Conditionally adaptive augmented Lagrangian method for physics-informed learning of forward and inverse problems using artificial neural networks

• URL: http://arxiv.org/abs/2508.15695v1
• Date: Thu, 21 Aug 2025 16:22:40 GMT
• Title: Conditionally adaptive augmented Lagrangian method for physics-informed learning of forward and inverse problems using artificial neural networks
• Authors: Qifeng Hu, Shamsulhaq Basir, Inanc Senocak,
• Abstract summary: We present several advances to the physics and equality constrained artificial neural networks (PECANN) framework.<n>We generalize the augmented Lagrangian method (ALM) to support multiple independent penalty parameters.<n>We reformulate pointwise constraint enforcement and Lagrange multipliers as expectations over constraint terms.
• Score: 0.24578723416255746
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We present several advances to the physics and equality constrained artificial neural networks (PECANN) framework that substantially improve its capability to learn solutions of canonical partial differential equations (PDEs). First, we generalize the augmented Lagrangian method (ALM) to support multiple independent penalty parameters, enabling simultaneous enforcement of heterogeneous constraints. Second, we reformulate pointwise constraint enforcement and Lagrange multipliers as expectations over constraint terms, reducing memory overhead and permitting efficient mini-batch training. Third, to address PDEs with oscillatory, multi-scale features, we incorporate Fourier feature mappings and show that a single mapping suffices where multiple mappings or more costly architectures were required in related methods. Fourth, we introduce a time-windowing strategy for long-time evolution in which the terminal state of each window is enforced as an initial-condition constraint for the next, ensuring continuity without discrete time models. Crucially, we propose a conditionally adaptive penalty update (CAPU) strategy for ALM, which preserves the principle that larger constraint violations incur stronger penalties. CAPU accelerates the growth of Lagrange multipliers for selectively challenging constraints, enhancing constraint enforcement during training. We demonstrate the effectiveness of PECANN-CAPU on problems including the transonic rarefaction problem, reversible advection of a passive by a vortex, high-wavenumber Helmholtz and Poisson equations, and inverse identification of spatially varying heat sources. Comparisons with established methods and recent Kolmogorov-Arnold network approaches show that PECANN-CAPU achieves competitive accuracy across all cases. Collectively, these advances improve PECANN's robustness, efficiency, and applicability to demanding problems in scientific computing.

