用积分约束提升神经网络对物理突变的处理能力
Coupled Integral PINN for Discontinuity
- 引入辅助网络与积分约束，增强对间断的建模能力
- 在伯格斯、浅水方程等模型上表现优于传统PINN
- 无需网格和数值通量计算，适合复杂间断问题
物理信息神经网络（PINN）通过最小化控制方程残差来求解前向偏微分方程，但在冲击波等间断现象上表现不佳。相比之下，有限体积法（FVM）通过积分守恒原理处理间断，可接受弱解。受此启发，我们提出耦合积分物理信息神经网络（CI-PINN），在标准PINN基础上增加一个用于积分势的辅助网络，并引入耦合积分约束。该方法在不依赖网格、无需数值通量积分与重构的情况下，显著提升了冲击附近区域的鲁棒性。我们在伯格斯方程、Buckley--Leverett方程、欧拉系统和浅水方程等多个前向基准测试中验证了CI-PINN的有效性。
原文摘要 · Abstract (English)
Physics-Informed Neural Networks (PINNs) solve forward PDEs by minimizing residual losses from the governing equations with initial and boundary conditions, but they often struggle with discontinuities such as shocks. In contrast, finite volume methods (FVM) handle discontinuities by enforcing integral conservation, which admits weak solutions. Motivated by this, we propose a Coupled Integral PINN (CI-PINN) that augments a standard PINN with an auxiliary network for integral potentials and coupled integral constraints. This improves robustness near shocks while avoiding meshing and the numerical flux integration/reconstruction used in classical schemes. We validate CI-PINN on forward benchmarks including Burgers, Buckley--Leverett, the Euler system, and the Shallow-Water equations.
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