提出投影方法，让物理神经网络严格守恒积分量。
Guaranteeing Conservation of Integrals with Projection in Physics-Informed Neural Networks
- 通过求解约束优化问题，设计投影公式保证线性与二次积分守恒。
- 相比软约束，积分误差降低三到四数量级，微幅提升方程求解精度。
- 改善损失函数的条件数，加速收敛，适用于任意可解积分守恒场景。
我们提出一种新颖的投影方法，确保物理信息神经网络（PINNs）中积分量的守恒。尽管PINNs使用软约束来施加偏微分方程（PDEs）结构，赋予训练灵活性，但也可能导致解违反物理定律。为此，我们引入一种投影方法，可分别或联合保证线性与二次积分的守恒。通过求解约束非线性优化问题推导出投影公式。结果显示，采用该投影的PINN（称为PINN-Proj）在积分守恒误差上比软约束降低了三到四个数量级，同时方程求解误差仅轻微增加。我们还发现，该投影通过改善损失景观的条件数提升了收敛性。该方法为在任意可解条件下保证任何积分守恒提供了通用框架。
原文摘要 · Abstract (English)
We propose a novel projection method that guarantees the conservation of integral quantities in Physics-Informed Neural Networks (PINNs). While the soft constraint that PINNs use to enforce the structure of partial differential equations (PDEs) enables necessary flexibility during training, it also permits the discovered solution to violate physical laws. To address this, we introduce a projection method that guarantees the conservation of the linear and quadratic integrals, both separately and jointly. We derived the projection formulae by solving constrained non-linear optimization problems and found that our PINN modified with the projection, which we call PINN-Proj, reduced the error in the conservation of these quantities by three to four orders of magnitude compared to the soft constraint and marginally reduced the PDE solution error. We also found evidence that the projection improved convergence through improving the conditioning of the loss landscape. Our method holds promise as a general framework to guarantee the conservation of any integral quantity in a PINN if a tractable solution exists.
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