Collocation point strategies in physics-informed neural networks: A comparative study on the Burgers' equation

Solving the Elastic Wave Equation with Physics-Informed Neural Networks: A Robust and Critical Assessment

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.

Internal solitary wave simulation using physics-informed neural networks with Boltzmann distribution-based adaptive sampling strategy

PINN-MG: a multigrid-inspired hybrid framework combining the iterative method and physics-informed neural networks

Iterative methods are widely used for solving partial differential equations (PDEs). However, the difficulty in eliminating global low-frequency errors significantly limits their convergence speed. In recent years, neural networks (NNs) have emerged as a novel approach for solving PDEs, with studies showing that they exhibit faster convergence for low-frequency components. Building on these complementary frequency-convergence characteristics of iterative methods and NNs, and drawing inspiration from multigrid methods, we propose a hybrid solving framework consisting of a combination of iterative methods and NN-based solvers, termed physics-informed neural network multigrid (PINN-MG, abbreviated as PMG). In this framework, the iterative methods eliminate local high-frequency oscillation errors, while PINNs correct global low-frequency errors. Throughout the solving process, high- and low-frequency components alternately dominate the error, with each being addressed by the iterative methods and PINNs, respectively, thereby accelerating the convergence. We validate the proposed PMG framework on the linear Poisson equations and nonlinear Helmholtz equations. The results show significant acceleration of the PMG built on the Gauss-Seidel (GS), pseudo-time, and generalized minimal residual (GMRES) methods. A detailed analysis of the convergence process validates the rationality of the framework. To further evaluate the advantages of PMG, we apply it to indefinite Helmholtz equations, a class of problems in which traditional solvers often diverge due to the divergence of the low-frequency components. The PMG framework effectively overcomes this divergence, improving both the stability and the convergence speed. We propose the PMG framework as a data-free hybrid solver that does not rely on any pretraining and, more importantly, provides a unified mechanism to tightly couple the NN methods with classical iterative solvers, achieving an organic and iterative integration of the two paradigms.

Adaptive regularized hybrid physics-informed neural networks for large-gradient problems

The proposed ARH-PINNs can accurately resolve large-gradient fields such as shock waves, while effectively suppressing non-physical oscillations and retaining low numerical dissipation.

Solving the FitzHugh-Nagumo Equation Using Physics-Informed Neural Networks (PINNs) and Deep Galerkin Methods

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.

Tikhonov-Regularized Physics-Informed Neural Networks for Terminal-State Distributed Optimal Control of Parabolic Partial Differential Equations

A regularized PINNs framework that incorporates Tikhonov regularization to solve terminal-state tracking optimal control constrained by parabolic partial differential equations is proposed, establishing a consistency result showing that PINNs minimizers nearly attain the continuous regularized objective under residual and quadrature approximation assumptions.