# A cusp-capturing PINN for elliptic interface problems

arXiv: https://arxiv.org/abs/2210.08424 · HTML: https://mlsanity.com/paper/2210.08424/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 43 references that are in the map.
- First posted on arXiv 16 Oct 2022.

## Builds on (most cited first, 5 of 5)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [A Discontinuity Capturing Shallow Neural Network for Elliptic Interface Problems](https://mlsanity.com/paper/2106.05587/) (2021)
- [A Shallow Ritz Method for Elliptic Problems with Singular Sources](https://mlsanity.com/paper/2107.12013/) (2021)

## Built on by (most cited first, 17 of 17)

- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [BiLO: Bilevel Local Operator Learning for PDE Inverse Problems](https://mlsanity.com/paper/2404.17789/) (2024)
- [Physics-informed neural network solves minimal surfaces in curved spacetime](https://mlsanity.com/paper/2509.10866/) (2025)
- [Learning Domain-Independent Green's Function For Elliptic Partial Differential Equations](https://mlsanity.com/paper/2401.17172/) (2024)
- [Data-Guided Physics-Informed Neural Networks for Solving Inverse Problems in Partial Differential Equations](https://mlsanity.com/paper/2407.10836/) (2024)
- [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)
- [Learning Singularity-Encoded Green's Functions with Application to Iterative Methods](https://mlsanity.com/paper/2509.11580/) (2025)
- [Physics-Informed Machine Learning for Two-Phase Moving-Interface and Stefan Problems](https://mlsanity.com/paper/2512.14010/) (2025)
- [A Multiple Transferable Neural Network Method with Domain Decomposition for Elliptic Interface Problems](https://mlsanity.com/paper/2502.19893/) (2025)
- [Bayesian BiLO: Bilevel Local Operator Learning for Efficient Uncertainty Quantification of Bayesian PDE Inverse Problems with Low-Rank Adaptation](https://mlsanity.com/paper/2507.17019/) (2025)
- [Multilevel Training for Kolmogorov Arnold Networks](https://mlsanity.com/paper/2603.04827/) (2026)
- [Dirichlet-Neumann learning algorithm for solving elliptic interface problems](https://mlsanity.com/paper/2301.07361/) (2023)
- [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)
- [Identifiability Limits of Physics-Informed Inference for Spatial Stochastic Dynamics from Static Snapshots](https://mlsanity.com/paper/2607.01749/) (2026)
- [A Hybrid Iterative Deep Ritz Method for Elliptic Interface Problems](https://mlsanity.com/paper/2609.27325/) (2026)
- [CEENs: Causality-enforced evolutional networks for solving time-dependent partial differential equations](https://mlsanity.com/paper/2610.04405/) (2026)

## Abstract (arXiv)

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.