Learning universal approximations for partial differential equations with Physics-Informed Broad Learning System

2026-06-18 · Source: Machine Learning · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Scientific Machine Learning · Depth: Expert, quick

Summary

The Physics-Informed Broad Learning System (PIBLS) is a novel backpropagation-free framework designed to solve partial differential equations (PDEs) by reformulating the process as a direct least-squares optimization. This approach addresses the computational cost of traditional numerical solvers and the slow convergence and optimization instability often seen in Physics-Informed Neural Networks (PINNs). An improved algorithm within PIBLS efficiently handles nonlinear PDEs, with mathematical proof establishing its universal approximation property for these equations. Experimental results on both linear and nonlinear PDEs show that PIBLS is one to three orders of magnitude faster than conventional PINNs, while also achieving significantly higher solution accuracy. This framework offers a computationally efficient paradigm for scientific machine learning, providing a practical, high-speed alternative for real-time simulation and design optimization tasks.

Key takeaway

For Machine Learning Engineers developing PDE solvers, you should consider PIBLS as a high-speed, accurate alternative to traditional PINNs. This framework's backpropagation-free, least-squares optimization approach can significantly reduce computational costs and improve solution fidelity. Evaluate PIBLS for your real-time simulation and design optimization tasks to achieve one to three orders of magnitude faster results with higher accuracy.

Key insights

PIBLS offers a backpropagation-free, least-squares optimization approach for PDEs, achieving faster, more accurate solutions than PINNs.

Principles

• Reformulate PDE solving as least-squares.
• Leverage backpropagation-free learning.
• Ensure universal approximation for PDEs.

Method

PIBLS solves PDEs by reformulating them into a direct least-squares optimization problem, utilizing a backpropagation-free architecture and an improved algorithm for nonlinear equations.

In practice

• Accelerate real-time PDE simulations.
• Optimize engineering designs faster.
• Improve accuracy in scientific ML.

Topics

• Physics-Informed Neural Networks
• Partial Differential Equations
• Broad Learning System
• Scientific Machine Learning
• Least-Squares Optimization
• Real-time Simulation

Best for: AI Scientist, Research Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.