# A Discontinuity Capturing Shallow Neural Network for Elliptic Interface Problems arXiv: https://arxiv.org/abs/2106.05587 ยท HTML: https://mlsanity.com/paper/2106.05587/ ## Key facts - Cited by 20 papers in the map (data through 14 Oct 2026). - Builds on 10 of its 26 references that are in the map. - First posted on arXiv 10 Jun 2021. ## Builds on (most cited first, 10 of 10) - [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014) - [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015) - [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017) - [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017) - [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017) - [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017) - [Approximating Continuous Functions by ReLU Nets of Minimal Width](https://mlsanity.com/paper/1710.11278/) (2017) - [A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems](https://mlsanity.com/paper/1912.01825/) (2019) - [PFNN: A Penalty-Free Neural Network Method for Solving a Class of Second-Order Boundary-Value Problems on Complex Geometries](https://mlsanity.com/paper/2004.06490/) (2020) - [Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEs](https://mlsanity.com/paper/1911.02109/) (2019) ## Built on by (most cited first, 20 of 20) - [Deep Network Approximation: Achieving Arbitrary Accuracy with Fixed Number of Neurons](https://mlsanity.com/paper/2107.02397/) (2021) - [A cusp-capturing PINN for elliptic interface problems](https://mlsanity.com/paper/2210.08424/) (2022) - [Binary structured physics-informed neural networks for solving equations with rapidly changing solutions](https://mlsanity.com/paper/2401.12806/) (2024) - [JAX-DIPS: Neural bootstrapping of finite discretization methods and application to elliptic problems with discontinuities](https://mlsanity.com/paper/2210.14312/) (2022) - [A Shallow Ritz Method for Elliptic Problems with Singular Sources](https://mlsanity.com/paper/2107.12013/) (2021) - [The Hard-Constraint PINNs for Interface Optimal Control Problems](https://mlsanity.com/paper/2308.06709/) (2023) - [Green Multigrid Network](https://mlsanity.com/paper/2407.03593/) (2024) - [Physics-Informed Machine Learning for Two-Phase Moving-Interface and Stefan Problems](https://mlsanity.com/paper/2512.14010/) (2025) - [Error-correcting neural networks for semi-Lagrangian advection in the level-set method](https://mlsanity.com/paper/2110.11611/) (2021) - [An efficient neural-network and finite-difference hybrid method for elliptic interface problems with applications](https://mlsanity.com/paper/2210.05523/) (2022) - [A Multiple Transferable Neural Network Method with Domain Decomposition for Elliptic Interface Problems](https://mlsanity.com/paper/2502.19893/) (2025) - [A shallow physics-informed neural network for solving partial differential equations on surfaces](https://mlsanity.com/paper/2203.01581/) (2022) - [Stochastic Scaling in Loss Functions for Physics-Informed Neural Networks](https://mlsanity.com/paper/2208.03776/) (2022) - [Conservative Physics-Informed Neural Networks for Non-Conservative Hyperbolic Conservation Laws Near Critical States](https://mlsanity.com/paper/2305.12817/) (2023) - [A categorical embedding discontinuity-capturing shallow neural network for anisotropic elliptic interface problems](https://mlsanity.com/paper/2503.15441/) (2025) - [A Hybrid Kernel-Free Boundary Integral Method with Operator Learning for Solving Parametric Partial Differential Equations In Complex Domains](https://mlsanity.com/paper/2404.15242/) (2024) - [Walk-on-Interfaces: A Monte Carlo Estimator for an Elliptic Interface Problem with Nonhomogeneous Flux Jump Conditions and a Neumann Boundary Condition](https://mlsanity.com/paper/2508.16767/) (2025) - [Smooth Piecewise Cutting for Neural Operator to Handle Discontinuities and Sharp Transitions](https://mlsanity.com/paper/2605.19823/) (2026) - [DeepOHeat-v2: Self-Improving Operator Learning for Fast and Trustworthy Thermal Optimization in 3D-IC Design](https://mlsanity.com/paper/2608.16080/) (2026) - [A Hybrid Iterative Deep Ritz Method for Elliptic Interface Problems](https://mlsanity.com/paper/2609.27325/) (2026) ## Abstract (arXiv) 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.