Aug 2026· Engineering Reports· Vol 8· 0 citations· 31 references
TL;DR
A Physics Constraint‐Guided Network (PCGN) for deep PDE solving is proposed, to introduce physical information at three levels: feature representation, optimization, and output correction.
Abstract
Partial differential equations (PDEs) are key tools for modeling continuous physical processes, but traditional solvers are costly for high‐dimensional problems with complex boundaries. Existing neural network solvers usually add physical constraints only as loss terms, which limits physical constraint embedding learning and can cause local violations. To address this issue, this paper proposes a Physics Constraint‐Guided Network (PCGN) for deep PDE solving. Its main novelty is to introduce physical information at three levels: feature representation, optimization, and output correction. First, governing equations, boundary conditions, and initial conditions are encoded into propagatable constraint features, and neighborhood propagation improves local consistency. Second, adaptive residual balancing adjusts different constraint terms, reducing instability from uneven residual scales. Third, a differentiable constraint projection layer corrects predictions toward feasible solutions. Experiments on Burgers' equation and Darcy flow show that PCGN achieves lower absolute and relative errors than existing deep learning solvers, while improving training stability and physical consistency.
Experiments show that FC-VPINN achieves approximately one-order-of-magnitude lower prediction errors than the traditional PINN and reduces memory usage to 40% of that required by the baseline, demonstrating improved accuracy and computational efficiency in multidimensional problems.
Wenjie Zhang, Yu-Bo Li, Wei-Dong Cui et al.· Chinese Physics B· 0 citations
Developing foundational neural simulators for Partial Differential Equations (PDEs) requires robust generalization across diverse physical parameters and boundary conditions. However, current deep learning approaches largely face a structural trade-off between condition-agnostic deployment and physical fidelity. Purely data-driven operators infer the underlying physics implicitly and thus lack the explicit constraints needed to ensure physically valid solutions across varying domains, rendering the learning problem ill-posed. On the other hand, Physics-Informed Neural Networks (PINNs) enforce rigorous physical constraints but necessitate costly, instance-specific optimization. Furthermore, the massive scale of emerging foundational operators has severely degraded their inference speeds, making them computationally uncompetitive with traditional numerical solvers. To address this bottleneck between condition-agnostic deployment, physical rigor, and inference efficiency, we propose a \textit{Generalized Neural Operator}. By formalizing the classical conditions for well-posedness within neural operators, our framework demonstrates the theoretical benefits of explicitly conditioning on PDE parameters and boundary conditions. To implement this synthesis without compromising computational speed, we introduce three novel architectural components: a parameter-gated mixture of kernels for efficient parameter generalization, a generalized boundary transfer operator that projects arbitrary boundary constraints into a unified latent Dirichlet representation, and a specialized training objective to ensure stability. Extensive experiments demonstrate that our theoretically grounded approach achieves superior generalization across heterogeneous physical regimes while maintaining strict inference efficiency comparable to conventional numerical baselines.
This work proposes a novel physics-informed broad learning system (PI-BLS), the first physics-informed learning framework based on broad RdNNs that achieves competitive and often superior performance with reduced training time and model parameters compared with conventional PINNs.
Pinki Khatun, M. Sajid, Abhinav Jha et al.· arXiv.org· 0 citations
The growing application of physics-informed neural networks (PINNs) for solving parametric partial differential equations (PDEs) in fluid dynamics has demonstrated their potential for modeling complex multiscale flows; however, conventional PINNs often exhibit spectral bias and slow, unstable convergence, limiting accuracy in boundary layers and wakes. This research presents novel physics-informed feature decomposition in residual dense block neural networks (PI-RDB-NN), which embeds physical constraints directly into the network architecture rather than relying solely on soft constraints. PI-RDB-NN uses hierarchical residual dense blocks for multi-scale feature extraction, allocates feature channels to velocity and pressure in a 2:1 ratio consistent with two-dimensional incompressible Navier–Stokes physics, and enforces mass conservation via a learnable divergence-aware projection applied at the feature level. The model is evaluated on National Advisory Committee for Aeronautics (NACA) 0012 airfoil flow at Reynolds numbers (Re)=5000 and Re=1000 using a hybrid loss combining PDE residuals, boundary conditions, and sparse computational fluid dynamics (CFD) data. PI-RDB-NN reduces PDE residual and divergence error by 91.2% and 71.7% vs traditional PINNs (Re=5000) and by 85.5% and 85.4% vs a physics-informed Deep Operator Network (DeepONet) baseline (Re=1000). These physics consistency gains improve aerodynamic force predictions and CFD agreement, confirmed by velocity, wake, and pressure coefficient (Cp) distributions. Consistent accuracy across both Reynolds regimes supports the framework's generality, with three-dimensional and unsteady extensions identified as future work.
Sarmad Iftikhar, Ishfaq Ahmad, Diltaj Ali et al.· The Physics of Fluids· 0 citations
Energy-based approaches provide a natural and physically consistent framework for a large class of partial differential equations arising in solid and fluid mechanics, where the governing equations follow from variational principles. In contrast to residual-based physics-informed neural networks (PINNs) and their weak-form variants, which enforce the strong or weak form of the equations through loss minimization, the Deep Energy Method (DEM) directly computes the solution as the minimizer of an energy or incremental potential functional. This eliminates the need for residual weighting, avoids high-order derivatives, and enables the direct enforcement of physical constraints through the variational structure.In this work, we systematically revisit the Deep Energy Method, placing it in the broader context of physics-informed learning and variational modeling. We clarify the relationship between DEM, PINNs, and VPINNs, and identify the class of problems for which energy minimization provides intrinsic advantages in terms of stability, robustness and interpretability. Particular emphasis is placed on incremental variational formulations, which allow DEM to be applied to nonlinear, history-dependent and time-dependent problems, including phase-field fracture and dissipative systems. The variational structure underlying DEM further provides a natural foundation for optimization and inverse problems, where the energy functional acts as a physics-based constraint rather than a residual penalty. Through a series of numerical examples, we demonstrate that DEM offers a principled and effective alternative to residual-based methods for variational problems, highlighting its strengths and limitations relative to existing physics-informed approaches.
The integration of deep learning with partial differential equation (PDE) solvers has emerged as a transformative paradigm in computational science, offering unprecedented capabilities for modeling complex biophysical systems. This paper presents a comprehensive framework for PDE solvers enhanced by deep learning, specifically tailored for genomic fluid dynamics applications. We propose a hybrid architecture combining Physics-Informed Neural Networks (PINNs) with operator learning techniques to efficiently solve nonlinear PDEs governing blood flow in viscoelastic arteries, with emphasis on the influence of external magnetic fields and genomic-scale parameter variations. Our methodology leverages the complementary strengths of neural networks and physics-based constraints, enabling accurate prediction of pressure and radius disturbances in arterial fluid flow while maintaining computational efficiency. The framework incorporates symbolic regression for model interpretability and demonstrates superior performance compared to conventional numerical solvers across multiple test cases. Experimental results indicate that our approach achieves up to 85% reduction in computational time while maintaining solution accuracy within 2% error margins. This research contributes to the growing field of scientific machine learning by providing a robust, interpretable, and efficient solution for genomic fluid dynamic simulations with potential applications in personalized medicine and biomedical device design.
C. Manigandan, S. Jeyarani, S.Selvakumar et al.· International journal of com...· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
