用新方法求解偏微分方程，速度比传统神经网络快100~1000倍。
Learning universal approximations for partial differential equations with Physics-Informed Broad Learning System
- 不依赖反向传播，直接通过最小二乘法求解方程。
- 在线性和非线性方程上速度比PINN快1到3个数量级，精度更高。
- 适合需要实时仿真的工程与科学计算场景。
偏微分方程（PDEs）在建模复杂物理、生物和工程系统中起核心作用。传统数值求解器虽稳健但因网格依赖导致计算成本高昂，而近年的物理信息神经网络（PINNs）虽为无网格方案，却常面临收敛慢与优化不稳定问题。本文提出物理信息广义学习系统（PIBLS），一种无需反向传播的新框架，将PDE求解重构为直接的最小二乘优化。我们改进了其中算法以高效处理非线性PDE，并提供了严格的数学证明，确立PIBLS对这类方程具有通用逼近能力。在线性和非线性PDE实验中，PIBLS相比传统PINNs提速1至3个数量级，同时显著提升解的精度。该框架为科学机器学习提供了一种计算高效的范式，适用于实时仿真与设计优化任务。
原文摘要 · Abstract (English)
Partial differential equations (PDEs) play a central role in modeling complex physical, biological, and engineering systems. While traditional numerical solvers are robust, they often incur prohibitive computational costs due to mesh dependencies, whereas recent Physics-Informed Neural Networks (PINNs) offer a mesh-free alternative but frequently suffer from slow convergence and optimization instability. To bridge this gap, this article proposes the Physics-Informed Broad Learning System (PIBLS), a novel backpropagation-free framework that reformulates PDE solving as a direct least-squares optimization. We improved an algorithm within this framework to handle nonlinear PDEs efficiently and provide a rigorous mathematical proof establishing the universal approximation property of PIBLS for these equations. Experiments on linear and nonlinear PDEs demonstrate that PIBLS is one to three orders of magnitude faster than conventional PINNs while achieving significantly higher solution accuracy. This framework provides a computationally efficient paradigm for scientific machine learning, offering a practical, high-speed alternative for real-time simulation and design optimization tasks.
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