Wei-Fan Hu

Yu-Hau Tseng, Te-Sheng Lin, Wei-Fan Hu et al.

In this paper, we propose a cusp-capturing physics-informed neural network (PINN) to solve discontinuous-coefficient elliptic interface problems whose solution is continuous but has discontinuous first derivatives on the interface. To find such a solution using neural network representation, we introduce a cusp-enforced level set function as an additional feature input to the network to retain the inherent solution properties; that is, capturing the solution cusps (where the derivatives are discontinuous) sharply. In addition, the proposed neural network has the advantage of being mesh-free, so it can easily handle problems in irregular domains. We train the network using the physics-informed framework in which the loss function comprises the residual of the differential equation together with certain interface and boundary conditions. We conduct a series of numerical experiments to demonstrate the effectiveness of the cusp-capturing technique and the accuracy of the present network model. Numerical results show that even using a one-hidden-layer (shallow) network with a moderate number of neurons and sufficient training data points, the present network model can achieve prediction accuracy comparable with traditional methods. Besides, if the solution is discontinuous across the interface, we can simply incorporate an additional supervised learning task for solution jump approximation into the present network without much difficulty.

Wei-Fan Hu, Te-Sheng Lin, Yu-Hau Tseng et al.

A new and efficient neural-network and finite-difference hybrid method is developed for solving Poisson equation in a regular domain with jump discontinuities on embedded irregular interfaces. Since the solution has low regularity across the interface, when applying finite difference discretization to this problem, an additional treatment accounting for the jump discontinuities must be employed. Here, we aim to elevate such an extra effort to ease our implementation by machine learning methodology. The key idea is to decompose the solution into singular and regular parts. The neural network learning machinery incorporating the given jump conditions finds the singular solution, while the standard five-point Laplacian discretization is used to obtain the regular solution with associated boundary conditions. Regardless of the interface geometry, these two tasks only require supervised learning for function approximation and a fast direct solver for Poisson equation, making the hybrid method easy to implement and efficient. The two- and three-dimensional numerical results show that the present hybrid method preserves second-order accuracy for the solution and its derivatives, and it is comparable with the traditional immersed interface method in the literature. As an application, we solve the Stokes equations with singular forces to demonstrate the robustness of the present method.

Wei-Fan Hu, Yi-Jun Shih, Te-Sheng Lin et al.

In this paper, we introduce a shallow (one-hidden-layer) physics-informed neural network for solving partial differential equations on static and evolving surfaces. For the static surface case, with the aid of level set function, the surface normal and mean curvature used in the surface differential expressions can be computed easily. So instead of imposing the normal extension constraints used in literature, we write the surface differential operators in the form of traditional Cartesian differential operators and use them in the loss function directly. We perform a series of performance study for the present methodology by solving Laplace-Beltrami equation and surface diffusion equation on complex static surfaces. With just a moderate number of neurons used in the hidden layer, we are able to attain satisfactory prediction results. Then we extend the present methodology to solve the advection-diffusion equation on an evolving surface with given velocity. To track the surface, we additionally introduce a prescribed hidden layer to enforce the topological structure of the surface and use the network to learn the homeomorphism between the surface and the prescribed topology. The proposed network structure is designed to track the surface and solve the equation simultaneously. Again, the numerical results show comparable accuracy as the static cases. As an application, we simulate the surfactant transport on the droplet surface under shear flow and obtain some physically plausible results.

Wei-Fan Hu, Shi-Xiang Zhong, Po-Wen Hsieh et al.

Singularly perturbed partial differential equations arise in many applications, including magnetohydrodynamic duct flows, chemical reaction transport systems, and Poisson Boltzmann electrostatics. These problems are characterized by sharp boundary layers and pronounced multiscale behavior, posing significant challenges for numerical methods. Existing approaches, particularly machine learning based methods, often rely on explicit asymptotic decompositions or specialized architectures, increasing implementation complexity and leading to optimization imbalance in stiff regimes. In this work, we propose a unified learning framework based on a weighted loss formulation within the standard physics informed neural network setting. The proposed method requires only prior knowledge of the boundary layer thickness, while the boundary layer locations are automatically identified during training. The resulting formulation avoids problem specific architectural modifications and remains applicable across different equation types. Numerical experiments on both scalar and coupled reaction diffusion and convection diffusion reaction systems, defined on regular and irregular domains, demonstrate robust performance for boundary layer thickness as small as $10^{-10}$ while maintaining high solution accuracy.

