发表机构
School of Automation, Central South University; Johns Hopkins University; China Institute of Water Resources and Hydropower Research（中南大学自动化学院; 约翰斯·霍普金斯大学; 中国水利水电科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出无反向传播的物理信息随机配置机（PI-SCM），通过解析雅可比矩阵等技术求解微分方程，训练速度较PINNs提升数个数量级，为实时科学机器学习提供高效基础。
AI 中文摘要
尽管物理信息神经网络（PINNs）已成为求解复杂微分方程的变革性范式，但它们依赖基于反向传播的梯度下降和自动微分（AD），这带来了显著的计算瓶颈和严重的非凸优化挑战。为克服这些根本性局限，我们提出了物理信息随机配置机（PI-SCM），这是一种用于微分方程正问题和逆问题的新型无反向传播框架。核心数学贡献在于对非线性微分算子的局部雅可比矩阵进行解析评估，这使得物理损失可实现线性化表示，并将其投影到统一的线性化代数子空间中。这种重构允许通过一系列广义最小二乘求解器显式确定最优网络权重，有效规避了传统非线性优化器的迭代陷阱。我们开发了一套渐进式算法套件，包括局部构造（PI-SC-I）、滑动窗口更新（PI-SC-II）和全局更新（PI-SC-III），并严格证明了它们的通用逼近性质。大量实验表明，PI-SCM实现了高保真预测精度和鲁棒参数辨识，同时与标准PINNs相比，训练过程加速了数个数量级。我们的工作为下一代实时科学机器学习应用提供了高效且可扩展的基础。
英文摘要
While Physics-Informed Neural Networks (PINNs) have emerged as a transformative paradigm for solving complex differential equations, their reliance on backpropagation-based gradient descent and automatic differentiation (AD) imposes significant computational bottlenecks and severe non-convex optimization challenges. To overcome these fundamental limitations, we propose the Physics-Informed Stochastic Configuration Machine (PI-SCM), a novel backpropagation-free framework for both forward and inverse problems in differential equations. The core mathematical contribution lies in the analytical evaluation of local Jacobians for nonlinear differential operators, which facilitates a linearized representation of the physical loss and projects it into a unified, linearized algebraic subspace. This reformulation allows for the explicit determination of optimal network weights via a sequence of generalized linear least squares solvers, effectively bypassing the iterative traps of traditional nonlinear optimizers. We develop a progressive algorithmic suite comprising localized construction (PI-SC-I), sliding-window updating (PI-SC-II), and global updating (PI-SC-III), and rigorously establish their universal approximation properties. Extensive experiments demonstrate that PI-SCM achieves high-fidelity predictive accuracy and robust parameter identification while accelerating the training process by orders of magnitude compared to standard PINNs. Our work provides a highly efficient and scalable foundation for next-generation, real-time Scientific Machine Learning applications.
Comments17 pages, 4 figures
