为神经网络求解偏微分方程提供误差认证，确保残差小就意味解准。
Rigorous Error Certification for Neural PDE Solvers: From Empirical Residuals to Solution Guarantees
- 建立残差控制与解空间误差的理论联系，证明残差趋零则解收敛。
- 给出确定性和概率性收敛结果，将残差/初边值误差转化为解误差上界。
- 首次实现从经验残差到解精度的可验证保证，适合可信科学计算研究者。
偏微分方程的不确定性量化传统上依赖离散化理论，通过网格细化控制解误差。物理信息神经网络从根本上偏离这一范式：它们通过最小化采样点上的残差损失来逼近解，引入了优化、采样、表示和过拟合等新误差源。因此，解空间中的泛化误差仍是一个开放问题。本文主要理论贡献在于建立了泛化边界，将残差控制与解空间误差相连接。我们证明，当神经近似位于解空间的紧子集内时，残差误差趋零可保证解收敛至真解。推导出确定性和概率性收敛结果，并提供可认证的泛化边界，将残差、边界和初值误差转化为明确的解误差保证。
原文摘要 · Abstract (English)
Uncertainty quantification for partial differential equations is traditionally grounded in discretization theory, where solution error is controlled via mesh/grid refinement. Physics-informed neural networks fundamentally depart from this paradigm: they approximate solutions by minimizing residual losses at collocation points, introducing new sources of error arising from optimization, sampling, representation, and overfitting. As a result, the generalization error in the solution space remains an open problem. Our main theoretical contribution establishes generalization bounds that connect residual control to solution-space error. We prove that when neural approximations lie in a compact subset of the solution space, vanishing residual error guarantees convergence to the true solution. We derive deterministic and probabilistic convergence results and provide certified generalization bounds translating residual, boundary, and initial errors into explicit solution error guarantees.
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