Deep learning-accelerated computational framework based on Physics Informed Neural Network for the solution of linear elasticity

Physics-informed Neural Implicit Flow neural network for parametric PDEs

The Physics-informed Neural Network (PINN) has been a popular method for solving partial differential equations (PDEs) due to its flexibility. However, PINN still faces challenges in characterizing spatio-temporal correlations when solving parametric PDEs due to network limitations. To address this issue, we propose a Physics-Informed Neural Implicit Flow (PINIF) framework, which enables a meshless low-rank representation of the parametric spatio-temporal field based on the expressiveness of the Neural Implicit Flow (NIF), enabling a meshless low-rank representation. In particular, the PINIF framework utilizes the Polynomial Chaos Expansion (PCE) method to quantify the uncertainty in the presence of noise, allowing for a more robust representation of the solution. In addition, PINIF introduces a novel transfer learning framework to speed up the inference of parametric PDEs significantly. The performance of PINIF and PINN is compared on various PDEs especially with variable coefficients and Kolmogorov flow. The comparative results indicate that PINIF outperforms PINN in terms of accuracy and efficiency.

Diminishing spectral bias in physics-informed neural networks using spatially-adaptive Fourier feature encoding

Physics-informed neural networks (PINNs) have recently emerged as a promising framework for solving partial differential equation (PDE) systems in computer mechanics. However, PINNs still struggle in simulating systems whose solution functions exhibit high-frequency patterns, especially in cases with wide frequency spectrums. Current methods apply Fourier feature mappings to the input to improve the learning ability of model on high-frequency components. However, they are largely problem-dependent which require proper selection of hyperparameters and introduces additional training difficulty into the optimization. To this end, we present a spatially adaptive Fourier feature encoding method accompanied by a tree-based sampling strategy in this work. Specifically, we propose to guide the Fourier feature mappings of input by gradually exposing the input coordinate from low to higher encoding frequencies during training through the feedback loop of loss. Meanwhile, we also propose to refine the sampling of residual points by presenting a novel tree-based sampling strategy. This method represents the input domain by a tree and formulates the sampling of residual points as a resource allocation problem which optimizes the sampling of residual points during training and assigns more computational capacity to the underfit region. The effectiveness of our proposed method is demonstrated in several challenging PDE problems, including Poisson equation, heat equation, Navier-Stokes equations, Reynolds-Averaged Navier-Stokes equations, and Maxwell equation. The results indicate that our method can better allocate the computational resources during training and enable the model to fit the local frequencies of target function adaptively.

Spectral integrated neural networks (SINNs) for solving forward and inverse dynamic problems

This study introduces an innovative neural network framework named spectral integrated neural networks (SINNs) to address both forward and inverse dynamic problems in three-dimensional space. In the SINNs, the spectral integration technique is utilized for temporal discretization, followed by the application of a fully connected neural network to solve the resulting partial differential equations in the spatial domain. Furthermore, the polynomial basis functions are employed to expand the unknown function, with the goal of improving the performance of SINNs in tackling inverse problems. The performance of the developed framework is evaluated through several dynamic benchmark examples encompassing linear and nonlinear heat conduction problems, linear and nonlinear wave propagation problems, inverse problem of heat conduction, and long-time heat conduction problem. The numerical results demonstrate that the SINNs can effectively and accurately solve forward and inverse problems involving heat conduction and wave propagation. Additionally, the SINNs provide precise and stable solutions for dynamic problems with extended time durations. Compared to commonly used physics-informed neural networks, the SINNs exhibit superior performance with enhanced convergence speed, computational accuracy, and efficiency.

Towards complex dynamic physics system simulation with graph neural ordinary equations

The great learning ability of deep learning facilitates us to comprehend the real physical world, making learning to simulate complicated particle systems a promising endeavour both in academia and industry. However, the complex laws of the physical world pose significant challenges to the learning based simulations, such as the varying spatial dependencies between interacting particles and varying temporal dependencies between particle system states in different time stamps, which dominate particles' interacting behavior and the physical systems' evolution patterns. Existing learning based methods fail to fully account for the complexities, making them unable to yield satisfactory simulations. To better comprehend the complex physical laws, we propose a novel model - Graph Networks with Spatial-Temporal neural Ordinary Differential Equations (GNSTODE) - that characterizes the varying spatial and temporal dependencies in particle systems using a united end-to-end framework. Through training with real-world particle-particle interaction observations, GNSTODE can simulate any possible particle systems with high precisions. We empirically evaluate GNSTODE's simulation performance on two real-world particle systems, Gravity and Coulomb, with varying levels of spatial and temporal dependencies. The results show that GNSTODE yields better simulations than state-of-the-art methods, showing that GNSTODE can serve as an effective tool for particle simulation in real-world applications. Our code is made available at https://github.com/Guangsi-Shi/AI-for-physics-GNSTODE.

Physics-informed kernel function neural networks for solving partial differential equations

This paper proposes an improved version of physics-informed neural networks (PINNs), the physics-informed kernel function neural networks (PIKFNNs), to solve various linear and some specific nonlinear partial differential equations (PDEs). It can also be considered as a novel radial basis function neural network (RBFNN). In the proposed PIKFNNs, it employs one-hidden-layer shallow neural network with the physics-informed kernel functions (PIKFs) as the customized activation functions. The PIKFs fully or partially contain PDE information, which can be chosen as fundamental solutions, green's functions, T-complete functions, harmonic functions, radial Trefftz functions, probability density functions and even the solutions of some linear simplified PDEs and so on. The main difference between the PINNs and the proposed PIKFNNs is that the PINNs add PDE constraints to the loss function, and the proposed PIKFNNs embed PDE information into the activation functions of the neural network. The feasibility and accuracy of the proposed PIKFNNs are validated by some benchmark examples.