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 · State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research, Beijing, China · State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research, Beijing 100038, China
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 ( 1×300 )
Unsupervised
21.3±0.57
3.69e−1±9.18e−2
Semi-supervised
27.5±0.89
8.05e−2±4.02e−3
PINN ( 4×64 )
Unsupervised
40.0±1.02
5.49e−3±3.03e−3
Semi-supervised
50.8±1.08
1.34e−3±5.22e−4
PISCN
Unsupervised
–
–
Semi-supervised
13.8±0.37
7.58e−2±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 ( 1×400 )
Unsupervised
40.5±1.23
2.62e−2±1.97e−2
Semi-supervised
53.2±1.51
1.13e−2±4.50e−3
PINN ( 4×64 )
Unsupervised
75.6±1.11
3.28e−2±1.43e−2
Semi-supervised
105±1.38
1.44e−2±5.06e−3
PIELM
Unsupervised
0.02±0.00
9.09e−3±3.83e−3
Semi-supervised
0.02±0.00
6.61e−3±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 ( 1×500 )
Unsupervised
31.8±0.79
5.40e−1±2.63e−2
Semi-supervised
36.2±0.94
4.65e−1±4.67e−2
VS-PINN ( 4×64 )
Unsupervised
65.0±1.06
5.23e−1±4.14e−2
Semi-supervised
71.6±1.35
7.75e−2±1.06e−1
PI-SC-I
Unsupervised
1.59±0.04
5.28e−1±3.50e−3
Semi-supervised
1.73±0.03
3.16e−1±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
∣μpred−μtrue∣
Time (s)
State RMSE
PINN ( 1×300 )
2.88e−1±1.09e−1
22.0±0.51
5.85e−2±1.35e−2
PINN ( 4×64 )
2.35e−4±2.01e−4
51.8±0.94
9.51e−4±3.42e−4
PI-SC-I
4.01e−1±4.69e−2
0.71±0.02
1.08e−1±2.05e−2
PI-SC-II
1.91e−3±1.59e−3
1.64±0.03
4.80e−3±2.10e−3
PI-SC-III
5.08e−4±2.75e−4
1.95±0.04
5.96e−4±2.00e−4
TABLE IV: Inverse results for the Van der Pol oscillator.
Algorithm
∣kpred2−ktrue2∣
Time (s)
State RMSE
PINN ( 1×400 )
1.09e+1±2.87e+0
50.6±1.19
2.82e−1±1.27e−1
PINN ( 4×64 )
6.72e−1±6.27e−1
101±2.13
2.36e−2±1.22e−2
PI-SC-I
3.72e+1±1.93e+1
1.49±0.03
2.17e−1±9.90e−2
PI-SC-II
7.14e−1±7.55e−1
3.14±0.07
5.04e−3±2.21e−3
PI-SC-III
8.42e−4±1.60e−3
3.51±0.10
6.70e−5±1.15e−4
TABLE V: Inverse results for the two-dimensional Helmholtz equation.
Activation
RMSE
Sine
5.66e−4±3.59e−4
Hyperbolic tangent
2.71e−3±8.92e−4
Sigmoid
1.68e−2±7.96e−3
TABLE VI: Semi-supervised PI-SC-III results with different activation functions.
Fig. 4: Evolution of the average true residual EL2 with the number of accepted hidden nodes for three activation functions.
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.
Zhiwen Yu, Derong Yang, Liujian Zhang +5
School of Computer Science and Engineering, South China University of Technology, Guangzhou 510006, China · Peng Cheng Laboratory, Shenzhen 518000, China · School of Future Technology, South China University of Technology, Guangzhou 510006, China +2
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.
Pinki Khatun, M. Sajid, Abhinav Jha +1
Department of Industrial Engineering, Università degli Studi di Firenze, Viale Morgagni 40/44, 50134 Firenze, Italy · Department of Mathematics, Indian Institute of Technology Indore, Indore 453552, India · Department of Mathematics, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar, 382055, Gujarat, India
Efficient and robust optimization is essential for neural networks, enabling scientific machine learning models to converge rapidly to very high accuracy -- faithfully capturing complex physical behavior governed by differential equations. In this work, we present advanced optimization strategies to accelerate the convergence of physics-informed neural networks (PINNs) for challenging partial (PDEs) and ordinary differential equations (ODEs). Specifically, we provide efficient implementations of the Natural Gradient (NG) optimizer, Self-Scaling BFGS and Broyden optimizers, and demonstrate their performance on problems including the Helmholtz equation, Stokes flow, inviscid Burgers equation, Euler equations for high-speed flows, and stiff ODEs arising in pharmacokinetics and pharmacodynamics. Beyond optimizer development, we also propose new PINN-based methods for solving the inviscid Burgers and Euler equations, and compare the resulting solutions against high-order numerical methods to provide a rigorous and fair assessment. Finally, we address the challenge of scaling these quasi-Newton optimizers for batched training, enabling efficient and scalable solutions for large data-driven problems.
Anas Jnini, Elham Kiyani, Khemraj Shukla +5
Department of Information Engineering and Computer Science, University of Trento, Trento, Italy · Division of Applied Mathematics, Brown University, Providence, RI, USA · Departament de Física, Universitat d’Alacant, Spain +3