Physics-Informed Stochastic Configuration Machine: A Backpropagation-Free Neural Network with Fast Training for Nonlinear Differential Equations

Yuehao Song; Zhong Chen; Lihui Cen; Liang Wu; Kai Zhang

math.NAAug 27, 2026

Physics-Informed Stochastic Configuration Machine: A Backpropagation-Free Neural Network with Fast Training for Nonlinear Differential Equations

Authors: Yuehao Song, Zhong Chen, Lihui Cen, Liang Wu, Kai Zhang

Organizations: School of Automation, Central South University, Changsha, China · School of Automation, Central South University, 932 South Lushan Road, Changsha 410083, China · Johns Hopkins University, Baltimore, USA · Johns Hopkins University, Baltimore, MD 21218, USA ·

Abstract

While Physics-Informed Neural Networks (PINNs) have emerged as a transformative paradigm for solving complex differential equations, their reliance on backpropagation-based gradient descent and automatic differentiation (AD) imposes significant computational bottlenecks and severe non-convex optimization challenges. To overcome these fundamental limitations, we propose the Physics-Informed Stochastic Configuration Machine (PI-SCM), a novel backpropagation-free framework for both forward and inverse problems in differential equations. The core mathematical contribution lies in the analytical evaluation of local Jacobians for nonlinear differential operators, which facilitates a linearized representation of the physical loss and projects it into a unified, linearized algebraic subspace. This reformulation allows for the explicit determination of optimal network weights via a sequence of generalized linear least squares solvers, effectively bypassing the iterative traps of traditional nonlinear optimizers. We develop a progressive algorithmic suite comprising localized construction (PI-SC-I), sliding-window updating (PI-SC-II), and global updating (PI-SC-III), and rigorously establish their universal approximation properties. Extensive experiments demonstrate that PI-SCM achieves high-fidelity predictive accuracy and robust parameter identification while accelerating the training process by orders of magnitude compared to standard PINNs. Our work provides a highly efficient and scalable foundation for next-generation, real-time Scientific Machine Learning applications.

Figures & tables

Algorithm	Paradigm	Time (s)	RMSE
PINN ( $1\times 300$ )	Unsupervised	$21.3\pm 0.57$	$3.69e{-1}\pm 9.18e{-2}$
	Semi-supervised	$27.5\pm 0.89$	$8.05e{-2}\pm 4.02e{-3}$
PINN ( $4\times 64$ )	Unsupervised	$40.0\pm 1.02$	$5.49e{-3}\pm 3.03e{-3}$
	Semi-supervised	$50.8\pm 1.08$	$1.34e{-3}\pm 5.22e{-4}$
PISCN	Unsupervised	–	–
	Semi-supervised	$13.8\pm 0.37$	$7.58e{-2}\pm 2.59e{-2}$

TABLE I: Forward results for the Van der Pol oscillator.

Fig. 1: Forward solution of the Van der Pol oscillator obtained by semi-supervised PI-SC-III. Top left: predicted state over $(t,U)$ . Top right: absolute pointwise error. Bottom: cross-sectional trajectories at $U=0.3$ and $U=0.6$ .

Algorithm	Paradigm	Time (s)	RMSE
PINN ( $1\times 400$ )	Unsupervised	$40.5\pm 1.23$	$2.62e{-2}\pm 1.97e{-2}$
	Semi-supervised	$53.2\pm 1.51$	$1.13e{-2}\pm 4.50e{-3}$
PINN ( $4\times 64$ )	Unsupervised	$75.6\pm 1.11$	$3.28e{-2}\pm 1.43e{-2}$
	Semi-supervised	$105\pm 1.38$	$1.44e{-2}\pm 5.06e{-3}$
PIELM	Unsupervised	$0.02\pm 0.00$	$9.09e{-3}\pm 3.83e{-3}$
	Semi-supervised	$0.02\pm 0.00$	$6.61e{-3}\pm 3.84e{-3}$

TABLE II: Forward results for the two-dimensional Helmholtz equation.

