LEARNING UNIVERSAL APPROXIMATIONS FOR PARTIAL DIFFERENTIAL EQUATIONS WITH PHYSICS-INFORMED BROAD LEARNING SYSTEM

Traditional numerical solvers for partial differential equations (PDEs) are often computationally expensive due to their reliance on complex mesh generation. While Physics-Informed Neural Networks (PINNs) offer a mesh-free alternative, they are frequently hindered by slow convergence, optimization instability, and the high computational cost of gradient-based backpropagation. This paper introduces the Physics-Informed Broad Learning System (PIBLS) to address these bottlenecks by reformulating PDE solving as a direct, backpropagation-free least-squares optimization problem.

PIBLS utilizes a Broad Learning System (BLS) architecture, which replaces the deep, hierarchical structure of standard neural networks with a shallow, wide architecture consisting of randomly generated feature and enhancement nodes. The framework operates in two main phases:

1. Initialization: Input coordinates are projected through fixed, randomly initialized weights and biases to create a basis matrix.
2. Optimization: Instead of iterative backpropagation, PIBLS solves for the optimal output weights using a direct least-squares method. For linear PDEs, this is solved analytically; for nonlinear PDEs, the authors employ an enhanced nonlinear least-squares perturbation algorithm that combines local search with stochastic global exploration.

To ensure the network satisfies physical laws, the authors derive analytical derivatives for the basis functions, allowing the residuals of the governing PDE, boundary conditions, and initial conditions to be computed exactly without automatic differentiation.

The authors provide a rigorous mathematical proof establishing that PIBLS possesses the universal approximation property within Sobolev spaces, guaranteeing its ability to approximate complex PDE solutions. Experimental results demonstrate that PIBLS outperforms conventional PINNs, achieving solution accuracies that are significantly higher while reducing computational time by one to three orders of magnitude. This makes PIBLS a highly efficient candidate for real-time simulation and design optimization tasks.