Related papers

• Fast, Convex and Conditioned Network for Multi-Fidelity Vectors and Stiff Univariate Differential Equations [0.0]
  Accuracy in neural PDE solvers often breaks down due to poor optimisation caused by ill-conditioning.<n>We show that components in governing equations can produce highly ill-conditioned activation vectors.<n>We introduce Shifted Gaussian, a simple yet effective activation filtering step that increases matrix rank and expressivity while preserving convexity.
  arXiv  Detail & Related papers  (2025-08-08T00:51:38Z)
• PhysicsCorrect: A Training-Free Approach for Stable Neural PDE Simulations [4.7903561901859355]
  We present PhysicsCorrect, a training-free correction framework that enforces PDE consistency at each prediction step.<n>Our key innovation is an efficient caching strategy that precomputes the Jacobian and its pseudoinverse during an offline warm-up phase.<n>Across three representative PDE systems, PhysicsCorrect reduces prediction errors by up to 100x while adding negligible inference time.
  arXiv  Detail & Related papers  (2025-07-03T01:22:57Z)
• Single-loop Algorithms for Stochastic Non-convex Optimization with Weakly-Convex Constraints [49.76332265680669]
  This paper examines a crucial subset of problems where both the objective and constraint functions are weakly convex.<n>Existing methods often face limitations, including slow convergence rates or reliance on double-loop designs.<n>We introduce a novel single-loop penalty-based algorithm to overcome these challenges.
  arXiv  Detail & Related papers  (2025-04-21T17:15:48Z)
• Time Marching Neural Operator FE Coupling: AI Accelerated Physics Modeling [3.0635300721402228]
  This work introduces a novel hybrid framework that integrates physics-informed deep operator network with FEM through domain decomposition.<n>To address the challenges of dynamic systems, we embed a time stepping scheme directly into the DeepONet, substantially reducing long-term error propagation.<n>Our framework shows accelerated convergence rates (up to 20% improvement in convergence rates compared to conventional FE coupling approaches) while preserving solution fidelity with error margins consistently below 3%.
  arXiv  Detail & Related papers  (2025-04-15T16:54:04Z)
• Joint Transmit and Pinching Beamforming for Pinching Antenna Systems (PASS): Optimization-Based or Learning-Based? [89.05848771674773]
  A novel antenna system ()-enabled downlink multi-user multiple-input single-output (MISO) framework is proposed.<n>It consists of multiple waveguides, which equip numerous low-cost antennas, named (PAs)<n>The positions of PAs can be reconfigured to both spanning large-scale path and space.
  arXiv  Detail & Related papers  (2025-02-12T18:54:10Z)
• One-Shot Safety Alignment for Large Language Models via Optimal Dualization [64.52223677468861]
  This paper presents a perspective of dualization that reduces constrained alignment to an equivalent unconstrained alignment problem. We do so by pre-optimizing a smooth and convex dual function that has a closed form. Our strategy leads to two practical algorithms in model-based and preference-based settings.
  arXiv  Detail & Related papers  (2024-05-29T22:12:52Z)
• Efficiently Training Deep-Learning Parametric Policies using Lagrangian Duality [55.06411438416805]
  Constrained Markov Decision Processes (CMDPs) are critical in many high-stakes applications.<n>This paper introduces a novel approach, Two-Stage Deep Decision Rules (TS- DDR) to efficiently train parametric actor policies.<n>It is shown to enhance solution quality and to reduce computation times by several orders of magnitude when compared to current state-of-the-art methods.
  arXiv  Detail & Related papers  (2024-05-23T18:19:47Z)
• Scaling physics-informed hard constraints with mixture-of-experts [0.0]
  We develop a scalable approach to enforce hard physical constraints using Mixture-of-Experts (MoE) MoE imposes the constraint over smaller domains, each of which is solved by an "expert" through differentiable optimization. Compared to standard differentiable optimization, our scalable approach achieves greater accuracy in the neural PDE solver setting.
  arXiv  Detail & Related papers  (2024-02-20T22:45:00Z)
• An adaptive augmented Lagrangian method for training physics and equality constrained artificial neural networks [0.9137554315375919]
  We apply our PECANN framework to solve forward and inverse problems that have an expanded and diverse set of constraints. We show that ALM with its conventional formulation to update its penalty parameter and Lagrange multiplier stalls for such challenging problems. We propose an adaptive ALM in which each constraint is assigned a unique penalty parameter that evolve adaptively according to a rule inspired by the adaptive subgradient method.
  arXiv  Detail & Related papers  (2023-06-08T03:16:21Z)
• Message Passing Neural PDE Solvers [60.77761603258397]
  We build a neural message passing solver, replacing allally designed components in the graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
  arXiv  Detail & Related papers  (2022-02-07T17:47:46Z)
• Physics and Equality Constrained Artificial Neural Networks: Application to Partial Differential Equations [1.370633147306388]
  Physics-informed neural networks (PINNs) have been proposed to learn the solution of partial differential equations (PDE) Here, we show that this specific way of formulating the objective function is the source of severe limitations in the PINN approach. We propose a versatile framework that can tackle both inverse and forward problems.
  arXiv  Detail & Related papers  (2021-09-30T05:55:35Z)
• Adaptive Subcarrier, Parameter, and Power Allocation for Partitioned Edge Learning Over Broadband Channels [69.18343801164741]
  partitioned edge learning (PARTEL) implements parameter-server training, a well known distributed learning method, in wireless network. We consider the case of deep neural network (DNN) models which can be trained using PARTEL by introducing some auxiliary variables.
  arXiv  Detail & Related papers  (2020-10-08T15:27:50Z)
• Combining Deep Learning and Optimization for Security-Constrained Optimal Power Flow [94.24763814458686]
  Security-constrained optimal power flow (SCOPF) is fundamental in power systems. Modeling of APR within the SCOPF problem results in complex large-scale mixed-integer programs. This paper proposes a novel approach that combines deep learning and robust optimization techniques.
  arXiv  Detail & Related papers  (2020-07-14T12:38:21Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.