Fig. 2: Forward solution of the two-dimensional Helmholtz equation obtained by PI-SC-III. Top left: predicted state over $(x,y)$ . Top right: absolute pointwise error. Bottom: cross-sectional profiles at $x=-0.5$ and $x=0.5$ .

Algorithm	Paradigm	Time (s)	RMSE
VS-PINN ( $1\times 500$ )	Unsupervised	$31.8\pm 0.79$	$5.40e{-1}\pm 2.63e{-2}$
	Semi-supervised	$36.2\pm 0.94$	$4.65e{-1}\pm 4.67e{-2}$
VS-PINN ( $4\times 64$ )	Unsupervised	$65.0\pm 1.06$	$5.23e{-1}\pm 4.14e{-2}$
	Semi-supervised	$71.6\pm 1.35$	$7.75e{-2}\pm 1.06e{-1}$
PI-SC-I	Unsupervised	$1.59\pm 0.04$	$5.28e{-1}\pm 3.50e{-3}$
	Semi-supervised	$1.73\pm 0.03$	$3.16e{-1}\pm 2.47e{-2}$

TABLE III: Forward results for the Allen–Cahn equation.

Fig. 3: Forward solution of the Allen–Cahn equation obtained by semi-supervised PI-SC-III. Top left: predicted state over $(t,x)$ . Top right: absolute pointwise error. Bottom: spatial profiles at $t=0.1$ and $t=0.4$ .

Algorithm	$\|\mu_{pred}-\mu_{true}\|$	Time (s)	State RMSE
PINN ( $1\times 300$ )	$2.88e{-1}\pm 1.09e{-1}$	$22.0\pm 0.51$	$5.85e{-2}\pm 1.35e{-2}$
PINN ( $4\times 64$ )	$2.35e{-4}\pm 2.01e{-4}$	$51.8\pm 0.94$	$9.51e{-4}\pm 3.42e{-4}$
PI-SC-I	$4.01e{-1}\pm 4.69e{-2}$	$0.71\pm 0.02$	$1.08e{-1}\pm 2.05e{-2}$
PI-SC-II	$1.91e{-3}\pm 1.59e{-3}$	$1.64\pm 0.03$	$4.80e{-3}\pm 2.10e{-3}$
PI-SC-III	$5.08e{-4}\pm 2.75e{-4}$	$1.95\pm 0.04$	$5.96e{-4}\pm 2.00e{-4}$

TABLE IV: Inverse results for the Van der Pol oscillator.

Algorithm	$\|k^{2}_{pred}-k^{2}_{true}\|$	Time (s)	State RMSE
PINN ( $1\times 400$ )	$1.09e{+1}\pm 2.87e{+0}$	$50.6\pm 1.19$	$2.82e{-1}\pm 1.27e{-1}$
PINN ( $4\times 64$ )	$6.72e{-1}\pm 6.27e{-1}$	$101\pm 2.13$	$2.36e{-2}\pm 1.22e{-2}$
PI-SC-I	$3.72e{+1}\pm 1.93e{+1}$	$1.49\pm 0.03$	$2.17e{-1}\pm 9.90e{-2}$
PI-SC-II	$7.14e{-1}\pm 7.55e{-1}$	$3.14\pm 0.07$	$5.04e{-3}\pm 2.21e{-3}$
PI-SC-III	$8.42e{-4}\pm 1.60e{-3}$	$3.51\pm 0.10$	$6.70e{-5}\pm 1.15e{-4}$

TABLE V: Inverse results for the two-dimensional Helmholtz equation.

Activation	RMSE
Sine	$5.66e{-4}\pm 3.59e{-4}$
Hyperbolic tangent	$2.71e{-3}\pm 8.92e{-4}$
Sigmoid	$1.68e{-2}\pm 7.96e{-3}$

TABLE VI: Semi-supervised PI-SC-III results with different activation functions.

Fig. 4: Evolution of the average true residual

\mathcal{E}_{L}^{2}

with the number of accepted hidden nodes for three activation functions.

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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.

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