AI Networking Cookbook: Practical recipes for AI-assisted network automation and development
TL;DR
This study systematically compares PINNs preprocessing methods, with particular focus on three universal approaches (the non-dimensionalization, inner normalization layers, and synergistic normalization method for data and equation), and provides a practical reference for selecting preprocessing methods in different physical scenarios.
Physics-Informed Neural Networks (PINNs) have recently emerged as a promising approach for solving Partial Differential Equations (PDEs), offering a meshfree alternative that integrates physical principles into the learning process. This presents a new paradigm compared to traditional discretization methods and purely data-driven machine learning techniques. While promising, PINNs are not a panacea; they inherit challenges such as spectral bias and unstable convergence. Moreover, their potential in seismology remains largely unexplored. In this work, we provide a robust and critical assessment of PINNs for solving the elastic wave equation in seismology. We investigate the performance of PINNs on problems with varying degrees of complexity across various seismic sources and parameter models, from constant to highly heterogeneous settings. A pivotal aspect of our work involves investigating whether embedding physical principles directly into the network architecture enhances convergence and accuracy. We test an extensive range of neural architecture designs, from unrestricted, uninformed PINNs to highly specialized ones. We find that integrating an understanding of wave physics into the network design significantly improves accuracy. For instance, introducing a custom wavelet or plane wave layer, coupled with encoder and decoder layers, consistently yields a relative $L_2$ error approximately half that of the standard PINN, as evidenced across numerous experiments. We further demonstrate that this novel architecture enhances accuracy when applied to the acoustic wave equation, underlying the versatility of our network. Another key contribution of our research is the successful conditioning of PINNs on seismic source locations. This signifies a considerable advancement towards rapid seismic hazard detection and seismic analysis.
This study examines the influence of six collocation point sampling strategies—random uniform, uniform grid, Latin hypercube sampling, Sobol sequence, staggered triangular lattice, and a hybrid combined approach—on the performance of PINNs applied to the one-dimensional Burgers’ equation.
When the coefficients in the partial differential equations (PDEs) vary with spatial position, as in the FitzHugh-Nagumo (FHN) equations, the numerical solution is more difficult. In this work, a novel framework that combines Physics-Informed Neural Networks (PINNs) and Deep Galerkin Methods (DGM) is proposed to overcome these difficulties, which marks a paradigm shift in solving PDEs using AI. The neural network structure is the same in both methods, with hidden layers, and is over-trained using the Adam optimization algorithm. The optimization of collocation point selection based on Sobol sequences is an important component that is helpful for PDEs with sharp gradients and assures optimal distribution of collocation points. We have shown that both techniques are highly accurate in our comparative analysis for a variety of parameter regimes. This shows the complementary strengths of DGM, as it has reduced training times and, for some configurations, PINNs performs better, indicating a computational advantage of DGM. Overall, the framework demonstrates the potential of deep learning for solving complex PDEs involving variable coefficients and provides flexible and scalable solutions that can effectively capture complex boundary conditions and geometries. The results have implications for new research areas in computational neuroscience, fluid mechanics, and material science. Future avenues for improving the ability to calculate the accuracy of the solution and computational efficiency could be adaptive sampling and multi-fidelity modeling.
Physics-informed neural networks (PINNs) represent a growing frontier in using artificial intelligence to solve partial differential equations (PDEs). Automatic differentiation (AD) plays a central role in this paradigm, which is mesh-free and replaces traditional iterative solvers with gradient-based optimization in continuous space. However, the inherent limitations of AD, particularly in handling higher-order derivatives and discontinuous solutions, pose significant challenges for complex problems. This has motivated a growing number of researchers to explore discretization-based constraints as an alternative path. Yet, the respective applicability of these two paradigms remains largely unexplored. In this work, we conduct systematic experiments across a wide spectrum of problems, from simple linear Poisson to high-Mach hypersonic flows with strong discontinuities. Through a rigorous decomposition of approximation, optimization, and truncation errors, we systematically elucidate the fundamental trade-offs and error-governing mechanisms of both paradigms, as well as two representative network architectures: multi-layer perceptron (MLP) and graph neural network (GNN). Our results reveal a consistent trend: as nonlinearity strengthens, the accuracy advantage of discretization-based constraints becomes increasingly pronounced, with smaller optimization errors compensating for the truncation errors. Moreover, the more complex the nonlinearity and boundary conditions, the greater the advantage of GNN over MLP. These insights offer a robust practical guideline for configuring neural PDE solvers in demanding engineering applications. Our source data and code are available at https://github.com/guoxing0809/neuropde_analysis.
Nonlinear partial differential equations (PDEs) play a vital role in modeling complex fluid dynamics phenomena such as turbulence, heat transfer, and flow instability. Traditional numerical approaches, including finite difference and finite element methods, often require extensive computational resources and may struggle with large-scale or real-time simulations. Recent advancements in deep learning have introduced efficient alternatives for solving nonlinear PDEs through data-driven and physics-informed approaches. This paper presents a comprehensive study of deep learning-based numerical solutions for nonlinear PDEs in fluid dynamics. The proposed framework integrates neural network architectures with physics informed constraints to accurately approximate fluid behavior while reducing computational complexity. Various deep learning models, including Physics-Informed Neural Networks (PINNs), Deep Operator Networks, and Fourier Neural Operators, are analyzed and compared with conventional numerical techniques. Experimental results demonstrate improved prediction accuracy, faster convergence, and enhanced scalability in solving complex fluid flow problems. The study further discusses practical applications, current limitations, and future research opportunities in AI-driven scientific computing for fluid dynamics.
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.
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