Physics-informed neural networks (PINNs) are a promising approach that combines the power of neural networks with the interpretability of physical modeling. PINNs have shown good practical performance in solving partial differential equations (PDEs) and in hybrid modeling scenarios, where physical models enhance data-driven approaches. However, it is essential to establish their theoretical properties in order to fully understand their capabilities and limitations. In this study, we highlight that classical training of PINNs can suffer from systematic overfitting. This problem can be addressed by adding a ridge regularization to the empirical risk, which ensures that the resulting estimator is risk-consistent for both linear and nonlinear PDE systems. However, the strong convergence of PINNs to a solution satisfying the physical constraints requires a more involved analysis using tools from functional analysis and calculus of variations. In particular, for linear PDE systems, an implementable Sobolev-type regularization allows to reconstruct a solution that not only achieves statistical accuracy but also maintains consistency with the underlying physics.

翻译：物理信息神经网络（PINNs）是一种融合神经网络能力与物理建模可解释性的前沿方法。PINNs在求解偏微分方程以及物理模型增强数据驱动方法的混合建模场景中展现出良好的实际性能。然而，为充分理解其能力与局限性，必须建立其理论性质。本研究指出，PINNs的经典训练可能遭受系统性过拟合问题。通过向经验风险添加岭正则化可解决该问题，从而确保所得估计量对线性和非线性偏微分方程组均具有风险一致性。但要使PINNs强收敛至满足物理约束的解，则需要运用泛函分析与变分法工具进行更深入的分析。特别地，对于线性偏微分方程组，可实现的Sobolev型正则化能够重构既达到统计精度又保持与底层物理一致性的解。