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Deep learning of free boundary and Stefan problems

arXiv 2006.05311 ↗First posted 4 Jun 2020math.NA cs.LG cs.NAcited by 35

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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.

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

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Citing papers per year: 2021: 5; 2022: 6; 2023: 6; 2024: 8; 2025: 4; 2026: 6.

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  1. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 608
  2. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 473
  3. 2022Respecting causality is all you need for training physics-informed neural networkscited by 156
  4. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 92
  5. 2021Improved architectures and training algorithms for deep operator networkscited by 86
  6. 2021One-Shot Transfer Learning of Physics-Informed Neural Networkscited by 49
  7. 2023A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositionscited by 48
  8. 2023On the Limitations of Physics-informed Deep Learning: Illustrations Using First Order Hyperbolic Conservation Law-based Traffic Flow Modelscited by 14
  9. 2024Data-Driven Physics-Informed Neural Networks: A Digital Twin Perspectivecited by 12
  10. 2022Deep neural network expressivity for optimal stopping problemscited by 9
  11. 2024Learning to Optimally Stop Diffusion Processes, with Financial Applicationscited by 9
  12. 2021Bayesian neural networks for weak solution of PDEs with uncertainty quantificationcited by 8
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  1. 2024ANaGRAM: A Natural Gradient Relative to Adapted Model for efficient PINNs learningcited by 7
  2. 2021Applying physics-based loss functions to neural networks for improved generalizability in mechanics problemscited by 5
  3. 2023Controlled physics-informed data generation for deep learning-based remaining useful life prediction under unseen operation conditionscited by 4
  4. 2023Physics-informed neural networks modeling for systems with moving immersed boundaries: application to an unsteady flow past a plunging foilcited by 4
  5. 2022Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditionscited by 3
  6. 2023Multiphysics discovery with moving boundaries using Ensemble SINDy and Peridynamic Differential Operatorcited by 3
  7. 2024Densely Multiplied Physics Informed Neural Networkscited by 3
  8. 2025Physics-Informed Machine Learning for Two-Phase Moving-Interface and Stefan Problemscited by 3
  9. 2023A Novel Training Framework for Physics-informed Neural Networks: Towards Real-time Applications in Ultrafast Ultrasound Blood Flow Imagingcited by 1
  10. 2024Kolmogorov n-Widths for Multitask Physics-Informed Machine Learning (PIML) Methods: Towards Robust Metricscited by 1
  11. 2022Phy-Taylor: Physics-Model-Based Deep Neural Networkscited by 0
  12. 2024The Principle of Minimum Pressure Gradient: An Alternative Basis for Physics-Informed Learning of Incompressible Fluid Mechanicscited by 0
  13. 2024Physically Informed Synchronic-adaptive Learning for Industrial Systems Modeling in Heterogeneous Media with Unavailable Time-varying Interfacecited by 0
  14. 2024Transformed Physics-Informed Neural Networks for The Convection-Diffusion Equationcited by 0
  15. 2025Fractional-Boundary-Regularized Deep Galerkin Method for Variational Inequalities in Mixed Optimal Stopping and Controlcited by 0
  16. 2025Taylor-Model Physics-Informed Neural Networks (PINNs) for Ordinary Differential Equationscited by 0
  17. 2025Physics-Informed Extreme Learning Machine (PIELM) for Tunnelling-Induced Soil-Pile Interactionscited by 0
  18. 2026Efficient temporal prediction of compressible flows in irregular domains using Fourier neural operatorscited by 0
  19. 2026Reinforcement Learning for Speculative Trading under Exploratory Frameworkcited by 0
  20. 2026Continual Learning as a Multiphase Moving-Boundary Problemcited by 0
  21. 2026CoFINN: Conservation Flux Informed Neural Networks for Physics Problems Governed by Conservation Lawscited by 0
  22. 2026A Data-Free Physics-Informed Neural Operator for Level-Set Interface Advectioncited by 0
  23. 2026Kolmogorov-Arnold Networks for Free-Boundary Partial Differential Equationscited by 0