Physics-Informed Broad Learning System: An Efficient Backpropagation-Free Framework for Solving Partial Differential Equations

2026-07-28

AI Summary

This work addresses the limitations of traditional physics-informed neural networks (PINNs)—notably slow training, high computational cost, and poor scalability—by introducing a novel framework that integrates broad learning systems (BLS) into physics-informed modeling. The proposed physics-informed broad learning system (PI-BLS) embeds differential operators and initial/boundary conditions directly into a linear output layer and solves the resulting least-squares problem via pseudoinverse computation, thereby enabling single-stage, deterministic training without backpropagation. By eliminating nonlinear iterative optimization, PI-BLS satisfies physical constraints through a single linear solve. Empirical evaluations on multiple forward partial differential equation benchmarks demonstrate that PI-BLS achieves accuracy comparable to or better than PINNs, using significantly fewer parameters and drastically reduced training time, thus substantially enhancing both efficiency and scalability.

Abstract

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.

Problem

• physics-informed neural networks
• partial differential equations
• computational cost
• training efficiency
• scalability

Innovation

• Physics-Informed Learning
• Broad Learning System
• Backpropagation-Free
• Partial Differential Equations
• Linear Optimization