用李对称性精准求解偏微分方程初边值问题，更快更准。
LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries
- 基于李对称性构造解，物理定律直接嵌入模型中。
- 在典型线性齐次PDE上，速度与精度均优于PINNs。
- 适合需要高可靠性的科学计算与工程建模场景。
初边值问题（IBVPs）是物理与工程中广泛现象建模的核心框架。本文提出一种新方法——LieSolver，利用李对称性精确构造满足偏微分方程（PDE）的解。通过对称变换，模型将底层物理规律嵌入其中，仅需初始与边界数据即可学习解函数。因此，边界损失可直接反映全域误差，实现对适定IBVP的严格误差估计。我们实现了LieSolver并应用于线性齐次PDE，结果表明其在速度与精度上均优于物理信息神经网络（PINNs），同时模型更为紧凑。整体而言，该方法显著提升了PDE约束问题预测的效率与可靠性。
原文摘要 · Abstract (English)
Initial-boundary value problems (IBVPs) provide the essential framework for modelling a wide range of phenomena in physics and engineering. We introduce a novel method for efficiently solving IBVPs using Lie symmetries to enforce the associated partial differential equation (PDE) exactly by construction. By leveraging symmetry transformations, our model embeds the underlying physical laws and learns the solution solely from initial and boundary data. Consequently, the boundary loss directly quantifies domain-wide error, enabling rigorous error estimation for well-posed IBVPs. We implement LieSolver and demonstrate its application to linear homogeneous PDEs, showing that it outperforms physics-informed neural networks (PINNs) in both speed and accuracy while yielding compact models. Overall, our approach significantly enhances the efficiency and reliability of predictions for PDE-constrained problems.
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