让神经网络求解微分方程时自动遵守物理定律，杜绝虚假解。
Learning Under Laws: A Constraint-Projected Neural PDE Solver that Eliminates Hallucinations
- 通过约束投影将网络输出限制在物理可行域内，确保每一步更新合法。
- 在伯格斯和欧拉系统上实现机器精度的守恒律与熵增控制。
- 适合需要高可信度物理模拟的科研与工程场景，如流体仿真。
神经网络可逼近偏微分方程的解，但常违背其应遵循的物理规律——凭空生出质量、冲击波漂移、破坏守恒或熵增。我们提出约束投影学习（CPL）框架，使训练过程严格遵循物理法则。该方法通过将网络输出投影到守恒律、朗金-尤戈尼奥特平衡、熵与正性约束的交集，确保每次更新均物理可接受。投影操作可微，仅增加约10%计算开销，完全兼容反向传播。进一步引入总变差阻尼（TVD）抑制小振荡，并采用滚动课程训练强化长时预测一致性。结果表明，该方法彻底消除硬性与软性违反：守恒律达机器精度，总变差增长为零，熵与误差保持有界。在伯格斯（Burgers）与欧拉（Euler）系统上，CPL生成稳定且符合物理的解，且不损失精度。与其期待神经求解器自发尊重物理，不如让这一行为成为学习过程的内在属性。
原文摘要 · Abstract (English)
Neural networks can approximate solutions to partial differential equations, but they often break the very laws they are meant to model-creating mass from nowhere, drifting shocks, or violating conservation and entropy. We address this by training within the laws of physics rather than beside them. Our framework, called Constraint-Projected Learning (CPL), keeps every update physically admissible by projecting network outputs onto the intersection of constraint sets defined by conservation, Rankine-Hugoniot balance, entropy, and positivity. The projection is differentiable and adds only about 10% computational overhead, making it fully compatible with back-propagation. We further stabilize training with total-variation damping (TVD) to suppress small oscillations and a rollout curriculum that enforces consistency over long prediction horizons. Together, these mechanisms eliminate both hard and soft violations: conservation holds at machine precision, total-variation growth vanishes, and entropy and error remain bounded. On Burgers and Euler systems, CPL produces stable, physically lawful solutions without loss of accuracy. Instead of hoping neural solvers will respect physics, CPL makes that behavior an intrinsic property of the learning process.
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