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

arXiv: https://arxiv.org/abs/2508.15695 · HTML: https://mlsanity.com/paper/2508.15695/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 57 references that are in the map.
- First posted on arXiv 21 Aug 2025.

## Builds on (most cited first, 20 of 20)

- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks](https://mlsanity.com/paper/2406.02917/) (2024)
- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Residual-based attention in physics-informed neural networks](https://mlsanity.com/paper/2307.00379/) (2023)
- [Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networks](https://mlsanity.com/paper/2406.11045/) (2024)
- [A Dual-Dimer Method for Training Physics-Constrained Neural Networks with Minimax Architecture](https://mlsanity.com/paper/2005.00615/) (2020)
- [Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodology](https://mlsanity.com/paper/2211.08939/) (2022)
- [Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusion](https://mlsanity.com/paper/2109.14860/) (2021)
- [Training neural networks under physical constraints using a stochastic augmented Lagrangian approach](https://mlsanity.com/paper/2009.07330/) (2020)
- [An adaptive augmented Lagrangian method for training physics and equality constrained artificial neural networks](https://mlsanity.com/paper/2306.04904/) (2023)
- [Critical Investigation of Failure Modes in Physics-informed Neural Networks](https://mlsanity.com/paper/2206.09961/) (2022)
- [Non-overlapping, Schwarz-type Domain Decomposition Method for Physics and Equality Constrained Artificial Neural Networks](https://mlsanity.com/paper/2409.13644/) (2024)

## Built on by (most cited first, 3 of 3)

- [A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learning](https://mlsanity.com/paper/2509.14198/) (2025)
- [AL-CoLe: Augmented Lagrangian for Constrained Learning](https://mlsanity.com/paper/2510.20995/) (2025)
- [Boundary condition enforcement with PINNs: a comparative study and verification on 3D geometries](https://mlsanity.com/paper/2512.14941/) (2025)

## Abstract (arXiv)

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.