Extended Interface Physics-Informed Neural Networks Method for Moving Interface Problems

2025/08/02 by Ran Bi, Weibing Deng, Bi, Ran +3
Engineering · Mathematics · Physics and Astronomy · #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.2508.01463

openalex publication_date 2025/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Physics-informed neural networks (PINNs) have emerged as an effective class of mesh-free methods for solving partial differential equations (PDEs), particularly on complex geometries. In this paper, we introduce an Extended Interface Physics-Informed Neural Network (XI-PINN) framework designed to solve parabolic moving interface problems. The proposed method employs a level set function--which can be either analytically prescribed or learned via a neural network--to capture the moving interface. Furthermore, we establish an a priori error analysis for the XI-PINN method and derive error bounds for the approximation. Extensive numerical experiments are provided to validate the accuracy and robustness of the framework, and its applicability is further demonstrated by solving the Oseen equations.

Citations

• Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective
• XI-DeepONet: An operator learning method for elliptic interface problems
• DRM Revisited: A Complete Error Analysis
• Challenges in Training PINNs: A Loss Landscape Perspective
• Solving Poisson Problems in Polygonal Domains with Singularity Enriched Physics Informed Neural Networks
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• When and why PINNs fail to train: A neural tangent kernel perspective
• When and why PINNs fail to train: A neural tangent kernel perspective
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• Solving Parabolic Moving Interface Problems with Dynamical Immersed Spaces on Unfitted Meshes: Fully Discrete Analysis
• Loss landscapes and optimization in over-parameterized non-linear systems and neural networks
• Loss landscapes and optimization in over-parameterized non-linear systems and neural networks
• Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators
• Neural Tangent Kernel: Convergence and Generalization in Neural Networks
• Improvements to the Levenberg-Marquardt algorithm for nonlinear least-squares minimization
• Solving Elliptic Optimal Control Problems via Neural Networks and Optimality System

Related

• Physics-informed neural networks for solving moving interface flow problems using the level set approach
• A Physics Informed Neural Network (PINN) Methodology for Coupled Moving Boundary PDEs
• Convergence of physics-informed neural networks applied to linear second-order elliptic interface problems
• The Hard-Constraint PINNs for Interface Optimal Control Problems
• Physics-Informed Neural Network Frameworks for the Analysis of Engineering and Biological Dynamical Systems Governed by Ordinary Differential Equations