# Walk-on-Interfaces: A Monte Carlo Estimator for an Elliptic Interface Problem with Nonhomogeneous Flux Jump Conditions and a Neumann Boundary Condition

arXiv: https://arxiv.org/abs/2508.16767 · HTML: https://mlsanity.com/paper/2508.16767/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 60 references that are in the map.
- First posted on arXiv 22 Aug 2025.

## Builds on (most cited first, 12 of 12)

- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [On the Inductive Bias of Neural Tangent Kernels](https://mlsanity.com/paper/1905.12173/) (2019)
- [Towards Understanding the Spectral Bias of Deep Learning](https://mlsanity.com/paper/1912.01198/) (2019)
- [Spectral Bias in Practice: The Role of Function Frequency in Generalization](https://mlsanity.com/paper/2110.02424/) (2021)
- [A Discontinuity Capturing Shallow Neural Network for Elliptic Interface Problems](https://mlsanity.com/paper/2106.05587/) (2021)
- [A cusp-capturing PINN for elliptic interface problems](https://mlsanity.com/paper/2210.08424/) (2022)
- [A Derivative-Free Method for Solving Elliptic Partial Differential Equations with Deep Neural Networks](https://mlsanity.com/paper/2001.06145/) (2020)
- [Grid-Free Monte Carlo for PDEs with Spatially Varying Coefficients](https://mlsanity.com/paper/2201.13240/) (2022)
- [Solving Poisson Equations using Neural Walk-on-Spheres](https://mlsanity.com/paper/2406.03494/) (2024)
- [Walk on Stars: A Grid-Free Monte Carlo Method for PDEs with Neumann Boundary Conditions](https://mlsanity.com/paper/2302.11815/) (2023)
- [A Practical Walk-on-Boundary Method for Boundary Value Problems](https://mlsanity.com/paper/2305.04403/) (2023)
- [Precise Gradient Discontinuities in Neural Fields for Subspace Physics](https://mlsanity.com/paper/2505.20421/) (2025)

## Abstract (arXiv)

Elliptic interface problems arise in numerous scientific and engineering applications, modeling heterogeneous materials in which physical properties change discontinuously across interfaces. In this paper, we present textitWalk-on-Interfaces (WoI), a grid-free Monte Carlo estimator for a class of Neumann elliptic interface problems with nonhomogeneous flux jump conditions. Our Monte Carlo estimators maintain consistent accuracy throughout the domain and, thus, do not suffer from the well-known close-to-source evaluation issue near the interfaces. We also presented a simple modification with reduced variance. Estimation of the gradient of the solution can be performed, with almost no additional cost, by simply computing the gradient of the Green's function in WoI. Taking a scientific machine learning approach, we use our estimators to provide training data for a deep neural network that outputs a continuous representation of the solution. This regularizes our solution estimates by removing the high-frequency Monte Carlo error. All of our estimators are highly parallelizable, have a O(1 / √W) convergence rate in the number of samples, and generalize naturally to higher dimensions. We solve problems with many interfaces that have irregular geometry and in up to dimension six. Numerical experiments demonstrate the effectiveness of the approach and to highlight its potential in solving problems motivated by real-world applications.