PI-BLS用伪逆一步求解PDE，比传统PINNs快得多，参数更少，效果还不差，适合做物理模拟的朋友试试。
物理信息神经网络(PINNs)需依赖梯度优化，训练慢且成本高。新提出的PI-BLS框架将微分算子和边界条件嵌入线性输出层优化，通过伪逆直接求解，完全消除反向传播。在多个前向偏微分方程基准上，PI-BLS以更少训练时间和参数达到与PINNs相当或更优的精度。
Physics-Informed Broad Learning System: An Efficient Backpropagation-Free Framework for Solving Partial Differential Equations
Physics-informed neural networks (PINNs) have emerged as a powerful paradigm for solving partial differential equations (PDEs) by embedding governing physical laws into deep neural networks. However, their reliance on computationally expensive gradient-based optimization and deep architectures often results in slow training, high computational cost, and limited scalability. In this work, we propose a novel physics-informed broad learning system (PI-BLS), the first physics-informed learning framework based on broad RdNNs. The proposed formulation embeds the governing differential operator and the associated initial and boundary constraints directly into a linear output-layer optimization problem, thereby replacing nonlinear gradient-based training with a deterministic least-squares solution obtained via the pseudoinverse. Consequently, the entire learning process is reduced to a single linear optimization stage while preserving the underlying physical constraints. As a result, PI-BLS offers an efficient learning paradigm for a physics-informed learning framework for solving PDEs that eliminates iterative backpropagation while preserving the underlying physical constraints. Experimental results on representative forward PDE benchmarks demonstrate that PI-BLS achieves competitive and often superior performance with reduced training time and model parameters compared with conventional PINNs.