# Deep learning of free boundary and Stefan problems

arXiv: https://arxiv.org/abs/2006.05311 · HTML: https://mlsanity.com/paper/2006.05311/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jun 2020.

## Built on by (most cited first, 30 of 35)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
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- [One-Shot Transfer Learning of Physics-Informed Neural Networks](https://mlsanity.com/paper/2110.11286/) (2021)
- [A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions](https://mlsanity.com/paper/2302.14227/) (2023)
- [On the Limitations of Physics-informed Deep Learning: Illustrations Using First Order Hyperbolic Conservation Law-based Traffic Flow Models](https://mlsanity.com/paper/2302.12337/) (2023)
- [Data-Driven Physics-Informed Neural Networks: A Digital Twin Perspective](https://mlsanity.com/paper/2401.08667/) (2024)
- [Deep neural network expressivity for optimal stopping problems](https://mlsanity.com/paper/2210.10443/) (2022)
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- [ANaGRAM: A Natural Gradient Relative to Adapted Model for efficient PINNs learning](https://mlsanity.com/paper/2412.10782/) (2024)
- [Applying physics-based loss functions to neural networks for improved generalizability in mechanics problems](https://mlsanity.com/paper/2105.00075/) (2021)
- [Controlled physics-informed data generation for deep learning-based remaining useful life prediction under unseen operation conditions](https://mlsanity.com/paper/2304.11702/) (2023)
- [Physics-informed neural networks modeling for systems with moving immersed boundaries: application to an unsteady flow past a plunging foil](https://mlsanity.com/paper/2306.13395/) (2023)
- [Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions](https://mlsanity.com/paper/2205.03672/) (2022)
- [Multiphysics discovery with moving boundaries using Ensemble SINDy and Peridynamic Differential Operator](https://mlsanity.com/paper/2303.15631/) (2023)
- [Densely Multiplied Physics Informed Neural Networks](https://mlsanity.com/paper/2402.04390/) (2024)
- [Physics-Informed Machine Learning for Two-Phase Moving-Interface and Stefan Problems](https://mlsanity.com/paper/2512.14010/) (2025)
- [A Novel Training Framework for Physics-informed Neural Networks: Towards Real-time Applications in Ultrafast Ultrasound Blood Flow Imaging](https://mlsanity.com/paper/2309.04755/) (2023)
- [Kolmogorov n-Widths for Multitask Physics-Informed Machine Learning (PIML) Methods: Towards Robust Metrics](https://mlsanity.com/paper/2402.11126/) (2024)
- [Phy-Taylor: Physics-Model-Based Deep Neural Networks](https://mlsanity.com/paper/2209.13511/) (2022)
- [The Principle of Minimum Pressure Gradient: An Alternative Basis for Physics-Informed Learning of Incompressible Fluid Mechanics](https://mlsanity.com/paper/2401.07489/) (2024)
- [Physically Informed Synchronic-adaptive Learning for Industrial Systems Modeling in Heterogeneous Media with Unavailable Time-varying Interface](https://mlsanity.com/paper/2401.14609/) (2024)
- [Transformed Physics-Informed Neural Networks for The Convection-Diffusion Equation](https://mlsanity.com/paper/2409.07671/) (2024)
- [Fractional-Boundary-Regularized Deep Galerkin Method for Variational Inequalities in Mixed Optimal Stopping and Control](https://mlsanity.com/paper/2505.19309/) (2025)
- [Taylor-Model Physics-Informed Neural Networks (PINNs) for Ordinary Differential Equations](https://mlsanity.com/paper/2507.03860/) (2025)
- [Physics-Informed Extreme Learning Machine (PIELM) for Tunnelling-Induced Soil-Pile Interactions](https://mlsanity.com/paper/2510.00698/) (2025)
- [Efficient temporal prediction of compressible flows in irregular domains using Fourier neural operators](https://mlsanity.com/paper/2601.01922/) (2026)

## Abstract (arXiv)

Free boundary problems appear naturally in numerous areas of mathematics, science and engineering. These problems present a great computational challenge because they necessitate numerical methods that can yield an accurate approximation of free boundaries and complex dynamic interfaces. In this work, we propose a multi-network model based on physics-informed neural networks to tackle a general class of forward and inverse free boundary problems called Stefan problems. Specifically, we approximate the unknown solution as well as any moving boundaries by two deep neural networks. Besides, we formulate a new type of inverse Stefan problems that aim to reconstruct the solution and free boundaries directly from sparse and noisy measurements. We demonstrate the effectiveness of our approach in a series of benchmarks spanning different types of Stefan problems, and illustrate how the proposed framework can accurately recover solutions of partial differential equations with moving boundaries and dynamic interfaces. All code and data accompanying this manuscript are publicly available at urlhttps://github.com/PredictiveIntelligenceLab/DeepStefan.