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

Abstract: 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.

• The paper presents a novel PIBLS approach that integrates broad learning with direct least-squares optimization to solve both linear and nonlinear PDEs.
• It achieves machine-level precision—with L2 errors as low as 10⁻¹⁶ for linear problems—and offers up to 50 times faster computation than traditional PINNs.
• The universal approximation proof in Sobolev spaces and the mesh-free solution process underscore its practical value for real-time simulation and complex engineering applications.

Physics-Informed Broad Learning System for Universal Approximation of PDEs

Introduction

This paper presents the Physics-Informed Broad Learning System (PIBLS), a novel computational framework for solving both linear and nonlinear partial differential equations (PDEs). By integrating the Broad Learning System (BLS) architecture and reformulating the PDE solving process as a direct least-squares optimization task, PIBLS circumvents the limitations of mesh-based traditional solvers and the convergence difficulties of gradient-based Physics-Informed Neural Networks (PINNs). The paper systematically details the mathematical formulation, theoretical guarantees, and practical advantages of PIBLS, substantiated by rigorous universal approximation proofs and comprehensive numerical benchmarks across diverse PDE scenarios.

Methodological Framework

PIBLS leverages a width-expansion neural architecture grounded in BLS, eschewing deep hierarchical feature extraction in favor of broad basis construction via randomly generated Feature and Enhancement Nodes, whose weights are fixed upon initialization. The sole learnable parameters are output weights $W_{\text{out}}$, optimized via least-squares, thus eliminating iterative backpropagation. The network projects spatial-temporal coordinates into expressive feature representations, enriched through enhancement nodes, with the final solution expressed as $\hat{u}(x) = A W_{\text{out}}$.

Key design features include:

• Direct Least-Squares Optimization: For linear PDEs, the optimal solution is attained analytically by solving a stacked system incorporating interior, boundary, and initial constraints. For nonlinear PDEs, an enhanced nonlinear least-squares (NLSQ-perturb) algorithm is used, integrating trust-region reflective optimization and stochastic perturbations to enable robust global convergence.
• Analytical Differentiation: Closed-form derivatives of the network outputs are computed, facilitating efficient evaluation of all physical residuals without reliance on automatic differentiation.
• Physics-Informed Constraints: The solution process embeds the physical laws directly in the loss formulation, ensuring compliance with the underlying PDE, boundary, and initial condition operators.

Universal Approximation Theory

Theoretically, PIBLS is proven to possess universal approximation capability in Sobolev spaces $H^s(\Omega)$ ($s \in \{1,2\}$) for any compact domain $\Omega \subset \mathbb{R}^D$ with Lipschitz boundaries. By treating the random mappings generated by BLS as diffeomorphic embeddings, the analysis demonstrates that the concatenated feature space is both expressive and injective.

The approximation proceeds via the following steps:

• Smooth Embedding: The random mapping from input coordinates $x$ to feature nodes $z(x)$ is shown to be a smooth immersion and injective for sufficiently large width, ensuring the basis functions span a rich functional space.
• Enhancement Nodes: Finite linear combinations of enhancement node activations further increase capacity, guaranteeing that any target solution $u^*$ in $H^s(\Omega)$ can be approximated arbitrarily well.
• Norm Equivalence: Under Sobolev norm equivalence through diffeomorphisms, convergence in the function space is guaranteed as both node counts and training samples increase.

This establishes a rigorous foundation for PIBLS as a universal solver for complex PDEs, including those with challenging nonlinearities and irregular boundary conditions.

Empirical Evaluation and Numerical Results

A comprehensive suite of 11 benchmark PDEs (1D, 2D, time-dependent, and nonlinear) demonstrates PIBLS efficacy. The evaluation includes direct comparisons with PINN models (parameter-matched and deep architectures), Physics-Informed Extreme Learning Machines (PIELM and locELM), and high-resolution Finite Element Method (FEM) baselines.

Key empirical findings include:

• Accuracy: PIBLS achieves machine-level precision (L2 errors as low as $10^{-16}$ for linear problems, and below $\hat{u}(x) = A W_{\text{out}}$0 for nonlinear PDEs), consistently outperforming PINN, PIELM, and even fine-mesh FEM implementations by several orders of magnitude.
• Computational Efficiency: Solution times are one to three orders of magnitude faster than PINN architectures (e.g., solving problems in tenths of a second versus tens to thousands of seconds for traditional PINNs), demonstrating a significant reduction in computational complexity and resource demands.
• Robustness Across Domains: PIBLS maintains its advantage on high-dimensional, spatio-temporal, and nonlinear benchmarks, including those where gradient-based methods failed to converge or stagnated at suboptimal error rates.
• Parameter Sensitivity: Saturation studies show solution accuracy stabilizes at low error levels with moderate network widths and training point counts; nonlinear case accuracy is more sensitive to random weight initialization, but can be robustly controlled by tuning the initialization range.

Strong claims substantiated by experimental evidence:

• PIBLS achieves solution accuracy up to 11 orders of magnitude higher than PINNs on linear test cases while converging in less than 1 second.
• On nonlinear benchmarks, PIBLS maintains accuracy superiority by two or more orders over locELM and FEM, while being up to 50 times faster than locELM and unaffected by PINN failure modes.

Practical and Theoretical Implications

PIBLS introduces a paradigm shift for scientific machine learning and computational PDE solving. Practically, it offers a high-speed, mesh-free universal solver suitable for real-time simulation, control design, and optimization in physical, biological, and engineering modeling. The separation of feature generation from weight optimization enables stability, scalability, and ease of deployment, with minimal hyperparameter tuning and deterministic convergence properties.

Theoretical implications extend to the validation of width-based neural architectures for PDEs, supporting broader adoption of shallow, broad-feature techniques in physics-informed learning. The universal approximation proof establishes PIBLS as a robust alternative to deep architectures, emphasizing the sufficiency of expressive basis construction and analytical solvers.

Future Directions:

The paper suggests exploration of adaptive strategies for automatic weight initialization, particularly for nonlinear PDEs, to further strengthen accuracy and reduce manual intervention. Extension to stochastic PDEs, high-dimensional inverse problems, and adaptive sampling strategies for collocation points represents promising avenues for advancing the PIBLS framework.

Conclusion

PIBLS constitutes a fundamentally robust, efficient, and mathematically guaranteed approach for PDE approximation. Its capacity for machine-level precision, speed, and universal representational power establishes it as a practical alternative to both traditional numerical methods and deep PINN architectures. The framework's simplicity and rigor render it suitable for widespread applications in scientific machine learning, real-time simulation, and computational engineering, with substantial potential for further methodological enhancements and theoretical developments (2606.19754).