent work will prove the matching lower bounds in the minimax sense, meaning that it is highly possible that our theory reveals the true underlying mechanism of the double descent phenomenon. We can also explain scaling law from this theory.

cs.LG↗