为PINNs提供可信赖的误差认证方法，助力真实场景下物理方程求解。
Prediction error certification for PINNs: Theory, computation, and application to Stokes flow
- 基于半群理论构建误差估计框架，改进稳定性参数计算方法。
- 首次实现对圆柱绕流中斯托克斯流的严格误差认证。
- 适合关注神经网络可靠性与科学计算可信性的研究者。
严格的误差估计是数值分析的核心课题。随着物理信息神经网络（PINNs）在求解偏微分方程中的广泛应用，已有多种方法用于量化其预测误差。本文基于作者先前提出的基于半群的误差估计框架，进一步拓展其适用范围。此前该框架受限于输入-状态稳定性相关量的计算，仅适用于学术案例；本文通过改进误差界并提出数值近似策略，有效解决了这一瓶颈。新框架使PINN预测在更广泛的实际问题中具备可认证性，文中以圆柱绕流的斯托克斯流为例进行了数值验证，展示了其在真实场景下的可行性与有效性。
原文摘要 · Abstract (English)
Rigorous error estimation is a fundamental topic in numerical analysis. With the increasing use of physics-informed neural networks (PINNs) for solving partial differential equations, several approaches have been developed to quantify the associated prediction error. In this work, we build upon a semigroup-based framework previously introduced by the authors for estimating the PINN error. While this estimator has so far been limited to academic examples - due to the need to compute quantities related to input-to-state stability - we extend its applicability to a significantly broader class of problems. This is accomplished by modifying the error bound and proposing numerical strategies to approximate the required stability parameters. The extended framework enables the certification of PINN predictions in more realistic scenarios, as demonstrated by a numerical study of Stokes flow around a cylinder.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。