arXiv subjects

Yameng Zhu

Publications and source records attributed to Yameng Zhu.

3 recordsLinked to original sources

arXiv2026

A Two-stage Adaptive Lifting PINN Framework for Solving Viscous Approximations to Hyperbolic Conservation Laws

Training physics-informed neural networks (PINNs) for hyperbolic conservation laws near the inviscid limit is challenging because shock discontinuities invalidate classical strong-form residuals, while small-viscosity regularization creates narrow internal layers that aggravate spectral bias. We propose a Two-stage Adaptive Lifting PINN (TAL-PINN), a fully autonomous framework that augments the physical coordinates with a learned auxiliary field constructed through $r$-adaptive coordinate transformations. The auxiliary geometry is extracted automatically from a computationally inexpensive high-viscosity coarse solution, providing a pre-conditioning mechanism that guides the network toward steep-gradient regions without prior knowledge of shock dynamics. The coordinate construction and lifted training can also be embedded in a viscosity-continuation procedure. Theoretically, we derive a scalar a posteriori $L^2$ error estimate that reveals the viscosity dependence of residual-based error control, establish a variance-reduction criterion for coordinate-induced importance sampling, and use neural tangent kernel analysis to explain how adaptive augmentation enriches tangent features and accelerates residual decay. Numerical experiments on scalar equations and the Euler system show faster loss decay and improved small-viscosity accuracy. Ablation studies further show that lifting yields additional gains beyond viscosity continuation and adaptive sampling alone.

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arXiv2026

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

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.

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arXiv2024

R-adaptive DeepONet: Learning Solution Operators for PDEs with Discontinuous Solutions Using an R-adaptive Strategy

DeepONet has recently been proposed as a representative framework for learning nonlinear mappings between function spaces. However, when it comes to approximating solution operators of partial differential equations (PDEs) with discontinuous solutions, DeepONet poses a foundational approximation lower bound due to its linear reconstruction property. Inspired by the moving mesh (R-adaptive) method, we propose an R-adaptive DeepONet method, which contains the following components: (1) the output data representation is transformed from the physical domain to the computational domain using the equidistribution principle; (2) the maps from input parameters to the solution and the coordinate transformation function over the computational domain are learned using DeepONets separately; (3) the solution over the physical domain is obtained via post-processing methods such as the (linear) interpolation method. Additionally, we introduce a solution-dependent weighting strategy in the training process to reduce the final error. We establish an upper bound for the reconstruction error based on piecewise linear interpolation and show that the introduced R-adaptive DeepONet can reduce this bound. Moreover, for two prototypical PDEs with sharp gradients or discontinuities, we prove that the approximation error decays at a superlinear rate with respect to the trunk basis size, unlike the linear decay observed in vanilla DeepONets. Therefore, the R-adaptive DeepONet overcomes the limitations of DeepONet, and can reduce the approximation error for problems with discontinuous solutions. Numerical experiments on PDEs with discontinuous solutions, including the linear advection equation, the Burgers' equation with low viscosity, and the compressible Euler equations of gas dynamics, are conducted to verify the advantages of the R-adaptive DeepONet over available variants of DeepONet.

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