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Antiderivative-enhanced adaptive sampling algorithm for physics-informed neural networks
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DOI：10.1088/1402-4896/ae30b5.png)
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Abstract
En 中文
Physics-informed neural networks (PINNs) effectively integrate physical equation constraints into the model to simultaneously learn data distributions and underlying physical laws. However, conventional uniform sampling is computationally expensive and inefficient for complex systems. The residual-based adaptive refinement distribution (RARD) algorithm, proposed by Wang et al, improves sampling but relies heavily on residual estimation, neglecting higher-order derivatives and leading to potential suboptimal sampling distributions. To address these issues, this study proposes the Residual-based Adaptive Antiderivative Approximation (RA-ADAF) approach. Within a fully connected neural network (FCNN) framework, RA-ADAF introduces antiderivative approximation layers (ADAF) with integral structure modeling capability, enhancing the representation of both the target function and its derivatives. Unlike the RARD strategy that solely relies on residual estimation for adaptive sampling, RA-ADAF combines residual-driven adaptive sampling with derivative-informed structural modeling. This dual mechanism improves model fitting in error-concentrated regions and enhances the representation of higher-order derivative behavior. The effectiveness of the RA-ADAF method is validated through a series of numerical experiments, including comparisons on the Burgers equation, Allen–Cahn equation, wave equation and 2D Poisson equation, as well as ablation studies. Experimental results demonstrate that RA-ADAF consistently achieves lower relative L2 error compared to the RARD method across multiple test cases. Moreover, it exhibits faster error convergence and more stable training behavior, providing a more reliable strategy for solving partial differential equations (PDEs) involving complex physical phenomena.
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