Wei-Fan Hu, Te-Sheng Lin, Yu-Hau Tseng et al.

In this paper, we propose a categorical embedding discontinuity-capturing shallow neural network for anisotropic elliptic interface problems. The architecture comprises three hidden layers: a discontinuity-capturing layer, which maps domain segments to disconnected sets in a higher-dimensional space; a categorical embedding layer, which reduces the high-dimensional information into low-dimensional features; and a fully connected layer, which models the continuous mapping. This design enables a single neural network to approximate piecewise smooth functions with high accuracy, even when the number of discontinuous pieces ranges from tens to hundreds. By automatically learning discontinuity embeddings, the proposed categorical embedding technique avoids the need for explicit domain labeling, providing a scalable, efficient, and mesh-free framework for approximating piecewise continuous solutions. To demonstrate its effectiveness, we apply the proposed method to solve anisotropic elliptic interface problems, training by minimizing the mean squared error loss of the governing system. Numerical experiments demonstrate that, despite its shallow and simple structure, the proposed method achieves accuracy and efficiency comparable to traditional grid-based numerical methods.

Ming-Chih Lai, Che-Chia Chang, Wei-Syuan Lin et al.

In this paper, a shallow Ritz-type neural network for solving elliptic equations with delta function singular sources on an interface is developed. There are three novel features in the present work; namely, (i) the delta function singularity is naturally removed, (ii) level set function is introduced as a feature input, (iii) it is completely shallow, comprising only one hidden layer. We first introduce the energy functional of the problem and then transform the contribution of singular sources to a regular surface integral along the interface. In such a way, the delta function singularity can be naturally removed without introducing a discrete one that is commonly used in traditional regularization methods, such as the well-known immersed boundary method. The original problem is then reformulated as a minimization problem. We propose a shallow Ritz-type neural network with one hidden layer to approximate the global minimizer of the energy functional. As a result, the network is trained by minimizing the loss function that is a discrete version of the energy. In addition, we include the level set function of the interface as a feature input of the network and find that it significantly improves the training efficiency and accuracy. We perform a series of numerical tests to show the accuracy of the present method and its capability for problems in irregular domains and higher dimensions.

Wei-Fan Hu, Te-Sheng Lin, Ming-Chih Lai

In this paper, a new Discontinuity Capturing Shallow Neural Network (DCSNN) for approximating $d$-dimensional piecewise continuous functions and for solving elliptic interface problems is developed. There are three novel features in the present network; namely, (i) jump discontinuities are accurately captured, (ii) it is completely shallow, comprising only one hidden layer, (iii) it is completely mesh-free for solving partial differential equations. The crucial idea here is that a $d$-dimensional piecewise continuous function can be extended to a continuous function defined in $(d+1)$-dimensional space, where the augmented coordinate variable labels the pieces of each sub-domain. We then construct a shallow neural network to express this new function. Since only one hidden layer is employed, the number of training parameters (weights and biases) scales linearly with the dimension and the neurons used in the hidden layer. For solving elliptic interface problems, the network is trained by minimizing the mean square error loss that consists of the residual of the governing equation, boundary condition, and the interface jump conditions. We perform a series of numerical tests to demonstrate the accuracy of the present network. Our DCSNN model is efficient due to only a moderate number of parameters needed to be trained (a few hundred parameters used throughout all numerical examples), and the results indicate good accuracy. Compared with the results obtained by the traditional grid-based immersed interface method (IIM), which is designed particularly for elliptic interface problems, our network model shows a better accuracy than IIM. We conclude by solving a six-dimensional problem to demonstrate the capability of the present network for high-dimensional applications.