发表机构
The University of Texas at Austin（德克萨斯大学奥斯汀分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对标准PINNs的局限性，提出Gen-PINNs框架，通过生成器与判别器结合的方式，在非线性、高阶等含尖锐前沿的PDE求解中，显著提升了精度与收敛性。
AI 中文摘要
物理信息神经网络（Physics-Informed Neural Networks, PINNs）是一种广泛应用的无数据机器学习方法，用于求解偏微分方程（Partial Differential Equations, PDEs）。随着生成式对抗网络（Generative Adversarial Networks, GANs）的最新进展，对抗学习在建模复杂的数据驱动问题方面展现出强大能力；然而，GAN在确定性物理信息PDE求解中的应用仍然有限。本研究首先指出标准PINNs在求解PDE时的局限性，包括频谱偏差、损失不平衡和优化器停滞。随后，我们提出生成式对抗物理信息神经网络（Generative Adversarial Physics-Informed Neural Networks, Gen-PINNs），这是一个统一的确定性残差对抗框架，旨在改进对具有尖锐或激波前沿行为的PDE的无数据求解。生成器使用动态加权的物理信息损失分量学习潜在的PDE解，而单独的判别器则针对理想零残差状态评估互补的PDE残差特征。该框架进一步开发并适配了若干方法学组件，包括用于解析高频空间内容的傅里叶表示、用于量化频率相关解误差的正交频谱诊断，以及用于物理、初始条件、边界条件和对抗损失目标的改进型基于梯度的动态加权系统。我们在非线性和高阶PDE（包括Burgers方程、Allen-Cahn方程和Kuramoto-Sivashinsky方程）上将Gen-PINNs与标准PINNs进行了测试。结果表明，在尖锐前沿、刚性和高阶PDE求解中，Gen-PINNs在精度和收敛性方面均有显著提升，凸显了确定性残差对抗学习作为求解具有挑战性的非线性PDE的有效方法的潜力。
英文摘要
Physics-Informed Neural Networks (PINNs) are a widely used data-free method for solving Partial Differential Equations (PDEs) using machine learning. With recent advances in Generative Adversarial Networks (GANs), adversarial learning has shown strong capabilities for modeling complex data-driven problems; however, the use of GANs in deterministic physics-informed PDE solutions remains limited. In this work, we first identify limitations of standard PINNs for solving PDEs, including spectral bias, loss imbalance, and optimizer stagnation. We then propose Generative Adversarial Physics-Informed Neural Networks (Gen-PINNs), a unified deterministic residual-adversarial framework designed to improve data-free solutions of PDEs with sharp or shock-front behavior. The generator learns the underlying PDE solution using dynamically weighted physics-informed loss components, while separate discriminators evaluate complementary PDE residual features against ideal zero-residual states. The framework further develops and adapts several methodological components, including a Fourier representation for resolving high-frequency spatial content, an orthonormal spectral diagnostic for quantifying frequency-dependent solution errors, and a modified gradient-based dynamic weighting system for physics, initial-condition, boundary-condition, and adversarial loss objectives. Gen-PINNs is tested against standard PINNs on nonlinear and higher-order PDEs, including the Burgers, Allen-Cahn, and Kuramoto-Sivashinsky equations. The results demonstrate substantial improvements in accuracy and convergence across sharp-front, stiff, and higher-order PDE solutions, highlighting the potential of deterministic residual-adversarial learning as an effective approach for solving challenging nonlinear PDEs.
